ChemWiki - User contributions [en]

Rep:Mod:Mtn113

2015-12-18T09:49:01Z

Mtn113: /* Frequency Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units <ref name="lab" />.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). If all frequencies are positive, this would suggest that a minimum point was reached in the optimisation and not an inflection point or a maximum point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. If there was a maximum point present, the negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy surface and implies that the energy of the structure at that point is at a local maximum, ie a transition state at a maximum energy level has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:45:59Z

Mtn113: /* Frequency Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units <ref name="lab" />.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy surface and implies that the energy of the structure at that point is at a local maximum, ie a transition state at a maximum energy level has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:45:24Z

Mtn113: /* Frequency Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units <ref name="lab" />.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a maximum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy surface and implies that the energy of the structure at that point is at a local maximum, ie a transition state at a maximum energy level has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:41:45Z

Mtn113: /* Optimisation via TS Berny */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units <ref name="lab" />.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy surface and implies that the energy of the structure at that point is at a local maximum, ie a transition state at a maximum energy level has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:39:39Z

Mtn113: /* 1,5-hexadiene (Anti2 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units <ref name="lab" />.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:39:13Z

Mtn113: /* 1,5-hexadiene (Gauche3 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units <ref name="lab" />. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:38:04Z

Mtn113: /* 1,5-hexadiene (Gauche3 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units </ref name="lab">. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:37:23Z

Mtn113: /* 1,5-hexadiene (Anti1 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref name="lab"><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:36:29Z

Mtn113: /* Conclusion */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reasons, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:35:13Z

Mtn113: /* 1,5-hexadiene (Gauche3 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above of gauche3 conformation, pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:33:29Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the reaction pathway and identifying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:32:16Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature, these transition states can be computationally modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
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</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T09:31:25Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to the instability of these transition states and their short lived nature. These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T03:16:58Z

Mtn113: /* Activation Energy Comparison */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway<ref>J. Cooley and R. Williams (1997), J. Chem. Educ., 74, 582. DOI:10.1021/ed074p582</ref>, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T03:14:22Z

Mtn113: /* Molecular Orbitals Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis <ref>H. Longuet-Higgins and E. Abrahamson (1965), J. Am. Chem. Soc., 87, 2045-2046. DOI: 10.1021/ja01087a033</ref>, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T03:08:06Z

Mtn113: /* Endo Product */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
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</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap<ref>M. Fox, R. Cardona and N. Kiwiet (1987), The Journal of Organic Chemistry, 52, 1469-1474. DOI: 10.1021/jo00384a016</ref> between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T03:05:09Z

Mtn113: /* Geometrical Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
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</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
DOI:10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451. DOI:10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T03:04:21Z

Mtn113: /* 1,5-hexadiene (Gauche3 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons<ref> Gung, B., Zhu, Z. and Fouch, R. (1995). Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols. J. Am. Chem. Soc., 117(6), pp.1783-1788.
DOI:10.1021/ja00111a016</ref>.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:58:17Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. (2007) Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:57:27Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP) <ref>Musgrave. C. Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations J. Phys. Chem. A, 2007, 111 (8), pp 1554-1561 DOI:10.1021/jp061633o</ref>, this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:53:05Z

Mtn113: /* Geometrical Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:51:35Z

Mtn113: /* Geometrical Analysis */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref name=":0">Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref name=":0" />/Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:40:28Z

Mtn113: /* 1,5-hexadiene (Anti1 Conformation) */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1<ref><span dir="ltr">Imperial College London, Computational Chemistry Wiki https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</span>
</ref> ). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:39:24Z

Mtn113: /* Background */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:36:53Z

Mtn113: /* Background */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation<ref><span dir="ltr">Organic Chemistry by Jonathan Clayden</span>, <span dir="ltr">Nick Greeves</span>, <span dir="ltr">Stuart Warren</span>
</ref> as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:34:50Z

Mtn113: /* Conclusion */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

Diels Alder Cycloaddition was conducted with cis-butadiene and ethene which showed the concerted mechanism for the reaction. However, regioselectivity of the reaction could not be elucidated from that reaction, hence another Diels Alder between maleic anhydride and cyclohexa-1,3-diene was conducted to study the regioselectivity through MO, geometrical analysis and activation energies. The optimisations to yield TS were done via the TS Berny method using Semi-Empirical AM1 level of theory. This elucidated the fact that endo pathway is more favourable due to a lower activation energy and stabilising secondary orbital overlap; and larger steric repulsions in the exo transition state. A possible extension would be to run the optimisations using a higher level of theory such as Semi-empirical PM6 which has more parameters or DFT/B3LYP/6-31G* which takes into consideration electronic interactions to yield more accurate energy data.

This computational exercise managed to simulate cyclic transition states in the above reactions which would otherwise be experimentally impossible to do. These computational results could then be used to complement and explain empirical observations for such reactions.

==References==

Rep:Mod:Mtn113

2015-12-18T02:13:14Z

Mtn113: /* Conclusion */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored transition states for two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

For the Cope Rearrangement, the molecule involved, 1,5-hexadiene, can exist in different conformers such as anti- and gauche- conformers. While it was thought that anti- conformation would possess a lower energy than gauche- conformation due to steric reason, it was found that the gauche conformation was stabilised by electronic orbital overlap. The Cope Rearrangement was confirmed to proceed via a concerted mechanism, as the TS imaginary vibration frequency showed a synchronous vibration. This transition state was obtained via optimisation with TS Berny method (chair), freezing coordinates (chair) as well as QST2 (boat)/QST3(boat) and these optimisations were conducted at HF/3-21G as well as DFT/B3LYP/6-31G*. While the differences between the geometries were insignificant across levels of theory, the differences become apparent when energies of TS are considered. From the activation energies of the chair and boat conformations, it was concluded that the chair conformation has a lower activation energy and hence reaction proceeds through the chair conformation.

==References==

Rep:Mod:Mtn113

2015-12-18T00:01:44Z

Mtn113: /* Introduction */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Using the Gaussian program, different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions, which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via one out of the two following methods: making a guess of the transition state and positioning the molecules as such before optimising the overall structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

==References==

Rep:Mod:Mtn113

2015-12-16T19:38:05Z

Mtn113:

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment explored two classes of pericyclic reactions: Cope Rearrangement and Diels Alder Cycloaddition.

==References==

Rep:Mod:VR1994

2015-12-16T19:32:04Z

Mtn113: Undo revision 521061 by Mtn113 (talk)

== Introduction ==
This computational study is based on the characterisation of transition structures on potential energy surfaces for two different types of reactions, the Cope Rearrangement and several Diels-Alder cycloaddition reactions. In particular, the Cope Rearrangement reaction of 1,5-hexadiene, the Diels-Alder cycloaddition of ethene and s-cis-1,2-butadiene and, finally, the Diels-Alder cycloaddition of cyclohexa-1,3-diene and maleic anhydride via both the endo and exo transition states. It is now well known that the transition state corresponds to the largest energy point (i.e. a maximum in energy) in the reaction coordinate and so these species can not be isolated as it is a transient species that immediately reverts back to the reactants or goes on to form the products. As a result of the fleeting existence of the transition state, the only method that one could use to be able to locate and study the transition state is to use computational chemistry i.e. via calculations using the Gaussview Program. There is a great interest in studying the transition state as it provides much chemical information that form the basis for any chemical reaction such as the activation energy, and thus the rates of reactions, as well as providing information on reaction mechanisms.

In this study, the method of using the Gaussview program to locate the transition states was adopted by the use of three main calculation methods: Hartree-Fock/321-G, B3LYP/6-31G(d) (Density Functional Theory) and Semi-Empirical/AM1 levels of theory.

In summary, these methods involved two different processes to optimize the structure of the transition state. The first of which involved the guessing of a transition structure and optimizing this guess using the TS (Berny) optimization method. The second method involves the use of the QST2 optimization method in which the all the atoms in the reactants and products were labelled and the subsequent calculation provided the transition structure via interpolation between the input reactant and product structures; this worked by finding a maxima in energy between these structures on the potential energy surface.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:22, 4 December 2015 (UTC) Nice well rounded intro

== The Cope Rearrangement of 1,5-hexadiene ==

=== Background ===

The reaction scheme for the Cope Rearrangement of 1,5-hexadiene is provided in Figure 1 below. The Cope Rearrangement is a [3,3]-sigmatropic rearrangement, which is a pericyclic reaction in which one sigma bond is broken whilst another sigma bond is formed. According to the Woodward-Hoffman rules, the Cope Rearrangement is a thermally allowed reaction ,as the total number of components that fit the equation (4q + 2)<sub>s</sub> + (4r)<sub>a</sub> is odd, since there are two <sub>π</sub>2<sub>a</sub> components and one <sub>σ</sub>2<sub>s</sub> component.

The mechanism of the Cope Rearrangement has widely been studied to understand whether the mechanism proceeds via a concerted or a non-concerted process. However, the modern opinion is in accord with the former mechanism process, a concerted mechanism. From the concerted mechanism, the Cope Rearrangement occurs via both a boat and a chair conformations of the transition state. In the following calculations, the local minima, associated with the reactants and products, as well as the maxima, corresponding to the transition states, were located and analysed for this Cope Rearrangement. All the calculations were carried at out the Hartree-Fock and Density Functional Theory levels of theory with the aim of attaining the activation energies of the Cope Rearrangement when the reaction proceeded via a boat or a chair transition state conformation.

[[File:Cope Rearragement.png|350px|thumb|center|Fig.1 The Cope Rearrangement of 1,5-hexadiene.]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:23, 4 December 2015 (UTC) Really good understanding of the chemistry

===Hartree-Fock and Density Functional Theory comparisons===
Before commencing the calculations for locating the transition states for the Cope Rearrangement it is important to discuss the theoretical points behind the HF and DFT levels of theory. The Density Functional Theory method uses far fewer approximations in its calculations which leads to more accurate results and energy values that are closer to the true value being given. HF tends to give values in energy that are far higher than the true value.
The Hartree-Fock and DFT calculations have a lot in common as the equations must be solved self-consistently. The Hartree-Fock method focuses on wavefunctions and orbitals, in particular it concentrates on the Slater Determinant. However, DFT equations concentrate on electron densities and as it includes an approximate treatment of electron correlation, it produces more accurate results than the Hartree-Fock theory. In the Hartree-Fock theory, the movement of the electrons are described by MO theory i.e. a linear combination of atomic orbitals. Hartree-Fock does not produce exact results as the theory is a significant approximation to the Schrödinger equation. This approximation is that the electrons, at all times, feel an average negative Coulomb energy when interacting with other electrons. Whilst this approximation simplifies matters, it leads to inaccurate results being presented.
In cases where this approximation leads to severe errors, the Density Functional Theory is used to help overcome any issues by accounting for electron correlation, which is the potential that electrons experience, in the equations.
A significant drawback of the DFT calculations is that it cannot be improved upon and thus more accurate results cannot be achieved.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:34, 4 December 2015 (UTC) very nice understanding of the methods, HF does not account for electron correlation because electrons of opposite spin can be in the same place at the same time in one spatial orbital which is wrong and there for the repulsion term is incorrect.

=== Optimising the Reactants and Products ===
In the following section, various conformers of 1,5-hexadiene will be studied. There is a possibility of many different conformers due to the ability of 1,5-hexadiene to be able to freely rotate about it's central C-C single bond.
==== Optimising the conformers of 1,5-hexadiene using the HF/3-21G level of theory====
To begin, a molecule of 1,5-hexadiene was prepared on the Gaussview program. This structure was then optimised at the HF/3-21G level of theory to give many different conformers, namely the anti1, anti2 and gauche3 conformers shown below (in the Jmol structures). Below, the energies and point groups (which was provided via the symmetrize tool on Gaussview) of the conformers are given.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white"| Conformer
! style="background: #0D4F8B; color: blue; background: white"| Jmol structure
! style="background: #0D4F8B; color: blue; background: white"| Energy (hartrees)
! style="background: #0D4F8B; color: blue; background: white"| Calculation Summary
! style="background: #0D4F8B; color: blue; background: white"| Point Group
! style="background: #0D4F8B; color: blue; background: white"| Point Group Summary
|-
| anti1
| <jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>connect(atomno=12)(atomno=13) DOUBLE; connect(atomno=4)(atomno=1) DOUBLE; frame 16</script>
<uploadedFileContents>REACT ANTI VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| -231.69260236
| [[File:React anti1 resultstable VR.jpg|250px|thumb|center]]
| C<sub>2</sub>
| [[File:Point group react gauche stableVR.jpg|320px|thumb|center]]
|-
| gauche3
| <jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>connect(atomno=14)(atomno=12) DOUBLE; connect(atomno=4)(atomno=1) DOUBLE; frame 16</script>
<uploadedFileContents>REACT GAUCHE VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| -231.69266121
| [[File:React gauche3 resultstable.jpg|250px|thumb|center]]
| C<sub>1</sub>
| [[File:Point Group react anti2 VR.jpg|320px|thumb|center]]
|-
| anti2
| <jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>connect(atomno=1)(atomno=3) DOUBLE; connect(atomno=14)(atomno=12) DOUBLE; frame 14</script>
<uploadedFileContents>REACT ANTI2 VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| -231.69253517
| [[File:React anti2 resultstable VR.jpg|250px|thumb|center]]
| C<sub>i</sub>
| [[File:Point group react ant1 stableVR.jpg|320px|thumb|center]]
|}
Firstly, the 1,5-hexadiene molecule was altered to ensure an "anti" linkage was formed with an anti-periplanar geometry (i.e. the dihedral angle is 180°), was formed. This structure was then optimised at the HF/3-21G level of theory to provide the anti1 and anti2 conformers via separate calculations. It can be seen that the energies and point groups of both of these conformers agree with the data provided in the appendix in the laboratory script. It can also be seen that the anti1 conformer is slightly more stable (i.e. lower in energy) than the anti2 conformer.

Next, the initial 1,5-hexadiene molecule was altered to provide a gauche linkage for the central four carbon atoms (i.e. the dihedral angle is 60°) and again it was optimised using the HF/3-21G level of theory to give the gauche3 conformer.

The expectation is that the anti1 and anti2 conformers would be lower in energy than gauch3 conformers as there are no steric clashes or interactions as the alkenyl groups attached to C3 and C4 are 180° apart in anti1 and anti2 conformers whilst only 60° apart in gauche3 leading to greater steric interactions. The greater the angle between the alkenyl groups, the less the steric repulsion between the groups. However, as can be seen from the table above, the gauche3 conformer has the lowest energy geometry which contradicts the initial expectation. The global minimum is in fact the gauche3 conformer as there a favourable interaction possible between the bonding π orbital of one alkenyl group and the vinyl proton of the other alkenyl group which lowers the total energy<ref name="Gauche3"/> ; in this case, there is a weak form of hydrogen bonding between the softly acidic vinyl C-H bond and the soft π bonding orbital (basic) <ref name="Anti1"/>.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:58, 4 December 2015 (UTC) interesting theory you are correct but that study is pretty old if you had looked in the orbitaks you should be able to see the twist of the gauche leads to favourable inphase overlap between the pi systems.

==== Optimising the anti2 conformer of 1,5-hexadiene using the B3LYP/6-31G(d) level of theory====
Next, the anti2 conformer was further optimised at the B3LYP/6-31G(d) level to give a reactant with the energy and Jmol structure given below.

{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Energy (hartrees)
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;"| Jmol structure
|-
| -234.61171062
| [[File:React anti2 reoptresultstableVR.jpg|300px|thumb|center]]
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=3) DOUBLE; connect(atomno=14)(atomno=12) DOUBLE; frame 14</script>
<uploadedFileContents>REACT ANTI2 REOPT 631GDVR.LOG</uploadedFileContents>
</jmolApplet></jmol>
|}
After re-optimising at the B3LYP/6-31G(d) level, the point group of the anti2 conformer remains the same, i.e. C<sub>i</sub>, which can be seen in Fig. 2 below.
[[File:Anti2 pointgroup reoptVR.jpg|375px|thumb|centre|Fig. 2 The point group summary of the anti2 conformer of 1,5-hexadiene when using the 6-31G(d) level of theory]]

====Comparison of the anti2 conformer of 1,5-hexadiene when using the HF/3-21G and B3LYP/6-31G(d) levels of theory====
Whilst the energy of the anti2 conformer cannot strictly be compared between the two levels of theory, the difference in geometries can be discussed. Below the difference in bond lengths and dihedral angles are discussed.
=====Bond Lengths=====
[[File:React anti2numberingVR.jpg|350px|thumb|center|Fig. 3 The numbering of the carbon atoms in the anti2 conformer of 1,5-hexadiene corresponding to the bond lengths given below.]]
The anti2 conformer was labelled as depicted in Fig. 3 and these labels are used to provide the bond lengths given in the table below.

{| class="wikitable"
! !! colspan="2" | Bond Lengths (Å)
|-
! Atoms !! HF/3-21G !! B2LYP/6-31G(d)
|-
| C1-C2 || 1.31615 || 1.33350
|-
| C2-C3 || 1.50895 || 1.50420
|-
| C3-C4 || 1.55264 || 1.54816
|}

Immediately, it can be seen that the HF level of theory gives a slightly shorter C1-C2 bond length (i.e. a C=C alkene bond) than B3LYP. The literature value of the C1-C2 bond length is 1.340 Å <ref name="lengths"/>; thus, one can see that the B3LYP level of theory is much more accurate in reproducing C1-C2 alkene bond length in 1,5-hexadiene. The literature value of the C2-C3 bond length is 1.508 Å <ref name="lengths"/> and that of C3-C4 is 1.538 Å <ref name="lengths"/>. So it can be seen that the HF level of theory is better at reproducing the C2-C3 bond length whilst B3LYP is better at reproducing the C3-C4 bond length. Overall, the B3LYP level of theory is better at reproducing a structure more accurately as it uses fewer approximations in it's calculations.

=====Dihedral angles=====
The dihedral angles of the anti2 conformer are given below when using the different levels of theory.
{| class="wikitable" border="1"
! !! colspan="2" | Dihedral Angle (°)
|-
! Atoms !! HF/3-21G !! B3LYP/6-31G(d)
|-
| C1-C2-C3-C4 || 114.65912 || 118.59923
|-
| C2-C3-C4-C5 || 179.99940 || 180.00000
|}
Initially, the dihedral angle (i.e. between C2-C3-C4-C5) was set to 180° and so the fact that the dihedral angle had been altered from 180° to 179.99940° shows the inconvenience when using the Hartree-Fock level of theory. This is further backed up by examining the angle between the carbons C1-C2-C3-C4. One would expect that at an alkene centre, there is a trigonal planar arrangement and so the dihedral angle should be 120°. However, the Hartree-Fock level of theory deviates from this expected trigonal planar by more than the B3LYP level of theory suggesting the greater appropriateness in using the B3LYP level of theory in order to produce accurate dihedral angles.

====Frequency analysis of the anti2 conformer of 1,5-hexadiene====
Next, a frequency analysis was conducted on the anti2 conformer using a B3LYP/6-31G(d) calculation and the calculation summary is provided below (Fig. 4).
[[File:Freq anti2 summaryVR.jpg|305px|thumb|left|Fig. 4 The calculation summary of the anti2 conformer of 1,5-hexadiene when performing a frequency analysis.]]
All vibrational frequencies, as can be seen from the IR spectra below (Fig. 5), are positive and real and so confirms that a minimum has been reached for the anti2 conformer. The absence of imaginary frequencies confirms that a minimum is reached and the calculation didn't result in a structure that corresponded to a transition state (i.e. an energy maximum). This frequency calculation corresponds to the second derivative of the potential energy; a negative second derivative (maximum i.e. transition states) would lead to the presence of imaginary frequencies whilst a positive second derivative (minimum) would provide positive vibrational frequencies.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:06, 4 December 2015 (UTC) This is true but you could have proved with the quantum harmonic oscilator equation

[[File:IRspectrum anti2 freqVR.jpg|750px|thumb|center|Fig. 5 The IR spectrum of the anti2 conformer of 1,5-hexadiene obtained using a frequency analysis with the 6-31G(d) level of theory.]]

====Thermochemical properties of the anti2 conformer of 1,5-hexadiene====
The thermochemical properties gained from performing a frequency analysis on the anti2 conformer are listed below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" |
! style="background: #0D4F8B; color: blue; background: white;" | Calculation at 298 K (Hartrees)
! style="background: #0D4F8B; color: blue; background: white;" | Calculation at 0 K (Hartrees)
|-
| Sum of Electronic and Zero-Point Energies
| -234.469215
| -234.469215
|-
| Sum of Electronic and Thermal Energies
| -234.461867
| -234.469215
|-
| Sum of Electronic and Thermal Enthalpies
| -234.460922
| -234.469215
|-
| Sum of Electronic and Thermal Free Energies
| -234.500800
| -234.469215
|}

===Optimisation of the "Chair" and "Boat" Transition Structures===
Having now optimised the anti2 conformer of 1,5-hexadiene and subsequently performing a frequency analysis, it is now appropriate to discuss the "Chair" and "Boat" transition structures in further detail to allow us to determine which reaction pathway gives lowest activation energy for the Cope Rearrangement.

====Optimising the allyl fragment====
Prior to analysing the transition structures, an allyl fragment must be drawn and optimised using the HF/3-21G level of theory. One needs two allyl fragments to be able to construct both the "chair" and "boat" transition structures. Below, the results are shown when the allyl fragment was optimised.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; frame 10</script>
<uploadedFileContents>REACT ALLYL OPT 321GVR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:Allyl calculationsummaryVR.jpg|350px|thumb|center]]
|}
N.B. the point group of the allyl fragment was C<sub>2v</sub> after optimisation.

====Optimising the "Chair" Transition Structure====
=====Using the HF/3-21G level of theory=====
Using the optimised allyl fragment, the "chair" transition structure was guessed by placing two allyl fragments 2.2 Å (i.e. the separation of the terminal carbons are 2.2 Å) apart in a staggered conformation. This guess structure was optimised to a transition state structure by optimising to a TS(Berny). The summary of the results are provided below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE; frame 24</script>
<uploadedFileContents>OPTFREQ CHAIR TS GUESS (B)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:(b)chairoptimisationa.jpg|350px|thumb|center]]
|}
The calculation had converged and so the "chair" transition structure had successfully been computed. Thus, it is now possible to attain the imaginary frequency corresponding to the Cope Rearrangement. The imaginary frequency, -817.94 cm<sup>-1</sup>, is shown below.
<jmol>
<jmolApplet>
<title>Vibration that leads to a reaction</title><color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 25;vibration 1.5;
</script>
<uploadedFileContents>OPTFREQ CHAIR TS GUESS (B)VR.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The presence of the imaginary frequency confirms that the structure calculated is present at a maximum in energy and hence corresponds to the "chair" transition structure.
An imaginary frequency represents a negative second derivative of the potential energy surface and so corresponds to a maximum in energy and thus a transition state structure. A transition structure only has one imaginary frequency. An imaginary frequency comes about due to a negative second derivative and it corresponds to a negative force constant i.e. a transition state. A negative second derivative means that the energy decreases in directions away from the point under consideration and thus it shows it is a maximum. The negative force constant gives rise to an imaginary frequency as the vibrational frequency depends on the square root of the force constant.
:<math>v = {1 \over 2 \pi} \sqrt{k \over m} </math>
k is the force constant. The square root of a negative number gives rise to an imaginary solution and hence an imaginary vibration.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:07, 4 December 2015 (UTC) apologies for above I see you have clearly explained it here

=====Using the Frozen Coordinate Method=====
The optimised allyl fragment can also be used to produce a "chair" transition state by using the Frozen Coordinate Method. In this case, the optimised allyl fragments are again placed 2.2 Å (i.e. the separation of the terminal carbons are 2.2 Å) apart in a staggered conformation. The distance between each set of terminal carbon atoms on the allyl fragments are frozen to 2.2 Å whilst the rest of the molecule to move. An optimisation is subsequently carried out at an HF/3-21G level of theory. Following this, the bond breaking and bond forming distances were optimised, using the HF/3-21G level of theory, further to produce the final transition state structure shown below. This was also a successful method in producing the "chair" transition state structure.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE; frame 18</script>
<uploadedFileContents>OPT CHAIR TS GUESS (C)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:Optchair(c) frozenVR.jpg|350px|thumb|center]]
|}
Once again, a single imaginary frequency is produced with a frequency of -817.89 cm<sup>-1</sup>. The vibration below leads to the Cope Rearrangement.
<jmol>
<jmolApplet>
<title>Vibration that leads to a reaction</title><color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 19;vibration 1.5;
</script>
<uploadedFileContents>OPT CHAIR TS GUESS (C)VR.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

=====Comparison of the "Chair" transition structures using the guessed transition structure method and the Frozen Coordinate Method=====
Below, a comparison of the bond breaking and bond forming distances of the "chair" transition structure using the guessed structure and Frozen Coordinate methods are given below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" |
! style="background: #0D4F8B; color: blue; background: white" | Bond forming length (Å)
! style="background: #0D4F8B; color: blue; background: white " | Bond forming length (Å)
|-
| T.S structure from guess method
| 1.38928
| 2.02009
|-
| T.S. structure from Frozen coordinate method
| 1.38929
| 2.02070
|}
Whilst the bond forming bond lengths are virtually exact when using either method, there is a small difference in the bond breaking lengths. The Frozen Coordinate method produced a bond breaking length which is slightly larger but the difference is not insignificant and thus very similar transition state structures have been produced by both methods.

====Optimising the "Boat" Transition Structure====
Now, the "boat" transition structure will be optimised.
=====Using the QST2 method=====
The boat transition structure was optimised using the QST2 method using the HF/3-21G level of theory. This method works by interpolating between the reactant and product structures and in this process, it is expected that an energy maxima will be reached corresponding to the transition structure.

Initially, the anti2 conformer was used as the starting point to generate the "boat" transition structure using the labelling provided in Fig. 6.
[[File:Firsttry NUMBERingVR.png|350px|thumb|left|Fig. 6 Numbering used for the initial conformer of 1,5-hexadiene.]]

However, optimising the anti2 conformer using a QST2 calculation provided an error resulting in the transition structure shown below (Fig. 7). This is extremely unfavorable as the reactant and product geometries are too different from the transition structure geometry and so they are not able to form the transition structure. The calculation failed as it did not produce the transition structure that was required.
[[File:TSWRONG (e)VR.png|350px|thumb|left|Fig. 7 Transition state observed using the conformer above.]]

Therefore, instead of using the anti2 conformer, the (imaginary) conformer shown below (Fig. 8) was used instead as it provided the reactants and products with a geometry that closely resembled the boat transition structure. The atoms were labelled as according to Fig. 8. In order to have a successful QST2 calculation, the reactants and products must share a similar geometry to the transition structure we are after i.e. the "boat" transition structure.

[[File:ReactantnumberingVR.png|500px|thumb|left|Fig. 8 The numbering of the conformer required to produce the correct Transition Structure.]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:10, 4 December 2015 (UTC) All your frequcenies for this part are correct well done.

The conformer above (Fig. 8) was then optimised using the QST2 calculation at the HF/3-21G level to produce the boat transition structure; the results are provided below.

{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=6)(atomno=5) PARTIALDOUBLE; connect(atomno=4)(atomno=5) PARTIALDOUBLE; connect(atomno=1)(atomno=2) PARTIALDOUBLE; connect(atomno=2)(atomno=3) PARTIALDOUBLE; frame 32</script>
<uploadedFileContents>REACTANT NUMBERING QST2 E PROPERVR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:Proper(e)boatVR.png|350px|thumb|center]]
|}
Once again, a single imaginary frequency, of -839.90<sup>-1</sup>, was produced which corresponds to a transition structure. The vibration shown below corresponds to the vibration that leads to the Cope Rearrangement.
<jmol>
<jmolApplet>
<title>Vibration that leads to a reaction</title><color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 33;vibration 1.5;
</script>
<uploadedFileContents>REACTANT NUMBERING QST2 E PROPERVR.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Intrinsic Reaction Coordinate (IRC) analysis====
Now the optimized "chair" transition structure will be analysed by the Intrinsic Reaction Coordinate calculation using the HF/3-21G level of theory. The point of the IRC calculation is provide information regarding which conformers are connected to the chair and boat transition structures. The IRC calculation had converged after sampling 44 points along the IRC even though a sampling of 50 points was initially specified in the calculation input. It must be stated that the calculation needs only to be run in the forward direction as the reaction coordinate is symmetrical. The accuracy of an IRC calculation can be improved by increasing the number of points to be sampled along the IRC coordinate but 50 points was found to be enough in order to allow a convergence of the calculation.
The results from the IRC calculation are provided below.
{| class="wikitable"
! style="background: #0D4F8B; color: black; background: white;" | [[File:TOTALENERGYVR.png|800px|thumb|center|Fig. 9 Total Energy IRC of the optimised "chair" transition structure.]]
|-
| style="background: #0D4F8B; color: black; background: white;" | [[File:RMSGRADIENTVR.png|800px|thumb|center|Fig. 10 RMS Gradient Norm along the IRC for the optimised "chair" transition structure.]]
|}
From the results of the IRC calculation, it can be seen that the RMS gradient (i.e. derivative of the total energy along the IRC) approaches and becomes zero with increasing IRC which confirms that calculation has been completed and has converged.
N.B. the IRC calculation must be run at the same level of theory as that used when optimising the transition state structure (here, the HF/3-21G level was used consistently).

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:13, 4 December 2015 (UTC) Nice IRC but you havent connect the confs that it connects and therefor. you could have done this by computing force constants at every step and then comparing the final one to the appendix. or re optimised the final structure.

====Activation Energies====
In order to be able to calculate the activation energies of this Cope rearrangement via either transition structure, the previously optimised chair and boat transition structures must be reoptimised at the B3LYP/6-31G(d) level. This was done by optimising to a TS(Berny)(as was done when computing the chair transition structure from a guess structure) whilst completing a frequency analysis. Below the results of the boat and chair transition structures arising from the reoptimisations are shown.

Below, the calculation summary and the imaginary vibration, of frequency -530.36<sup>-1</sup>, of the reoptimised boat structure is shown.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Imaginary frequency vibration
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol>
<jmolApplet>
<color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 13;vibration 1.5;
</script>
<uploadedFileContents>Boat optfreq B3YLP (g)VR.out</uploadedFileContents>
</jmolApplet>
</jmol>
| [[File:BOAT631gd CALCVR.png|350px|thumb|center]]
|}

Below, the calculation summary and the imaginary vibration, of frequency -566.36<sup>-1</sup>, of the reoptimised chair structure is shown.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Imaginary frequency vibration
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
|-
| <jmol>
<jmolApplet>
<color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 21;vibration 1.5;
</script>
<uploadedFileContents>Chair optfreq B3YLP (g)VR.out</uploadedFileContents>
</jmolApplet>
</jmol>
| [[File:Chairreport631VR.png|350px|thumb|center]]
|}

Having now reoptimised the chair and boat transition structures, it is possible to calculate the activation energies for the reactant, in the anti2 conformation, to convert to the product via both the chair and boat transition structures when using the HF/3-21G and B3LYP/6-31G(d) levels of theory at both 0 K and 298.15 K.
Before that, all the thermochemical data associated with the transition structures (both chair and boat) and the reactant, in the anti2 conformation, are provided below at HF/3-21G and B3LYP/6-31G(d) levels of theory used (and at 0 K and 298.15 K).
{| class="wikitable" style="margin: 1em auto 1em auto;"
! !! HF/3-21G Electronic Energy (a.u.)||HF/3-21G Sum of Electronic and Thermal Energies at 0 K (a.u.)||HF/3-21G Sum of Electronic and Thermal Energies at 298.15 K (a.u.)|| B3LYP/6-31G(d) Electronic Energy (a.u.)||B3LYP/6-31G(d) Sum of Electronic and Thermal Energies at 0 K (a.u.)|| B3LYP/6-31G(d) Sum of Electronic and Thermal Energies at 298.15 K (a.u.)
|-
|Chair||-231.61932241||-231.466697||-231.461338||-234.55697207||-234.414894||-234.408976
|-
|Boat||-231.60280241||-231.450930||-231.445301||-234.54309307||-234.402342||-234.396008
|-
|Anti 2 Reactant||-231.69253517||-231.539539||-231.532565||-234.61171062||-234.469215||-234.461867
|}
From this thermochemical data, the activation energies can be found and they have been listed in the table below.
{| class="wikitable" border="1"
|-
!
! colspan="2" width="250" align="center" | '''HF/3-21G'''
! colspan="2" width="250" align="center" | '''B3LYP/6-31G(d)'''
! width="125" align="center" | '''Expt.'''
|-
! ''' '''
! width="125" align="center" | '''0 K '''
! width="125" align="center" | '''298.15 K'''
! width="125" align="center" | '''0 K'''
! width="125" align="center" | '''298.15 K'''
! width="125" align="center" | '''0 K'''
|-
| ΔE (Chair)(kcal/mol)
| align="center" | 45.709011
| align="center" | 44.695584
| align="center" | 34.086916
| align="center" | 33.189579
| align="center" | 33.5 ± 0.5
|-
| ΔE (Boat)(kcal/mol)
| align="center" | 55.602945
| align="center" | 54.758945
| align="center" | 41.963409
| align="center" | 41.327115
| align="center" | 44.7 ± 2.0
|}
Much information can be gathered from this data. Firstly, the energy required to surmount the transition structure is greater at 0 K than at 298.15 K. This statement is rather straight forward as there is an absence of thermal energy at 0 K and so a greater energy input is required to surmount the activation barrier. The presence of thermal energy (E = kT) means that the activation barrier can be surmount much more easily as less energy must be input into the system.
Furthermore, the accuracy of the activation energies that are obtained is much greater when using the B3LYP/6-31G(d) level of theory as it uses fewer approximations in its calculations.
Lastly, our aim was to find out which transition structure provided a lower energy pathway for the Cope rearrangement. This can now be ascertained by observation of the activation energies above. It can clearly be seen that the activation energy for the Cope Rearrangement when passing through a chair transition structure is much lower than when passing through a boat transition structure. Thus one can conclude that the rate of the Cope Rearrangement is much greater when passing through the chair transition structure and so it is the preferred route for the reaction.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:24, 4 December 2015 (UTC) Your energies are correct and you have come to correct conclusions. You didnt however say how the geoms are approximated well for the lower basis set. but energies or very off. therefore for geoms a lower basis set can be sufficient. This was an extremly well written report and you have done basically everything asked of you and you have backed it up with a very good understanding of the theory

==The Diels-Alder cycloaddition==
===Background===
Two different Diels-Alder reactions were studied in this study. Firstly, the Diels-Alder cycloaddition of ethene and cis-butadiene (Fig. 11) was studied in depth followed by the Diels-Alder cycloaddition of maleic anhydride and cyclohexa-1,3-diene (Fig. 12). Diels-Alder reactions are [4+2]-cycloadditions in which a diene and a dienophile react in a concerted pericyclic reaction. The Diels-Alder cycloadditions are thermodynamically favorable as two new σ bonds are much stronger than two π bonds. For the reaction to occur most effectively, there must be an electron rich diene and an electron deficient dienophile as this allows the HOMO and LUMO to be closer in energy and so allows greater orbital interactions. Thus it is expected that the reaction between maleic anhydride, an electron poor dienophile, and cyclohexa-1,3-diene, a relatively electron diene, should be more facile than that between cis-butadiene and ethene.
In terms of the Woodward-Hoffmann rules, the Diels-Alder cycloaddition is a thermally allowed reaction. This is because the total number of components that fit the equation (4q + 2)<sub>s</sub> +(4r)<sub>a</sub> is odd. In this case there is one <sub>π</sub>4<sub>2</sub> component and one <sub>π</sub>2<sub>s</sub> component. It is important to note that q and r can only take integer values and that the suffixes 'a' and 's' stand for antarafacial and suprafacial respectively <ref name="WHANALYSIS"/>.
In all of the following calculations, semi-empirical/AM1 calculations will be carried out. There are many advantages of using this orbital method over using Hartree-Fock as the calculations converge more quickly and it is more applicable towards the organic molecules studied in these reactions.
Moreover, the endo and exo transition states of the reaction between maleic anhydride and cyclohexa-1,3-diene will be studied which forms the endo and exo products shown below in Fig. 12.
It should be noted that reactions between reactants in which the HOMO and LUMO are most closely matched in energy with the same symmetry will allow a reaction to occur.
[[File:Diels Alder1VR.png|350px|thumb|left|Fig. 11 The Diels-Alder reaction between ethene and cis-butadiene.]]
[[File:Diels Alder2VR.jpg|350px|thumb|left|Fig. 12 The Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene to form the endo and exo products.]]

Before commencing the studies on the Diels-Alder reactions, the semi-empirical/AM1 level of theory that will be used throughout the experiment should be introduced. The semi-empirical level of theory is based on the Hartree-Fock theory described previously. However, as described previously, the Hartree-Fock method makes many approximations and this can often lead to errors being computed. Thus, the semi-empirical level of theory, the results produced are fitted under a set of parameters such as to produce data that best agrees with the experimental data. The AM1 calculation is based on the Neglect of Differential Diatomic Overlap integral approximation which decreases the complexity of the calculations in comparison to the Hartree-Fock equations discussed previously.

===Diels-Alder cycloaddition of ethene and cis-butadiene===
====Optimising the ethene and cis-butadiene reactants====
Prior to studying the Diels-Alder cycloaddition of ethene and cis-butadiene, these reactants will first be optimised at the Semi-Empirical/AM1 level of theory. Using Gaussview, a molecule of cis-butadiene and ethene were built and optimised in an Opt+Freq job type at the level of theory stated above. The results of these calculations are provided in the next two sections.
=====Cis-butadiene=====
The cis-butadiene molecule was optimised, as stated above, to provide the results which are shown in the table below. N.B. The dihedral angle of the butadiene molecule had been changed to 0° to build a butadiene molecule in the cis conformation.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=4) DOUBLE; connect(atomno=8)(atomno=6) DOUBLE; frame 14</script>
<uploadedFileContents>CIS BUTADIENE OPT SE AM1VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:ButadieneresultsVR.png|350px|thumb|left]]
| C<sub>2v</sub>
| 0.04879719
|}

From the Opt+Freq job type carried out, the molecular orbitals of cis-butadiene can be obtained. The HOMO and LUMO of cis-butadiene are given below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of cis-butadiene
! style="background: #0D4F8B; color: blue; background: white" | LUMO of cis-butadiene
|-
| [[File:ButadieneHOMOVR.png|350px|thumb|left|Antisymmetric]]
| [[File:ButadieneLUMOVR.png|350px|thumb|left|Symmetric]]
|}
Thus, one can observe that the HOMO is antisymmetric with respect to the plane bisecting the C-C single bond whilst the LUMO is symmetric with respect to this plane.
=====Ethene=====
Similarly, a molecule of ethene was optimised at the Semi-Empirical/AM1 level of theory using an Opt+Freq job type. The results of this calculation are shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=1)(atomno=4) DOUBLE;frame 12</script>
<uploadedFileContents>ETHENE OPT SE AM1VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:EtheneresultsVR.png|350px|thumb|left]]
| D<sub>2h</sub>
| 0.02619028
|}

Once again, having done a frequency analysis, it is possible to analyse the molecular orbitals of ethene. The HOMO and LUMO are shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of ethene
! style="background: #0D4F8B; color: blue; background: white" | LUMO of ethene
|-
| [[File:EtheneHOMOVR.png|350px|thumb|left|Symmetric]]
| [[File:ETHENElumoVR.png|350px|thumb|left|Antisymmetric]]
|}
One can observe that the HOMO of ethene is symmetric with respect to the plane containing the molecule whilst the LUMO is antisymmetric.
====Optimising the transition state for the Diels-Alder cycloaddition of ethene and cis-butadiene====
Having now successfully optimised the reactants, it is possible to determine the transition state for this Diels-Alder cycloaddition by carrying out further calculations. The transition state structure was obtained by optimising to a TS(Berny) (this was used earlier when finding out the transition state for the Cope Rearrangement for 1,5-hexadiene from a guess structure). Using the optimised structures of ethene and cis-butadiene, a guess transition structure was built and the interfragment distance was changed to 2.20 Å. The optimisation was carried out the Semi-Empirical/AM1 level of theory using an Opt+Freq job type. The results of this calculation is shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=3)(atomno=2) PARTIALDOUBLE; connect(atomno=2)(atomno=1) PARTIALDOUBLE; connect(atomno=1)(atomno=6) PARTIALDOUBLE; connect(atomno=4)(atomno=5) PARTIALDOUBLE; frame 44</script>
<uploadedFileContents>DA BUTAETHENE TS OPTFREQ SE AM1 (II)vignesh.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:TSbutaethenecalcVR.png|350px|thumb|left]]
| 0.11165469
|}
Having done a frequency analysis, the molecular orbitals can be obtained and the HOMO and LUMO of the transition state are shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of transition structure
! style="background: #0D4F8B; color: blue; background: white" | LUMO of transition structure
|-
| [[File:TS(1)HOMOVR.png|350px|thumb|left|Antisymmetric]]
| [[File:TS(1)LUMOVR.png|350px|thumb|left|Symmetric]]
|}
The HOMO of the transition state is antisymmetric and the LUMO is symmetric with respect to the plane. It can be seen that the HOMO of cis-butadiene combines with the LUMO of ethene to form the HOMO, which is antisymmetric, of the transition state. To complete this argument, the LUMO of the transition state can bebuilt by combination of the HOMO of of ethene, which is symmetric, and the LUMO of cis-butadiene, which is antisymmetric.

(The LUMO of cis-butadiene is symmetric. Benefit of the doubt here, as you've said before that it is symmetric, but be careful [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 26 November 2015 (UTC))

From examination of the vibrational frequencies of the transition structure, one can see the presence of an imaginary frequency, which is shown below, and so this confirms that the structure obtained is indeed a transition state. This vibration, of frequency -956.40 cm<sup>-1</sup>, shows the synchronous bond formation at bond ends of each fragment as the fragments move together. The two newly forming C-C bonds form at the same time and hence the reaction is concerted.
<jmol>
<jmolApplet>
<title>The imaginary frequency that leads to the Diels-Alder cycloaddition of ethene and cis-butadiene.</title>
<color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 45;vibration 1.5;
</script>
<uploadedFileContents>DA BUTAETHENE TS OPTFREQ SE AM1 (II)vignesh.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
On the other hand, when observing the lowest positive vibrational frequency, of frequency 147.00 cm<sup>-1</sup>, it can be seen that this vibration is not allowed in any bond forming or bond breaking process and so does not lead to a reaction. In this vibration, neither fragment moves towards or away from each other and so no bond formation occurs. It can be concluded that all positive frequencies are perpendicular to the imaginary frequency and so none of them correspond to the fragments moving together and forming bonds.

(Perpendicular is the wrong word here. The modes are independent or uncoupled to each other [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 26 November 2015 (UTC))

<jmol>
<jmolApplet>
<title>The first positive vibrational frequency of the transition structure.</title>
<color>white</color><size>250</size>
<script>connect(atomno=1)(atomno=4) PARTIALDOUBLE; connect(atomno=4)(atomno=6) PARTIALDOUBLE; connect(atomno=9)(atomno=12) PARTIALDOUBLE; connect(atomno=12)(atomno=14) PARTIALDOUBLE</script>
<script>frame 46;vibration 1.5;
</script>
<uploadedFileContents>DA BUTAETHENE TS OPTFREQ SE AM1 (II)vignesh.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

Overall this reaction is allowed as the energies of the HOMO of the cis-butadiene and the LUMO of the ethene are close in energy to be able to interact. Thus, this interaction allows new σ bonds to form between the fragments. The HOMO of cis-butadiene and the LUMO also have the same symmetry and are in phase when the reaction occurs and so the reaction is allowed.

====Analysis of the geometry of the transition structure from the Diels-Alder cycloaddition of ethene and cis-butadiene====

[[File:NumberingTS.png|350px|thumb|left|Fig. 13 The numbering of the atoms of the transition state from the reaction of ethene and cis-butadiene]]
{| class="wikitable"
|+ Table. C-C bond lengths for the ethene + cis-butadiene transition structure using the numbering from Fig. 13
! style="text-align: center; color: blue; background: white" | Atoms
! style="text-align: center; color: blue; background: white" | Bond Length (Å)
|-
| style="text-align: center;" | C1-C2
| style="text-align: center;" | 1.38186
|-
| style="text-align: center;" | C2-C3
| style="text-align: center;" | 1.39748
|-
| style="text-align: center;" | C3-C4
| style="text-align: center;" | 1.38187
|-
| style="text-align: center;" | C5-C6
| style="text-align: center;" | 1.38291
|-
| style="text-align: center;" | C1-C6
| style="text-align: center;" | 2.11924
|-
| style="text-align: center;" | C4-C5
| style="text-align: center;" | 2.11915
|}

In this reaction, a C-C single bond is forming between C1 and C6 and between C4 and C5. It can be seen that the C-C distances in this case is 2.12 Å which is shorter than the originally inputted inter fragment distance of 2.20 Å. Thus it shows that C-C bonds are beginning to form in the transition structure due to the overlap of the HOMO of the cis-butadiene and the LUMO of the ethene bringing the fragments together.
This is further backed up by observing that the C1-C6 and C4-C5 bond lengths are much shorter than twice the Van der Waals radius for a carbon atom, which is around 1.70 Å <ref name="VdWradius"/>, which indicates that the sigma bonds are forming.
It should be noted that the sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.33 Å respectively <ref name="c-cbonds"/>. On examination of other bond lengths within the cyclic transition state. It can be seen that the bond length of C2-C3 is intermediate that of a double and a single bond showing that a π bond is now forming between these two carbons. The opposite is true for the bonds C1-C2,C3-C4 and C5-C6 in which a π bond is now breaking to leave a single bond which explains why these lengths are intermediate between a single and a double bond length.

(Just because the C-C distance is shorter than the input distance, this does not suggest a transition state necessarily. Distances shorter than the Van der Waals radii would indicate a bonding interaction of some kind. It is the single imaginary frequency that truly confirms that it is a transition state [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 26 November 2015 (UTC))

===The regioselectivity of the Diels-Alder cycloaddition===
The regioselectivity of the Diels-Alder reaction will be discussed by examining the reaction between maleic anhydride and cyclohexa-1,3-diene in which an endo and exo product can be formed by different approaches of the diene to the dienophile. By computation of which transition structure is lower in energy, we can state which product, endo or exo, is preferred in terms of energetics.
====Optimisation of cyclohexa-1,3-diene====
A molecule of cyclohexa-1,3-diene is optimised using the Semi-Empirical/AM1 level of theory. The results of this optimisation are given below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=4)(atomno=3) DOUBLE; connect(atomno=5)(atomno=6) DOUBLE; frame 22</script>
<uploadedFileContents>CYCLOHEXADIENE OPT SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:CyclohexaresultsVR.png|350px|thumb|left]]
| C<sub>2</sub>
| 0.02771129
|}
Once again, the molecular orbitals of cyclohexa-1,3-diene can be examined. In particular, the HOMO and LUMO are shown below. The HOMO is antisymmetric with respect to the plane whilst the LUMO is symmetric with respect to the plane.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of cyclohexa-1,3-diene
! style="background: #0D4F8B; color: blue; background: white" | LUMO of cyclohexa-1,3-diene
|-
| [[File:Cyclohexahomo.png|350px|thumb|left|Antisymmetric]]
| [[File:Cyclohexalumo.png|350px|thumb|left|Symmetric]]
|}

====Optimisation of maleic anhydride====
Similarly, an optimisation of the maleic anhydride molecule was carried out using the Semi-Empirical/AM1 level of theory. The results of the calculation are shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=2)(atomno=3) DOUBLE; connect(atomno=8)(atomno=1) DOUBLE; connect(atomno=9)(atomno=4) DOUBLE; frame 20</script>
<uploadedFileContents>MAL ANHYDRIDE OPT SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:MalcalcVR.png|350px|thumb|left]]
| C<sub>s</sub>
| -0.12182418
|}

(The symmetry of maleic anhydride should be C2v [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 26 November 2015 (UTC))

The molecular orbitals are once again examined and the HOMO and LUMO of maleic anhydride are shown below. The HOMO is symmetric with respect to the plane whilst the LUMO is antisymmetric with respect to the same plane.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of maleic anhydride
! style="background: #0D4F8B; color: blue; background: white" | LUMO of maleic anhydride
|-
| [[File:HOMOmalVR.png|350px|thumb|left|Symmetric]]
| [[File:LUMOmalVR.png|350px|thumb|left|Antisymmetric]]
|}

====Optimising the exo transition structure====
The optimised cyclohexa-1,3-diene and maleic anhydride molecules were combined and seperated by an interfragment distance of 2.2 Å. The components were organised such that it resembled an exo product i.e. a guess structure was created. This structure was subsequently optimised using the Semi-Empirical/AM1 level of theory by optimising to a TS(Berny) to finally form an exo transition state. The results of this calculation are shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=16)(atomno=17) partialDOUBLE; connect(atomno=2)(atomno=1) partialDOUBLE; connect(atomno=3)(atomno=2) partialDOUBLE; connect(atomno=3)(atomno=4) partialDOUBLE; connect(atomno=15)(atomno=23) DOUBLE; connect(atomno=18)(atomno=22) DOUBLE;frame 122</script>
<uploadedFileContents>EXO TS OPTFREQ SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:ExoTScalcVR.png|350px|thumb|left]]
| C<sub>1</sub>
| -0.05041975
|}
The HOMO and LUMO of the exo transition state are given below. The HOMO and LUMO are antisymmetric with respect to the plane.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of the exo transition state
! style="background: #0D4F8B; color: blue; background: white" | LUMO of the exo transition state
|-
| [[File:EXOHOMOVR.png|350px|thumb|left|Antisymmetric]]
| [[File:EXOTSlumoVR.png|350px|thumb|left|Antisymmetric]]
|}
Having also done a frequency analysis, it was possible to observe an imaginary frequency, of value -812.32 cm<sup>-1</sup>, and this shows that the calculation successfully converged to the formation of the exo transition state. The vibration below also shows the movement that occurs to allow the cycloaddition to take place. Again, it shows that the new C-C bonds are formed synchronously.
<jmol>
<jmolApplet>
<title>Vibration leading to the Diels-Alder cycloaddition</title>
<color>white</color>
<size>250</size>
<script>frame 123;vibration 1;</script>
<uploadedFileContents>EXO TS OPTFREQ SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Optimising the endo transition structure====
Using the same method as for the exo transition structure, the optimised maleic anhydride and cyclohexa-1,3-diene are placed 2.2 Å apart in an endo fashion, i.e. a guess structure, such that secondary orbital interactions are possible. This resulting structure was then optimised to a TS(Berny) using the Semi-Empirical/AM1 level of theory to give the results shown below.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Jmol Structure
! style="background: #0D4F8B; color: blue; background: white;" | Calculation Summary
! style="background: #0D4F8B; color: blue; background: white;" | Point Group
! style="background: #0D4F8B; color: blue; background: white;" | Energy (Hartrees)
|-
| <jmol><jmolApplet>
<color>white</color>
<size>300</size>
<script>connect(atomno=16)(atomno=17) partialDOUBLE; connect(atomno=2)(atomno=1) partialDOUBLE; connect(atomno=3)(atomno=2) partialDOUBLE; connect(atomno=3)(atomno=4) partialDOUBLE; connect(atomno=15)(atomno=23) DOUBLE; connect(atomno=18)(atomno=22) DOUBLE;frame 114</script>
<uploadedFileContents>ENDO TS OPTFREQ SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet></jmol>
| [[File:EndocalcVR.png|350px|thumb|left]]
| C<sub>1</sub>
| -0.05150477
|}
The HOMO and LUMO of the endo transition state are given below. Once again, the HOMO and LUMO are antisymmetric with respect to the plane.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" | HOMO of the endo transition state
! style="background: #0D4F8B; color: blue; background: white" | LUMO of the endo transition state
|-
| [[File:EndoHOMO.png|350px|thumb|left|Antisymmetric]]
| [[File:ENDOLUMOVR.png|350px|thumb|left|Antisymmetric]]
|}
An imaginary frequency, of value -806.55 cm<sup>-1</sup> is observed after a frequency calculation had been performed on the endo transition structure. This, therefore, confirms the formation of the transition state. The vibration shows the terminal ends of each fragment approach each other in a concerted manner and shows the two new C-C bonds being formed synchronously to lead to a cycloaddition reaction taking place.
<jmol>
<jmolApplet>
<title>Vibration leading to a Diels-Alder cycloaddition</title>
<color>white</color>
<size>250</size>
<script>frame 115;vibration 1;</script>
<uploadedFileContents>ENDO TS OPTFREQ SE AM1 (III)VR.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Comparison of the endo and exo transition structures====
Having now obtained the exo and endo transition states, one can compare the geometries and relative energies of the two transition states.
=====Energy of the conformers=====
Below the electronic energies as well as the sum of the electronic and zero-point energies and the sum of the electronic and thermal energies are given below. These values are provided for both of the reactants as well as the exo and endo transition structures at both 0 K and 298.15 K. From this we can calculate the activation energies for the process via the endo and exo transition states, these activation energies are given below.
{|class="wikitable" style="margin: 1em auto 1em auto;"
|+ Energies for exo, endo transition structures and reactants, cyclohexa-1,3-diene and maleic anhydride, at AM1 (Semi-Empirical)
|-
| ''' '''
!colspan="3"|'''AM1 Semi-Empirical'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (Hartrees)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (Hartrees)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (Hartrees)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Exo TS'''
| align="center" | -0.05041975
| align="center" | 0.134880
| align="center" | 0.144881
|-
| '''Endo TS'''
| align="center" | -0.05150477
| align="center" | 0.133493
| align="center" | 0.143682
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02771129
| align="center" | 0.152502
| align="center" | 0.157726
|-
| '''Maleic anhydride'''
| align="center" | -0.12182418
| align="center" | -0.063346
| align="center" | -0.058192
|}

{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white" |
! style="background: #0D4F8B; color: blue; background: white" | Electronic Energy (Hartrees)
! style="background: #0D4F8B; color: blue; background: white" | Relative energy (kcal/mol)
|-
| Exo Transition structure
| -0.05041975
| 0.680860
|-
| Endo Transition structure
| -0.05150477
| 0
|}
From inspection of the relative energies, the endo transition state is lower in energy and so the Diels-Alder cycloaddition between maleic anhydride and cyclohexa-1,3-diene takes place via an endo transition state, as it has a lower activation energy, to form an endo product. The relative energies were calculated using the electronic energies.
The exo transition state is higher in energy due to steric reason. There are steric interactions between the lone pairs of the ether oxygen on the maleic anhydride and the sp<sup>3</sup> hydrogens on the -CH<sub>2</sub> groups. This interaction is pronounced in the exo state as the these orbitals are eclipsing and clashing with one another which can be further increased in the transition state when the components approach each other more closely.
Moreover, the nodal properties of the HOMO, of both of the exo and endo transition structures, between the -(C=O)-O-(C=O)- and the remainder of the fragment were analysed. There are more nodes present between the -(C=O)-O-(C=O)- and the remainder of the fragment in the endo transition structure than the exo transition structure.

Below, the summary of the activation energies are provided.
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''AM1 Semi-Empirical'''
| width="125" align="center" | '''AM1 Semi-Empirical'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''ΔE (Exo TS) (kcal/mol)'''
| align="center" | 28.690339
| align="center" | 28.455651
|-
| '''ΔE (Endo TS) (kcal/mol)'''
| align="center" | 27.820612
| align="center" | 27.703267
|}
From examination of the activation energies, this data further confirms that the cycloaddition prefers to occur via an endo transition state as the activation energies are lower when passing through the endo transition structure.

=====Comparison of bond lengths=====
The C-C bond lengths were obtained for the endo and exo transition states by using the numbering system for transition structures as shown by the figures below. The tables below give the C-C bonds for the exo and endo transition structures for specific C-C bonds.
{| class="wikitable"
! style="background: #0D4F8B; color: blue; background: white;" | Numbering of the exo transition state
! style="background: #0D4F8B; color: blue; background: white;" | Numbering of the endo transition state
|-
| [[File:ExonumberingVR.png|400px]]
| [[File:Endo numberingV.png|400px]]
|}

{| class="wikitable"
|+ Bond lengths for exo transition state
! Atoms !! Bond Lengths (Å)
|-
| C5-C6 || 1.39674
|-
| C6-C1 || 1.39439
|-
| C1-C2 || 1.48975
|-
| C2-C3 || 1.52210
|-
| C1-C17 || 2.17056
|-
| C17-C16 || 1.41015
|-
| C15-C16 || 1.48817
|}

{| class="wikitable"
|+ Bond lengths for endo transition state
! Atoms !! Bond Lengths (Å)
|-
| C5-C6 || 1.39725
|-
| C1-C6 || 1.39307
|-
| C1-C2 || 1.49054
|-
| C2-C3 || 1.52298
|-
| C1-C17 || 2.16243
|-
| C17-C16 || 1.40850
|-
| C16-C15 || 1.48924
|}

(GaussView states a very high level of precision, but the error level is very high. As such, it's not correct to state a bond length beyond the second decimal place, really [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 17:17, 26 November 2015 (UTC))

Immediately, it can be seen that the new σ bond being formed (i.e. C1-C17) is significantly shorter in the endo transition structure than the exo transition structure. This again confirms that the endo transition state is the preferred transition state to pass through during the Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene as there are much greater orbital interactions which means that the fragments are pulled closer together to maximise these favorable interactions.
In both the exo and endo forms, the new σ C-C bond being formed is less than twice the VdW radius of a carbon atom, 1.70 Å, which shows that C-C bond is in the process of being formed in both processes and shows that reaction will occur via the exo and endo transition structures. Overall, there are favorable interactions between the maleic anhydride and the cyclohexa-1,3-diene in both transition states but these favorable interactions are greater in the endo form due to the addtionally stabilising secondary orbital interactions.
These secondary orbital interactions also provide another reasoning as to why the exo transition state is more strained.

(Use this paragraph as an opportunity to tie everything together as to why the endo TS is lower in energy. This includes the steric hindrance (evidenced by the distance between the side groups in the exo TS being shorter than 2xVdW). Then you can mention that this prevents as much of the favourable overlap as the endo TS has access to [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 26 November 2015 (UTC))

====Secondary Orbital Overlap Effects====
This interaction acts to stabilize a molecule through the overlap of π orbitals of the diene and the dienophile. This effect allows the endo transition state to be lower in energy than the exo transition state. This is because there is a favorable interaction between the π orbitals of the carbon atoms on the diene and the C=O π orbitals of the maleic anhydride. This secondary orbital interaction is a non-bonding interaction but it lowers the total energy of the transition state and thus lowers the activation energy which increases the rate of the cycloaddition process for the endo transition state. Thus the endo product is kinetically favoured whilst the exo product is thermodynamically favoured. The secondary orbital interaction is not possible in the exo transition state as the C=O π orbitals are directed in the opposite direction from the π system of the diene.

==Conclusions==
In summary, this computational experiment studied the Cope Rearrangement of 1,5-hexadiene and the Diels-Alder reactions of cis-butadiene and ethene as well as the reaction between cyclohexa-1,3-diene and maleic anhydride on the Gaussian program.

For the Cope Rearrangement, a variety of different levels of theory, such as the B3LYP/6-31G(d) and HF/3-21G levels of theory, were used throughout the study and the results provided by each of these calculations were compared and contrasted. It was found that the B3LYP/6-31G(d) level of theory is far superior in accuracy compared to the Hartree-Fock level of theory as the Density Functional Theory takes in account fewer approximations in its calculations to give a more accurate result. The transition states, both the chair and boat structures, for the Cope Rearrangement were discussed and the formation of the transition states from the calculations had been confirmed by the presence of a single imaginary vibrational frequency. This vibration involved the reactant undergoing the Cope Rearrangement whilst all other positive vibrations were not involved in the reaction process. Moreover, the activation energies for the Cope Rearrangement using the different levels of theory were compared and it was found that the B3LYP/6-31G(d) level of theory produced much greater accuracy in the results. It was found that the Cope Rearrangement was more favorable through the chair transition state than the boat transition state.

In terms of the Diels-Alder reactions that were examined, all calculations were carried out using the SE/AM1 level of theory which proved to be sufficient. The Diels-Alder cycloaddition reaction between ethene and cis-butadiene provided simplistic results which didn't provide much information on the Diels-Alder reactions apart from confirming the concerted nature of the reaction through the synchronous bond formation of the C-C bonds between the ethene and c-butadiene. However, the regioselectivity of the Diels-Alder reaction was studied in depth by inspecting the reaction between maleic anhydride and cyclohexa-1,3-diene. This study showed that the process was faster, i.e. had a lower activation energy, when passing through the endo transition structure due to stabilising secondary orbital interactions which are not present in the exo structure. Finally, when comparing the energetic and geometric properties of the endo and exo transition structures, it can be found out that the exo transition structure is more strained due to fewer stabilising interactions and more steric clashes between the oxygen lone pairs of the maleic anhydride and sp<sup>3</sup> hydrogens on the diene.

Having thoroughly evaluated this computational study, there are many factors that could have been implemented to produce more accurate and reliable results. Firstly, one could have tried the Frozen Coordinate Method for the formation of the transition states in the Diels-Alder reactions to allow for a broader comparison. Moreover, an extended study on the formation of the boat transition structure of the Cope Rearrangement could have been carrried out by optimising using the QST3 method to see any changes from the results of the calculation compared to the QST2 method.
Further research could have been done on studying both the Cope Rearrangement <ref name="hetero"/> and Diels-Alder reactions using aza-1,5-dienes and hetero dienes and dienophiles respectively. This can be studied to see if the preferred route for the reactions had changed. For example, it could be studied whether the Diels-Alder reaction is still preferred via the endo transition structure energetically when hetero dienes/dienophiles are used. The heteroatom used can be either oxygen, sulfur or nitrogen.

==References==
<references>
1. <ref name="Gauche3"> B.W. Gung, Z. Zhu, R.A. Fouch, J. Am. Chem. Soc., 1995, 117(6), pp 1783-1788. </ref>

2. <ref name="Anti1"> N. Nishio, Séminaires & Conférences Chimie École Doctorale, 2010, 459. </ref>

3. <ref name="lengths"> G. Schultz, I. Hargittai, J. Mol. Struct., 1995, 346, pp 63-69. </ref>

4. <ref name="WHANALYSIS"> D. Nasipuri, Stereochemistry of Organic Compounds: Principles and Applications, New Age International Ltd., India, 1st Ed., 1991.</ref>

5. <ref name="c-cbonds"> R. J. Ouellette, J. David Rawn, Organic Chemistry Structure, Mechanism, and Synthesis, Elseview Inc., USA, 1st Ed., 2014, p 27. </ref>

6. <ref name="VdWradius"> S. S. Batsanov, Inorg. Mat., 2001, 37(9), pp 871-885. </ref>

7. <ref name="hetero"> R. F. Winter, G. Rauhut, Chemistry: a European Journal 8, 2002, 3, pp. 641-649. </ref>

</references>

Rep:Mod:Mtn113

2015-12-16T19:30:51Z

Mtn113: /* Exo Product */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Exo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==

Rep:Mod:Mtn113

2015-12-16T19:27:45Z

Mtn113: /* Endo Product */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Endo new tsberny.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 EXO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==

File:Mtn113 Exo new tsberny.mol

2015-12-16T19:25:23Z

Mtn113:

File:Mtn113 Endo new tsberny.mol

2015-12-16T19:24:56Z

Mtn113:

Rep:Mod:Mtn113

2015-12-16T19:23:21Z

Mtn113: /* Obtaining the Transition State via TS Berny */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene cisbutadiene TS berny new DFT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ENDO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 EXO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==

File:Mtn113 Ethene cisbutadiene TS berny new DFT.mol

2015-12-16T19:23:03Z

Mtn113:

File:Mtn113 Ethene&cisbutadiene TS berny new DFT.mol

2015-12-16T19:22:23Z

Mtn113:

File:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.mol

2015-12-16T19:21:48Z

Mtn113:

Rep:Mod:Mtn113

2015-12-16T19:19:57Z

Mtn113: /* Optimisation of ethene and cis-butadiene */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Ethene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Cis butadiene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ENDO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 EXO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==

File:Mtn113 Cis butadiene.mol

2015-12-16T19:19:43Z

Mtn113:

File:Mtn113 Ethene.mol

2015-12-16T19:18:29Z

Mtn113:

File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

2015-12-16T19:13:42Z

Mtn113: Mtn113 uploaded a new version of File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

2015-12-16T19:12:05Z

Mtn113: Mtn113 uploaded a new version of File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

Rep:Mod:Mtn113

2015-12-16T19:08:33Z

Mtn113: /* Optimisation via QST2/QST3 */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 Opt chair QST3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CIS BUTADIENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ENDO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 EXO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==

File:Mtn113 Opt chair QST3.mol

2015-12-16T19:07:49Z

Mtn113:

File:Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol

2015-12-16T19:05:17Z

Mtn113: Mtn113 uploaded a new version of File:Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol

File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

2015-12-16T19:04:18Z

Mtn113: Mtn113 uploaded a new version of File:Mtn113 CHAIR TS REDUNDANT COORDINATE.mol

Rep:Mod:Mtn113

2015-12-16T18:59:35Z

Mtn113: /* Optimisation via QST2/QST3 */

= <br> Transition States and Reactivity =

== Introduction ==
Transition states possess the highest energy levels in a reaction pathway, reflected as saddle points in potential energy surfaces. While transition states cannot be isolated and observed experimentally due to These transition states can be modeled under different levels of theory. Different computational methods (levels of theory) exist, of which we would employ three: Hartree Fock/3-21G, Density Functional Theory/B3LYP/6-31G* or Semi-Empirical/AM1. The fastest and most basic method would be the semi-empirical method, more specifically the Austin Model 1(AM1), followed by the Hartree Fock (HF) method and Density Functional Theory (DFT) method which is the most accurate and widely used mode of calculation. This follows from the fact that the AM1 method does not work from atomic orbital basis sets, while the HF method assumes that many-electron wavefuntions take the form of a determinant of single-electron wavefunctions which means electronic correlations are neglected and thus electrons of the same spin can theoretically occupy the same orbital. As a result, energies estimated by HF tend to be overestimated as compared to DFT. However, in the DFT method, many-electron wavefunctions are replaced by considering electron density, and by including an electron exchange-correlation functional (B3LYP), this means that interactions between electrons are taken into account, reflecting more accurate (and usually lower) energy values. Because of the aforementioned differences in the basis sets of the different levels of theory, it is not meaningful to compare energy values across varying methods.

In this exercise, locating of transition states is done via making a guess of the transition state and optimising the structure with the TS Berny method; or by inputting the reactant and product structures and finding the transition state by studying the structure where the highest energy level was reached between the two inputs. Two classes of pericyclic reactions would be explored in this exercise, namely the Cope Rearrangement and Diels Alder Cycloaddition.

== The Cope Rearrangement ==
===Background===
[[File:Mtn113 Cope scheme.JPG]]<br>
The Cope Rearrangement reaction has been studied as a classic example of a [3,3]-sigmatropic rearrangement, which falls under a class of pericyclic reactions. Involving 4π electrons, under the Woodward-Hoffmann rules, the rearrangement occurs with a conrotatory displacement of terminal atomic orbitals. With the rotation around the single C-C bonds in 1,5-hexadiene, there are many possible conformations of the molecule and it is possible that the transition state goes through a Boat or a Chair conformation. It is largely agreed that this rearrangement proceeds via a concerted mechanism through the chair conformation preferentially due to its lower energy and this claim shall be elucidated through this computational experiment.

A 1,5-hexadiene was first drawn as the anti- conformation, which is expected to be the most stable conformation as the most sterically demanding groups are anti-periplanar to each other with a dihedral angle of 180. Another conformation was drawn with a 60 degree change in the dihedral angle, which was the synclinal (gauche) conformation. Both conformations are drawn and subsequently optimized using HF (3-21G) and DFT (B3LYP/6-31G*).

=== Conformational Analysis ===
==== 1,5-hexadiene (Anti1 Conformation) ====
The anti- conformation was symmetrized and was found to possess a C<sub>2</sub> point group symmetry (anti1). The energy was found to correspond to the reference value of -231.69260 hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti1
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2</sub>
|style= align=center style;|[[File:Mtn113 anti1.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI1.LOG| Anti1 log]]
|}

==== <br> 1,5-hexadiene (Gauche3 Conformation) ====

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Gauche3
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT GAUCHE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>1</sub>
|style= align=center style;|[[File:Mtn113 gauche3.jpeg|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 GAUCHE.LOG| Gauche3 log]]
|}

A gauche conformation was then drawn by varying the dihedral angle by 60 degrees. It was expected that the gauche conformation would possess a higher energy than the anti conformation due to steric repulsion due to closer proximity of alkene groups and the groups' electron density. The molecule was symmetrized to C<sub>1</sub> point group symmetry. After optimisation, it was found that this gauche3 conformation has the lowest energy of -231.69266 Hartree units. This experimental result proved to be contrary to the hypothesis, which only considered steric repulsions. This suggests that electronic reasons predominate over steric reasons in this case.

[[File:Mtn113_Gauche3_MO.JPG|thumb|centre|500x500px|Figure 1. Visualisation of HOMO ]]

As observed from the highest occupied molecular orbitals above, the pi electron clouds of the alkene groups are seen to overlap, leading to stabilising secondary orbital interactions. This overlap will lead to an overall lowering of the energies of the molecular orbitals for the pi electrons.

==== <br>1,5-hexadiene (Anti2 Conformation) ====
Another conformation was drawn to obtain the anti2 conformer. In order to reach this conformer quickly, the molecule was symmetrised and made to adopt the C<sub>i</sub> point group symmetry before optimisation was carried out. The energy value obtained was compared and verified with reference value to be -231.69254 Hartree units.

{| class="wikitable"
!Conformation
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Log File
|-
|style= align=center style;|Anti2
|style= align=center style;|HF/3-21G
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_hf.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 REACT ANTI2.LOG| Anti2 HF log]]
|-
|style= align=center style;|Anti2
|style= align=center style;|DFT (B3LYP)/
6-31G*
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|[[File:Mtn113_Anti2_dft.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[Media:Mtn113 DFT ANTI2.LOG| Anti2 DFT log]]
|}

===== <br> Comparing Level of Theory =====
The difference in computational methods is shown in this example by running optimisation of anti2 via both HF/3-21G and DFT/B3LYP/6-31G*. The stark discrepancy is shown in the energy values obtained for each of the methods. While it is meaningless to compare the quantified values, this result obtained is in accordance with what was predicted, where the less accurate method, HF, would be expected to overestimate the energy as compared to DFT. <br>
HF: -231.69254 Hartree units <br>
DFT: -234.61171 Hartree units
[[File:Mtn113 Anti2 labelled.JPG|thumb|400x400px|Figure 2. Labelled anti2 conformer ]]

{| class="wikitable"

!rowspan="2" style="text-align: center;" |Level of Theory
!rowspan="2" style="text-align: center;" |Point Group
!colspan="1" style="text-align: center;" |Dihedral Angle °
!colspan="3" style="text-align: center;" |Bond Lengths Å

|-
| style="text-align: center;" | C12-C1-C4-C7
| style="text-align: center;" | Terminal C13-C12
| style="text-align: center;" | C12-C1
| style="text-align: center;" | Central C1-C4
|-
|style= align=center style;|HF/3-21G
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|-180.00000
|style= align=center style;|1.32
|style= align=center style;|1.51
|style= align=center style;|1.55
|-
|style= align=center style;|DFT (B3LYP)/6-31G*
|style= align=center style;|C<sub>i</sub>
|style= align=center style;|180.00000
|style= align=center style;|1.33
|style= align=center style;|1.50
|style= align=center style;|1.55

|}

The geometrical analysis was carried out by comparing bond distances (rounded off to 2 decimal places to minimise errors in program) between adjacent carbon atoms and the dihedral angles. While there are slight differences, the discrepancies are not significant. This suggests that for simple molecules, computing with less precise basis sets can yield reasonable results at a lower computational cost and at a much faster rate.

===== <br>Frequency Analysis =====
An Opt+Freq calculation was run on the anti2 conformer to ensure that the lowest energy conformation was obtained in each method. This was done by checking the vibration frequencies from the conformation and making sure there were no imaginary frequencies present (negative values). This would suggest that a minimum point was reached in the optimisation and not an inflection point. This is because the second derivative of the energy gives a force constant k, that is included in the calculation of vibrational frequencies in the form of the root of the force constant. Thus, a negative second derivative obtained would give a negative k value, and thus the root will be imaginary. The Infrared Spectrum is also recorded and compared with reference spectrum.

{| class="wikitable"
!IR Spectrum
!Frequency calculation
|-
|style= align=center style;|[[File:Mtn113 Ir anti2.JPG|centre]]
|style= align=center style;|[[File:Mtn113 Freq anti2.JPG|thumb|centre|283x283px]]
|-
|}

All 42 vibrational frequencies were positive, showing that a minimum point has indeed been reached in the optimisation.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn 113 Results anti2.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 REACT FREQ DFT ANTI2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 REACT FREQ DFT ANTI2.LOG|DFT_Freq_Log]]
|-
|}

{| class="wikitable"
!Thermochemical data
|-
|style= align=center style;|[[File:Mtn113 Thermochem anti2.JPG|centre]]
|-
|}
Thermochemical data was obtained from the log file.
The first energy value obtained, which is the sum of electronic potential energy and zero point energy, was calculated at a temperature of 0 K. The rest of the energy values were calculated at 298.15 K and 1 atm. As the bonds in 1,5-hexadiene are modelled by a quantum harmonic oscillator, we can infer that at 0 K, the zero point energy would be measuring the vibrational energy that is present in the molecule at 0 K, because translational and rotational energies would be absent at 0 K.
The second energy value calculated at 298.15 K would include all modes of energy present: electronic, rotational, translational and vibrational modes.
The third energy value includes thermal enthalpy which is incorporated in the form of an additional term RT in H = E + RT (where R is the gas constant and T is temperature). At 0 K, RT = 0 and therefore H = E.
The last energy value takes into account the free energy of the system with the inclusion of the entropy term H = G + TS. As the calculations are conducted at a constant pressure, any increase in temperature will lead to an increase in enthalpy H correspondingly.

=== <br> Chair/ Boat Transition State Optimisation ===

In this part of the tutorial, the Chair and Boat transition states would be optimised in a variety of ways, all done at HF/3-21G level of theory. The chair transition states would be optimised using TS Berny and by freezing coordinates while the boat transition states would be optimised using QST2 and QST3 methods.

==== <br> Optimisation via TS Berny ====
The molecule 1,5-hexadiene was redrawn as two separate 3 carbon allylic fragments. They were optimised at HF/3-21G basis set and then combined and positioned such that they resemble a chair transition state and the terminal carbons were 2.2 Å apart. In this method, the structure drawn and positioned must be roughly accurate to the actual transition state if not the optimisation will not be successful. A Gaussian optimisation was set up using TS Berny and force constants were chosen to be calculated once. An additional keyword "Opt=noeigen" was added to ensure the program does not crash in the event that more than one imaginary frequency was located. As explained above, an imaginary frequency observed means that the second derivative of the energy measured is negative, which shows the curvature of the potential energy serface and implies that the energy of the structure is at a local maximum, ie a transition state with maximum potential energy has been reached.

{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;| [[File:Mtn 113 chk Ts berny results.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS OPT BERNY.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS OPT BERNY.LOG| Chair TS Berny log]]
|-
|}

{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state
|-
|style= align=center style;| -817.92
|style= align=center style;| [[File:Ts berny freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair ts berny.gif|500x500px|Figure 3. Visualisation of imaginary frequency]]
|-
|}
An imaginary frequency was observed to be at -817.92cm<sup>-1</sup>, confirming the transition state has been reached.

==== <br> Optimisation via frozen coordinates====
In this alternative pathway, the distance between terminal carbons are kept constant (or 'frozen') at 2.2 Å using the Redundant Coordinates editor, and the rest of the molecule was then optimised to a minimum with no force constants calculated. This might be a better way to conduct an optimisation since it does not rely as much on the way the molecule was drawn and positioned as the previous method with TS Berny. Once again, an imaginary frequency was obtained at -817.85cm<sup>-1</sup>, showing that a transition state had been obtained.
{| class="wikitable"
!Results Summary
!Optimised structure
!Log File
|-
|style= align=center style;|[[File:Mtn113 results Freeze coordinate.JPG|thumb|centre|283x283px]]
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CHAIR TS REDUNDANT COORDINATE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[Media:Mtn113 CHAIR TS REDUNDANT COORDINATE.LOG| Frozen coordinate log]]
|-
|}

{| class="wikitable"
!Imaginary frequency cm<sup>-1</sup>
!Frequency calculation
!Transition state

|-
|style= align=center style;| -817.85
|style= align=center style;| [[File:Mtn113 Freeze coordinate freq.JPG|thumb|centre|283x283px]]
|style= align=center style;|[[File:Mtn113 Chair freeze coordinate.gif]]
|-
|}

====Comparison of methods====
In comparison of the above two methods, it can be seen that the energy values obtained are very close (-231.61932244 vs -231.61932225); hence there is no difference in using either method in leading to the same chair transition state. When the geometries of the resulting molecules were compared, it can be seen that the frozen coordinates method seem to have a more symmetrical molecule as the intermolecular distances are closer to each other than those in TS berny. However, this does not seem to have a significant effect on the resulting transition state.

[[File:Mtn113 Molecule chair.JPG|thumb|centre|500x500px|Figure 3. Labelled chair TS]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! style="text-align: center;" | Atoms
! style="text-align: center;" | TS Berny/(Å)
! style="text-align: center;" | Frozen Coordinate/(Å)
|-
| style="text-align: center;" | C3-C14
| style="text-align: center;" | 2.02036
| style="text-align: center;" | 2.02042
|-
| style="text-align: center;" | C6-C11
| style="text-align: center;" | 2.02067
| style="text-align: center;" | 2.02052
|}

==== <br> Optimisation via QST2/QST3 ====
From the Synchronous Transit-Guided Quasi-Newton (STQN) method, an option QST2 was utilised in this optimisation for the boat conformation. In this QST2 method, unlike other previous methods, it does not require a guess for the transition state. Instead, two molecule specifications, one each for the reactant and product, are required as inputs for this optimisation. The first time the optimisation was ran, the program crashed as the reactant and product conformers did not resemble the actual conformers. Thus, some adjustments were made to the dihedral angle for the central four carbon atoms to 0 degrees and the inside angles for the inner three carbons to be 100 degrees. After that adjustment was made, the opt+freq calculations went through successfully and showed an imaginary frequency, which confirmed the presence of a transition state.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised structure
!Log File
|-
|[[File:Mtn113 Qst2 results.JPG|centre|300x300px]]
| style="text-align: center;" |-231.60280200
| style="text-align: center;" |-839.94
| style="text-align: center;" |
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 OPT CHAIR QST2 CHANGESHAPE.mol</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" |[[Media:OPT CHAIR QST2 CHANGESHAPE.LOG| QST2]]
|}

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst2 labelled.JPG|centre|400x400px]]
|[[File:Mtn113 Boat qst2.gif]]
|}

Subsequently, the optimisation was done again with the QST3 method instead, where the difference lies in that, in addition to the molecule specifications of the product and reactants, a guess was also made of the transition state. This guess was taken from the transition state found from the IRC pathway from QST2.

{| class="wikitable"
!Results Summary
!Energy/ Hartree units
!Imaginary Frequency cm<sup>-1</sup>
!Optimised Structure
!Log File
|-
|[[File:Mtn113 Qst3 results.JPG|centre|400x400px]]
| style="text-align: center;" |-231.60280243
| style="text-align: center;" |-839.93
| style="text-align: center;" |
| style="text-align: center;" |[[Media:Mtn113 OPT CHAIR QST3.LOG| QST3]]
|}

The suggested transition state is seen in the third column. From the results above, it can be seen that there is no significant difference between running the optimisation of the boat conformation with QST2 or QST3 as the energy and imaginary frequency values are almost equivalent.

{| class="wikitable"
!Labelled Optimised Boat Molecules
!Transition State Vibration
|-
|[[File:Mtn113 Qst3 labelled.JPG|centre|500x500px]]
|[[File:Mtn113 Boat qst3.gif]]
|}

=====<br> Intrinsic Reaction Coordinate (IRC) =====
However, it is still not known which conformer of 1,5-hexadiene that the transition state leads to yet. An IRC is a minimum energy pathway taken by the molecule from its transition state (a local maximum) down the path of least resistance on both sides towards a local minimum on the potential energy surface. An IRC was run in both directions for 70 steps from the transition state, but it was observed that the IRC would turn out to be symmetrical, hence IRC in only the forward direction could be calculated for future IRCs.

{| class="wikitable"
!IRC
|-
|[[File:Mtn113 Chair irc.JPG|600x600px]]
|-
|[[File:Mtn113 Chair IRC.gif|centre]]
|}

From the graphs, it can be seen that the energy at the transition state where it starts to run is the highest, and falling to a minimum (product) thereafter. From the gradient graph, it can be seen that it starts off from the transition state because it would be a maximum point (with gradient 0), increase to a maximum as it follows the path with the greatest slope, before falling to a local minimum with a gradient tending towards 0. The structure obtained at the 44th step was reoptimised and was found to possess an energy value of -231.69166702 Hartree atomic units, and a point group symmetry of C2. The log file can be found [[Media:Mtn113 CHAIR TS FREEZE COORDINATE FROM IRC.LOG|here]]. By comparison to reference energy values (-231.69167 a.u.), it can be concluded that the conformer obtained is gauche2.

===<br> Comparing level of theory ===
In this last part of the tutorial, the level of theory would be compared on the basis of the resulting structures from each method as well as investigating how close the activation energy values are to the reference values. The boat and chair conformations were reoptimised at a higher level of theory, DFT/B3LYP/6-31G* and a vibrational frequency analysis was run.

====Geometrical Analysis====
The log files for opt+freq at HF/3-21G for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE FREQ.LOG|here]].
The log files for opt+freq at DFT/B3LYP/6-31G* for boat conformation can be found [[Media:Mtn113 OPT CHAIR QST2 CHANGESHAPE DFT FREQ.LOG|here]].

The log file for opt+freq at HF/3-21G for chair conformation can be found [[Media:Mtn113 CHAIR TS OPT BERNY FREQ.LOG|here]].
The log file for opt+freq at DFT/B3LYP/6-31G* for chair conformation can be found [[Media:CHAIR TS OPT BERNY DFT FREQ.LOG|here]].

{| class="wikitable"
!Method
!Chair TS Seperation (Å)
!Boat TS Seperation (Å)
|-
|HF/321G
|2.02
|2.14
|-
|DFT/B3LYP/631G*
|1.97
|2.21
|}

From the geometrical analysis, the distances between the terminal carbons of the allylic fragments were shown to be similar and no significant discrepancies were detected.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466701
| align="center" | -231.461342
| align="center" | -234.556931
| align="center" | -234.414909
| align="center" | -234.408981
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445299
| align="center" | -234.543079
| align="center" | -234.402354
| align="center" | -234.396010
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611712
| align="center" | -234.469213
| align="center" | -234.461865
|}
From the energy values calculated, a constant discrepancy of 3 Hartee units was observed between the values obtained from the HF/3-21G and DFT/B3LYP/6-31G* methods.

{| class="wikitable"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
| width="125" align="center" | '''Activation Energy at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.08
| align="center" | 33.18
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
Activation energies refer to the minimum amount of energy that reactant molecules need to gain before they can reach the transition state energy, hence the difference between the TS energies and the reactants was calculated. This is calculated in kcal which is converted from a.u. by multiplying by 627.503. From the activation energies calculated and in comparison to the experimental values, it can be seen that computations run at a higher level of theory would produce activation energies that are much closer to experimental data. However, this comes at a higher computational cost and longer time required to run the computations. In all, it depends on the objective of the computations - if geometries of the resulting transition state are to be studied, computations can be run at lower levels of theory. However, if energy values are required for comparison and discussion, they have to be run at higher levels of theory to ensure accuracy. It can be concluded that in future computations, the geometries can be optimised first at a lower level of theory, followed by a re-optimisation of the product at a higher level of theory to obtain closer energy values to experimental data.

==<br> Diels Alder Cycloaddition==
===Background===
[[File:Mtn113 Diels alder scheme.JPG]]

In this part of the computation, knowledge gained from the tutorial was applied to the Diels Alder cycloaddition. Cycloadditions belong to a class of pericyclic reactions, and Diels Alder cycloadditions are characterised under [4n+2]π type reactions. These reactions occur between a diene and a dienophile in a concerted fashion, involving the breaking of two pi bonds and the formation of two sigma bonds. Two Diels Alder cycloadditions are investigated in this part of the exercise, namely the reaction between ethene and cis-butadiene and the reaction between maleic anhydride and cyclohexa-1,3-diene. Reactions would proceed when the Highest Occupied Molecular Orbital (HOMO) of the electron rich diene is close in energy to the Lowest Unoccupied Molecular Orbital (LUMO) of the electron poor dienophile to ensure sufficient overlap of the molecular orbitals and ensure a favourable reaction. Since maleic anhydride is an electron poor dienophile and cyclohexa-1,3-diene is more electron rich than cis-butadiene, the molecular orbitals are expected to be closer in energy and hence larger electronic interactions.

For this section, calculations are first calculated at the Semi-empirical level of theory to produce estimated structures that best agree with empirical data. To be more specific, the AM1 method employed here uses a Neglect of Differential Diatomic Overlap (NDDO) integral approximation which follows a set of parameters and is less complicated as compared to the Hartree-Fock method used in previous sections. Hence, for complex organic molecules, it makes sense to use the semi-empirical method to "guess" a transition state structure at a lower computing cost and time before using a higher theory level to compute it. It was found however that the AM1 method does not work as well for inorganic molecules, and a method with more parameters such as PM6 might have to be used instead.

===<br> Optimisation of ethene and cis-butadiene===
{| class="wikitable"
!Molecule
!Level of Theory
!Optimised Structure
!Point Group
!Results Summary
!Energy/ Hartree units
!Log File
|-
|style= align=center style;|Ethene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|D<sub>2h</sub>
|style= align=center style;|[[File:Mtn113 Ethene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.02619028
|style= align=center style;|[[Media:Mtn113 ETHENE.LOG| Ethene Log]]
|-
|style= align=center style;|Cis-butadiene
|style= align=center style;|Semi-empirical/AM1
|style= align=center style;|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 CIS BUTADIENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|C<sub>2v</sub>
|style= align=center style;|[[File:Mtn113 Butadiene results.JPG|thumb|centre|300x300px]]
|style= align=center style;|0.04879719
|style= align=center style;|[[Media:Mtn113 CIS BUTADIENE.LOG| Butadiene Log]]
|}

As mentioned above, the structures of ethene and cis-butadiene were drawn on Gaussview, cleaned and optimised separately using the Semi-empirical/AM1 method. The dihedral angle for cis-butadiene was changed to 0 degrees to ensure planarity of the molecule. The molecular orbitals of ethene and cis-butadiene were visualised to note the symmetries of the HOMO and LUMO. It can be seen from the diagrams below that for butadiene the HOMO is antisymmetric with respect to the plane of symmetry perpendicular to the central C-C bond. The LUMO on the other hand is symmetrical with respect to the same plane. Ethene, on the other hand, has a symmetric HOMO and antisymmetric LUMO.

{| class="wikitable"
! style=align=center style; |
! style=align=center style; | HOMO
! style=align=center style; | LUMO
|-
| Ethene
| [[File:Mtn113 Ethene HOMO.JPG|300px|thumb|left|Symmetric]]
| [[File:Mtn113 Ethene LUMO.JPG|300px|thumb|left|Antisymmetric]]
|-
| Butadiene
| [[File:Mtn113 Butadiene HOMO.JPG|300px|thumb|left|Antisymmetric]]
| [[File:Mtn113 Butadiene LUMO.JPG|300px|thumb|left|Symmetric]]
|}

===<br> Obtaining the Transition State via TS Berny ===
Once the reactants have been optimised, they were combined with an intermolecular distance of 2.20 Å. The transition state structure for this reaction was obtained by running an optimisation with TS Berny under semi-empirical/AM1 level of theory. After the computation was successful, it was reoptimised at a higher level of theory DFT/B3LYP/6-31G*. The results obtained from both the levels of theory are summarised as below. There were large differences in the imaginary frequencies obtained, as well as energy values of the transition states. However, IRC shows reaction pathways to be similar and lead to the desired products.
{| class="wikitable"
!Level of Theory
!Optimised Structure
!Results Summary
!Imaginary Frequency cm<sup>-1</sup>
!Log File
|-
|style= align=center style;|Semi-empirical/AM1
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny am1 results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-956.23
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW.LOG|AM1 TS Log]]
|-
|style= align=center style;|DFT/B3LYP/6-31G*
|
<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style= align=center style;|[[File:Mtn113 Tsberny dft results.JPG|thumb|centre|300x300px]]
|style= align=center style;|-526.70
|style= align=center style;|[[Media:Mtn113 ETHENE CISBUTADIENE TS BERNY NEW DFT.LOG|DFT TS Log]]
|}

{| class="wikitable"
!Level of Theory
!Reaction coordinate and animation
|-
|rowspan="2"|Semi-empirical/AM1
|style= align=center style;|[[File:Mtn113 Tsberny IRC am1.JPG|600x600px]]
|-
|style= align=center style;|[[File:Tsberny am1 IRC1 new.gif]]
|-
|rowspan="2"|DFT/B3LYP/6-31G*
|style= align=center style;|[[File:Mtn113 Tsberny IRC dft.JPG|600x600px]]
|-
|style= align=center style;|[[File:Mtn113 Ethene&cisbutadiene IRC1 new DFT.gif]]
|}

===Molecular Orbitals Analysis===
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Tsberny am1 HOMO.JPG|450px|thumb|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Tsberny am1 LUMO.JPG|400px|thumb|Symmetric]]

|}
The HOMO of the transition state is observed to be antisymmetric with respect to the plane that is perpendicular to the central C-C bond, while the LUMO is symmetric to the plane.
From Frontier Molecular Orbitals analysis, it can be inferred that only molecular orbitals of the same symmetry can combine. Thus, the antisymmetric HOMO is made from the HOMO of cis-butadiene and the LUMO of ethene. The symmetric LUMO is built from the LUMO of cis-butadiene and HOMO of ethene.

===Vibrational Frequency Analysis===
{| class="wikitable"
!Imaginary Frequency cm<sup>-1</sup>
!Visualisation
!Description
|-
|style|-956.23
|[[File:Mtn113 Ethene&cisbutadiene img.gif]]
|Synchronous

|-
|147.24
|[[File:Mtn113 Ethene&cisbutadiene real.gif]]
|Asynchronous
|}
The negative frequency suggests a transition state has been obtained and this is confirmed with the visualisation of the vibration. It can be seen that the terminal carbons that are involved in the C-C sigma bond formation move towards each other in a synchronous fashion. This also implies that the formation of the two new bonds occur in a concerted mechanism.

In contrast to this, the visualisation of the first positive frequency is also shown. This shows the fragments rotating about a perpendicular rotational axis and the fragments do not get close enough to form new bonds, hence this vibration does not lead to a successful reaction.

===Geometrical Analysis===
[[File:Mtn113 Ethene&cisbutadiene labelled.JPG]]

{| class="wikitable"
!
!C9-C12 /Å
!C12-C5 /Å
!C5-C3 /Å
!C3-C1 /Å
! Literature C=C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! Literature C-C value<ref>Pauling, L. (1931). THE NATURE OF THE CHEMICAL BOND. II. THE ONE-ELECTRON BOND AND THE THREE-ELECTRON BOND. J. Am. Chem. Soc., 53(9), pp.3225-3237.
doi:http://pubs.acs.org/doi/abs/10.1021/ja01360a004</ref>/ Å
! C Van Der Waals Radius <ref>Bondi, A. (1964). van der Waals Volumes and Radii. The Journal of Physical Chemistry, 68(3), pp.441-451.doi:http://pubs.acs.org/doi/abs/10.1021/j100785a001</ref>/ Å
|-
|'''Semi Empirical/AM1'''
|1.383
|2.119
| 1.382
| 1.397
|rowspan="2"|1.334
|rowspan="2"|1.544
|rowspan="2"|1.70

|-
|'''DFT/B3LYP/6-31G*'''
|1.386
|2.272
| 1.383
| 1.407
|}

From the above bond distances (rounded off to 3 decimal places to compare with literature values), there is no significant discrepancies besides the fact that at a higher level of theory, the transition state is observed when the two fragments are further apart (2.27211 vs 2.11915 Å). Both these values are smaller than the sum of van der Waals radii, which show some form of interaction in both cases. However, this discrepancy suggests that the actual through space interactions between the terminal carbons are stronger than expected and the highest energy transition state is reached at a distance that is further apart.

The rest of the bond distances are in agreement between the two levels of theory and shows clearly that the distances between C5-C3 and C3-C1 are almost equivalent, suggesting breaking of pi bond over C5-C3 and forming of the pi bond over C3-C1. The bond distances are found to be between the literature values of C-C and C=C bonds, suggesting partial double bond character in both bonds.

==Investigating Regioselectivity of Diels Alder Cycloaddition==
For more complicated organic molecules undergoing Diels Alder Cycloaddition, concepts of regioselectivity have to be taken into consideration. Two products can be formed - endo product which is the kinetically favoured product and the exo product which is the thermodynamically favoured product.

This is investigated by computing the reaction between maleic anhydride and cyclohexa-1,3-diene. As mentioned above, this reaction is expected to be more facile due to the dienophile (maleic anhydride) being more electron poor and diene (cyclohexa-1,3-diene) more electron rich. The transition states of both cases would be studied and activation energies and MOs would be analysed.

===Optimisation of Transition States===
The product of the Diels Alder cycloaddition was drawn and then had some bonds removed, and the resulting structure was then optimised under Semi-empirical AM1 to a TS (Berny). The fragments were positioned with an interfragment distance of 2.2 Å. The point group was found to be C<sub>1</sub>. The results from the optimisation, MOs and IRC are shown below.

====Endo Product====
The molecule was drawn such that maleic anhydride overlaps with the diene component on cyclohexa-1,3-diene to allow secondary orbital interactions.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 ENDO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05150480
| -806.39
|[[File:Mtn113 Endo img.gif]]
|[[Media:Mtn113 ENDO NEW TSBERNY.LOG|Endo Log]]
|}
An IRC was run to confirm visually that interactions between the fragments lead to the desired product through the transition state.<br>
[[File:Mtn113 Endo irc.gif]]<br>
As shown below, both HOMO and LUMO of endo transition states are antisymmetric with respect to the plane that is perpendicular to the central C-C bond. Secondary orbital overlap between maleic anhydride and diene can also be seen in the LUMO+2 MO between C=O π* and C=C π* orbitals, which does not directly contribute to the bonding in the molecule.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Endo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Endo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO +2
|[[File:Mtn113 Endo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

====Exo Product====
The molecule was drawn such that maleic anhydride does not overlap with the diene component and hence no secondary orbital interaction was possible.

{| class="wikitable"
!Optimised Structure
!Energy/ Hartrees
!Imaginary Frequency/ cm<sup>-1</sup>
!Imaginary Frequency Animation
!Log File
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>Mtn113 EXO NEW TSBERNY.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| -0.05041981
| -812.36
|[[File:Mtn113 Exo img.gif]]
|[[Media:Mtn113 EXO NEW TSBERNY.LOG|Exo Log]]
|}

An IRC was run to check visually that the reaction proceeds towards the desired product through the transition state.<br>
[[File:Mtn113 Exo irc.gif]]<br>
As with endo product MOs, both HOMO and LUMO of the exo product are antisymmetric with respect to the plane of symmetry. However, from LUMO+2 MO, an absence of secondary orbital interactions was observed due to the C=O π* and C=C π* orbitals in opposite orientations and hence too far apart to overlap. This lack of extra stability accounts for the higher energy value obtained for the transition state for the exo product.
{| class="wikitable"
!MO
!Visualisation
|-
|HOMO
|[[File:Mtn113 Exo product HOMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO
|[[File:Mtn113 Exo product LUMO.JPG|400px|thumb|left|Antisymmetric]]

|-
|LUMO+2
|[[File:Mtn113 Exo product LUMO 2.JPG|400px|thumb|left|Antisymmetric]]

|}

===Comparison of endo and exo product===
====Geometrical Analysis====
{|class="wikitable"
|rowspan="2"|'''Endo'''
|rowspan="2"| [[File:Mtn113 Endo labelled.JPG|200px]]
|'''C4-C1/Å'''
|'''C1-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C4/Å '''

|-
|1.41
|2.16
|1.39
|1.40
|1.39
|2.16
|-
|rowspan="2"|'''Exo'''
|rowspan="2"|[[File:Mtn113 Exo labelled.JPG|200px]]
|'''C1-C4/Å'''
|'''C4-C3/Å'''
|'''C3-C7/Å '''
|'''C7-C5/Å '''
|'''C5-C2/Å '''
|'''C2-C1/Å '''
|-
|1.41
|2.17
|1.39
|1.40
|1.40
|2.17
|}
Bond distances were measured (and rounded off to 2 decimal places to minimise errors in program) and it was observed that the distances between the atoms forming the new sigma bonds were shorter in the endo product than the exo product. This suggests that there is more significant steric repulsion between the side groups of the two fragments leading to the exo product than that present in endo product. Thus, this proves the steric explanation for the higher energy level of exo transition state as well as further preventing any secondary orbital overlap in the exo pathway. In each molecule, the distances between the atoms that are about to form new sigma bonds were observed to be around the same value, suggesting a synchronous fashion in bond formation.
The rest of the bonds were found to exist between C-C and C=C bond lengths which suggest partial double bond character which is to be expected in transition states.

====Activation Energy Comparison====
{| class="wikitable"
|-
| ''' '''
!colspan="3"|'''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (0 K)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (298.15 K)'''
|-
| '''Maleic anhydride'''
| align="center" | -0.12182415
| align="center" | -0.063349
| align="center" | -0.058195
|-
| '''Cyclohexa-1,3-diene'''
| align="center" | 0.02795792
| align="center" | 0.152538
| align="center" | 0.157033
|-
| '''Endo TS'''
| align="center" | -0.05150480
| align="center" | 0.133494
| align="center" | 0.143683
|-
| '''Exo TS'''
| align="center" | -0.05041981
| align="center" | 0.134881
| align="center" | 0.144882
|}

Upon inspection of the transition states, it can be seen that the endo transition state is lower in energy and thus it will be the kinetically favoured pathway, as the activation energy is lower to form the endo product than the exo product. The exo transition state possesses a higher energy probably due to some steric hindrance but mainly due to electronic reasons - absence of stabilising secondary orbital overlap.

{| class="wikitable"
|+ Summary of activation energies (in kcal mol<sup>-1</sup>)
| ''' '''
| width="125" align="center" | '''Semi-empirical/AM1'''
| width="125" align="center" | '''Semi-empirical/AM1'''
|-
| ''' '''
| width="125" align="center" | '''Activation Energy at 0 K '''
| width="125" align="center" | '''Activation Energy at 298.15 K'''
|-
| '''ΔE (Endo)'''
| align="center" | 27.8015204
| align="center" |28.1403720
|-
| '''ΔE (Exo)'''
| align="center" |28.6718671
| align="center" |28.8927481
|}

From the activation energies in kcal/mol, it is confirmed that the activation energy values leading to the endo product are lower than the values leading to the exo product, although the differences are not very significant. Perhaps running the reactions under a higher level of theory would elucidate more significant and quantifiable data, however this was not conducted due to insufficient time.

==Conclusion==
This computational experiment

==References==