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Mrd:gm1616

2018-05-11T15:44:15Z

Gm1616: /* EXERCISE 2: F - H - H system */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs. This is because there is a much larger energy in all of the systems than the activation energy required for hydrogen, so the hydrogen has a large KE as it approaches the F atom and so it will have a larger energy when it is repelled from the Florine, meaning there is more energy for the hydrogens' bond to reform.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below and the lower PE H + HF products were formed with no intersystem crossing taking place. There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation. The hydrogen's bond never reforms since the H atoms are never close enough to overcome the stronger H-F attraction and form a new bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:43:01Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs. This is because there is a much larger energy in all of the systems than the activation energy required for hydrogen, so the hydrogen has a large KE as it approaches the F atom and so it will have a larger energy when it is repelled from the Florine, meaning there is more energy for the hydrogens' bond to reform.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below and the lowest potential energy products were formed with there is no intersystem crossing and the system moves straight towards the products of H-F + H. There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation. The hydrogen's bond never reforms since the H atoms are never close enough to overcome the stronger H-F attraction and form a new bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:41:31Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs. This is because there is a much larger energy in all of the systems than the activation energy required for hydrogen, so the hydrogen has a large KE as it approaches the F atom and so it will have a larger energy when it is repelled from the Florine, meaning there is more energy for the hydrogens' bond to reform.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation. The hydrogen's bond never reforms since the H atoms are never close enough to overcome the stronger H-F attraction and form a new bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:37:50Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs. This is because there is a much larger energy in all of the systems than the activation energy required for hydrogen, so the hydrogen has a large KE as it approaches the F atom and so it will have a larger energy when it is repelled from the Florine, meaning there is more energy for the hydrogens' bond to reform.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:27:07Z

Gm1616: /* Reactive and unreactive trajectories */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates. There is greater overall energy in the system and the vibrations are much larger in the final molecules.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:25:50Z

Gm1616: /* Calculating the reaction path */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions and neither reactants nor products predominated as the reaction proceeded. The final momenta was of the same magnitude but with a positive sign.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:23:14Z

Gm1616: /* Trajectories from r1 = r2: locating the transition state */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating in the well only of the minimum energy pathway and has no gradient. This transition state can be seen below with the inter-nuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not tending towards products or reactants.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T15:21:36Z

Gm1616: /* Dynamics from the transition state region */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point on the minimum energy pathway. A reaction must pass through this point to form products.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the point along the minimum energy pathway will quickly fall and the newly favoured molecule will form. At a minima, the potential energy will increase with any change in reaction conditions or change in the reactant/product molecule's geometry.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:43:04Z

Gm1616: /* EXERCISE 2: F - H - H system */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:41:30Z

Gm1616: /* Reactive and unreactive trajectories */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Activated complex theory of bimolecular reactions. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:40:38Z

Gm1616: /* Reactive and unreactive trajectories */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI: 10.1021/ed051p709</ref>. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:38:53Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules <ref>Theoretical Study of the Validity of the Polanyi Rules for the Late-Barrier Cl + CHD3 Reaction. DOI:10.1021/jz301649w</ref> state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:36:39Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> so the H-F bond is stronger by 133 kJ/mol<ref>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:35:05Z

Gm1616:

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol<ref>1</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

<references/>

Mrd:gm1616

2018-05-11T11:34:24Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol<ref>1</ref> and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

Mrd:gm1616

2018-05-11T11:33:30Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

Mrd:gm1616

2018-05-11T11:33:13Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.
[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.
[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

Mrd:gm1616

2018-05-11T11:32:46Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa. This means that vibrational energy is better at promoting an endothermic reaction and translational energy is better at promoting an exothermic reaction.

Mrd:gm1616

2018-05-11T11:27:33Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol. The more momenta in the system, the more intersystem crossing occurs.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, there is no intersystem crossing and the system moves straight towards the products of H-F + H.There is reduced total energy so it is expected that the molecules will have less potential energy and so there will be less repulsion when the H-F atoms reach the minimum distance of separation, hence the hydrogen will not reform the bond with the other hydrogen since they never reach a large enough distance to overcome the stronger H-F attraction and replace it with the H-H bond.

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45. Any conditions where p-(HH) decreases and p(FH) increases by the same amount from 5.45 also resulted in a reaction.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T11:16:19Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

As the momentum of the H-H is varied between 3 and -3 for a reaction between F and H2, it can be seen that the momenta has an effect on whether the reaction is possible and how much inter-system crossing will occur. At the lowest momentum of -3, the system remains closer to the transition state and the system passes through the transition state many times. As the momenta is increased to a less negative value, the side of the reactants (H2) is favoured. However, by pHH=-1.3, the system once again remains close to the transition state and involves lots of crossing, before the product (HF + H) is favoured. AS momenta increases, there is no intersystem crossing and the products instantly form. This pattern repeats once more with pHH=1.5 favouring reactants, pHH=2.6 staying near the transition state and pHH=3 favouring products. All of the systems have a maximum potential energy of approximately -105 kJ/mol.

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T11:04:42Z

Gm1616: /* Reaction dynamics */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
In the reaction trajectory between F and H2, the reaction barrier is crossed three times as the F + H2 reforms and then H-F + H remains as the final product. This can be seen in the inter-nuclear momenta vs time graph below which shows the system being crossed when t=0.33, 0.42 and 0.57, of which the crossing at 0.33 and 0.57 correspond to the H-F bond forming and the H-F inter-nuclear momenta at these points increases to a displacement of 9 as the molecule vibrates periodically. Initially, the system has mainly kinetic energy as the H-H bond approaches the stationary F atom until the first H atom is closer to F than the equilibrium bond distance, at which point the nuclei from the atoms repel each other and most of the kinetic energy is converted to potential energy as the H atom is now stationary. However, as the H-F bond lengthens under the nuclei's repulsion, the second H atom is still approaching the first H atom until they reach the minimum separation distance and begin to repel. This repulsion moves the first H atom back towards the F atom and so the H-F bond reforms and the second H atom moves away from the H-F molecule as the potential energy it gained in the repulsion at the minimum separation distance is converted into kinetic energy. Throughout the reaction, the total energy never changes but its composition of kinetic energy and potential energy does change.

This could be proven experimentally by using heavier Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:26:28Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

The difference in these activation energies proves that when H2 + F is the forward reaction, the reaction is exothermic since H2 + F has a higher energy than the H-F + H products, and so ΔHreaction = - 20.5 kJ/mol.

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:21:37Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.
Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:21:19Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal. The transition state is closer in energy to H2 + F and so, according to Hammond's Postulate, this suggests the transition state has a geometry which is similar to these reactants too.

Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

The difference between the energy of the transition state and the energy of the reactant molecules gives the activation energy:

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:13:09Z

Gm1616:

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy (kJ/mol) !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal.

Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:11:15Z

Gm1616: /* PES inspection */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 kJ/mol (20.5 kJ/mol kinetic and -119.0 kJ/mol potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 kJ/mol (31.0 kJ/mol kinetic and -109.0 kJ/mol potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the H atom in between the H and F atoms oscillated periodically between the H-F bond and the H-H bond. For this asymmetric system, it is expected that the distance between all 3 atoms is not equal.

Energy of the transition state is -104.5 kJ/mol (12.5 kJ/mol kinetic and -117 kJ/mol potential).

Activation Energy of H2 + F = -98.5 - (-104.5) = 6 kJ/mol

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5 kJ/mol

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-11T10:04:36Z

Gm1616: /* H + H2 system */

==H + H2 system==

===Dynamics from the transition state region===
At the transition state, the PE surface has a gradient of 0 and it is a maximum point. Along the energy minimum, the PE surface has a gradient that is 0 on one distance axis and a increasing or decreasing gradient on the second distance axis.

The minima of the potential energy surface changes value as the collision occurs, and the maximum point of this minimum PE path is the transition state. There is a the largest curvature along the minimum PE path at the TS, so if a small change in the reactions conditions occurs that favours either reactants or products, the PE minima will quickly fall and the newly favoured molecule will form. At a minima, the PE increases with any change in reaction conditions.

===Trajectories from r1 = r2: locating the transition state===

The best estimate for the transition state position is with the distance of AB=BC=0.9212 (4dp) because at this geometry the TS remains oscillating on the ridge only and has no gradient. This transition state can be seen below with the internuclear distance-time graph which shows that the A-B and A-C are oscillating in phase, so A-B must also be oscillating in phase. This means that atoms A and C are vibrating periodically and in opposing directions around central atom B, so the system is not moving towards a reaction.

[[File:Gm_h_h2_internuc_graph.PNG]]

===Calculating the reaction path===

The differences between the MEP and dynamics calculation types on the PE surface plot are that there is no oscillating bond length in the MEP plot. Also, the MEP plot does not reach as large a bond length for the same amount of steps as the dynamic calculation type.

r1(t) = 0.5,
r2(t) = 8.75,
p1(t) = 1.1,
p2(t) = 2.4.

r1(t) = 8.75,
r2(t) = 0.5,
p1(t) = 2.4,
p2(t) = 1.1.

In the calculation where the final distances were set to the initial distances and the momenta were added with a negative sign, it could be seen that the final distances were much closer together, suggesting the system came back to the transition state conditions. The momenta was of the same magnitude but with a positive sign once again.

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1 !! p2 !! Total Energy !! Reactive? !! Potential Energy Surface Plot !! Description of trajectory
|-
| -1.25 || -2.5 || -99.018 || Yes || [[File:Gm_h_h2_momenta_1.PNG]] ||Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -1.5 || -2.0 || -100.456 || No || [[File:Gm_h_h2_momenta_2.PNG]] || The A-B bond oscillates as it vibrates and the B-C bond length decreases as C approaches AB but the transition state is never reached and the molecule remains unchanged and C moves away from AB.
|-
| -1.5 || -2.5 || -98.956 || Yes || [[File:Gm_h_h2_momenta_3.PNG]] || Initially, B-C's bond length oscillates and A-B's bond length decreases up until the transition state at which the reaction occurs and A-B's bond length oscillates and C-B's bond length increases.
|-
| -2.5 || -5.0 || -84.956 || Yes || [[File:Gm_h_h2_momenta_4.PNG]] || The barrier is recrossed so that the C-B bond is formed briefly after the transition state before the A-B bond reforms and the C-B bond distance increases as it move away from the A-B molecule that has an oscillating bond length.
|-
| -2.5 || -5.2 || -83.416 || Yes || [[File:Gm_h_h2_momenta_5.PNG]] || The barrier is recrossed so that the A-B bond is formed briefly after the transition state before the B-C bond reforms and the A-B bond distance increases once again as B-C's bond distance oscillates about a fixed length as it vibrates.
|-
|}

Transition state theory assumes that during a reaction, a quasi-equilibrium will be set up between the reactants and activated complexes in the transition state and the rate of reaction can be determined by the gradients and shape of the molecular potential profile that a moloecule follows as it crosses through the transition state. It assumes only classical behaviour of the molecules. The theory is good at predicting rates for generic reaction trajectories at high temperatures (since classical mechanics deviates less from quantum mechanics), but it assumes that the product state remains unchanged indefinitely and so it cannot be accurate for the barrier crossing trajectories since the product state exists temporarily for those conditions. Also, it does not take into account the concentration/surface area which can affect a reaction rate significantly.

==EXERCISE 2: F - H - H system==

===PES inspection===
The total energy for H2 + F is -98.5 (20.5 kinetic and -119.0 potential) when HH distance = 0.74, HF distance = 2, p(HH)=2 and p(FH)=-0.5.
The total energy for H-F + H is -78.0 (31.0 kinetic and -109.0 potential) when HH distance = 1, HF distance = 1.5, p(HH)=2 and p(FH)=-3.
This shows that the H2 + F is the exothermic process and the reverse reaction is endothermic.
The H-H bond strength is 432 kJ/mol and the H-F bond strength is 565 kJ/mol so the H-F bond is stronger by 133 kJ/mol and hence the reaction which breaks the H-F bond will be unfavourable since the energy required to break the bond is not regained by the H-H bond formation.

The approximate transition state was when the atoms has a bond length of 1.58 for H-F and 0.76 for H-H at which point the structure oscillated periodically between the H-F bond and the H-H bond.

Energy of the transition state is -104.5 (12.5 kinetic and -117 potential).

Activation Energy of H2 + F = -98.5 - (-104.5) = 6

Activation Energy of H-F + H = -78.0 - (-104.5) = 26.5

===Reaction dynamics===
The reaction between F and H2 involves barrier recrossing occurring twice before the final product of H-F and H forms. This is the energetically favoured reaction due to the stronger H-F bond forming and the weaker H-H bond breaking. Initially, the vibrating H-H molecule moves towards the Fluorine atom and then the system passes through the transition state and the H-F bond forms which is stable. This bond attraction between the H and F causes a large increase in momentum as the H moves towards the F atom. However, the H-H bond reforms as the other H atom continues to approach the H-F molecule, and the H-H bond reforms as the two H atoms move closer together anf further away from the F atom due to the vibration. This occurs once again to reform the H-F bond again and the HF molecule vibrates as the H atom moves away. The energy released from the new bond is seen in the momentum from the eject4ed H atom and the small vibrations of the F in the HF molecule.

This could be proven experimentally by using Deuterium to replace one of the hydrogens and then determining whether there is a change in activation energy or rate and hence confirming how the hydrogen is involved in the mechanism.

[[File:Gm_f_h2_momenta_graph.PNG]]

{| class="wikitable" border="1"
! p(H-H) !! Reactive? !! Potential Energy Surface Plot !! p(H-H) !! Reactive? !! Potential Energy Surface Plot
|-
| -3.0 || No || [[File:Gm_f_h2_momenta_-3.PNG]] || -2.8 || No || [[File:Gm_f_h2_momenta_-2.8.PNG]]
|-
| -2.4 || No || [[File:Gm_f_h2_momenta_-2.4.PNG]] || -1.3 || No || [[File:Gm_f_h2_momenta_-1.3.PNG]]
|-
| -1 || Yes || [[File:Gm_f_h2_momenta_-1.PNG]] || 1 || Yes || [[File:Gm_f_h2_momenta_1.PNG]] ||
|-
| 1.5 || No || [[File:Gm_f_h2_momenta_1.5.PNG]] || 2.6 || No || [[File:Gm_f_h2_momenta_2.6.PNG]] ||
|-
| 3.0 || Yes || [[File:Gm_f_h2_momenta_3.PNG]] ||
|-
|}

When the overall energy of the system was reduced with p(HF)=0.8 and p(HH)=-0.1 then the potential energy surface looked like the figure below. In this PE surface, the

[[File:Gm_f_h2_momenta_-0.1.PNG]]

For the inverse momenta method to find a reaction trajectory, the reaction proceeded to the products with the initial conditions of F-H=0.9, H-H=2.4, p(FH)=p-(HH)=5.45.

Polanyi's Empirical Rules state that vibrational energy is better at promoting a late energetic-barrier to reaction in comparison to translational energy which is more efficient at promoting an early energy barrier to reaction. If the barrier is late, than the transition state at that point will resemble the products more closely than the reactants and vice versa.

Mrd:gm1616

2018-05-10T21:24:23Z

Gm1616: /* PES inspection */

Mrd:gm1616

2018-05-10T15:46:42Z

Gm1616: /* Reaction dynamics */

Mrd:gm1616

2018-05-10T15:44:46Z

Gm1616: /* Reaction dynamics */

Mrd:gm1616

2018-05-10T15:36:04Z

Gm1616: /* PES inspection */

Mrd:gm1616

2018-05-10T15:19:35Z

Gm1616: /* PES inspection */

Mrd:gm1616

2018-05-10T15:16:17Z

Gm1616: /* PES inspection */

Mrd:gm1616

2018-05-10T15:12:35Z

Gm1616: /* EXERCISE 2: F - H - H system */

Mrd:gm1616

2018-05-10T15:05:21Z

Gm1616: /* Trajectories from r1 = r2: locating the transition state */

Mrd:gm1616

2018-05-10T14:38:13Z

Gm1616: /* Reaction dynamics */

Mrd:gm1616

2018-05-10T14:37:42Z

Gm1616: /* Reaction dynamics */

Mrd:gm1616

2018-05-10T14:05:38Z

Gm1616: /* Reaction dynamics */

Mrd:gm1616

2018-05-10T14:04:31Z

Gm1616: /* Reaction dynamics */

File:Gm f h2 momenta -0.1.PNG

2018-05-10T14:02:13Z

Gm1616:

Mrd:gm1616

2018-05-10T14:00:04Z

Gm1616: /* Reaction dynamics */

File:Gm f h2 momenta -3.PNG

2018-05-10T13:58:30Z

Gm1616:

File:Gm f h2 momenta 3.PNG

2018-05-10T13:58:21Z

Gm1616:

File:Gm f h2 momenta -2.8.PNG

2018-05-10T13:58:12Z

Gm1616:

File:Gm f h2 momenta 2.6.PNG

2018-05-10T13:58:03Z

Gm1616:

File:Gm f h2 momenta -2.4.PNG

2018-05-10T13:57:54Z

Gm1616:

File:Gm f h2 momenta -1.PNG

2018-05-10T13:57:44Z

Gm1616:

File:Gm f h2 momenta 1.PNG

2018-05-10T13:57:35Z

Gm1616:

File:Gm f h2 momenta 1.5.PNG

2018-05-10T13:57:24Z

Gm1616:

File:Gm f h2 momenta -1.3.PNG

2018-05-10T13:57:15Z

Gm1616: