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MRD:NW716

2018-05-18T16:04:52Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 Å and distance BC is around 0.75 Å. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets close to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to a high translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding particles, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/> The above reaction results do agree with Polanyi's empirical rules. However, sometimes a small change in one of the momenta might change the reaction from reactive to unreactive. Hence, Polanyi's empirical rules are a guideline to describe general required excitation for exothermic and endothermic reactions, but unable to predict whether the reaction is reactive for a certain initial conditions.

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T16:03:45Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 Å and distance BC is around 0.75 Å. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets close to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to a high translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/> The above reaction results do agree with Polanyi's empirical rules. However, sometimes a small change in one of the momenta might change the reaction from reactive to unreactive. Hence, Polanyi's empirical rules are a guideline to describe general required excitation for exothermic and endothermic reactions, but unable to predict whether the reaction is reactive for a certain initial conditions.

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:58:59Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 Å and distance BC is around 0.75 Å. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets close to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to a high translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:56:04Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 Å and distance BC is around 0.75 Å. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets close to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:51:16Z

Nw716: /* Transition State Approximation */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 Å and distance BC is around 0.75 Å. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:44:38Z

Nw716: /* Transition State Theory */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated using the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:43:37Z

Nw716: /* Reaction with Different Momenta */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta Combinations===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The translational kinetic energy in the system is dominantly in the AB distance coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC distance coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB distance coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:38:48Z

Nw716: /* Minimum Energy Pathway and Dynamics Calculations */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods. MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly to 0.74 as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When the parameters of the final position are used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position of this reaction is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:36:30Z

Nw716: /* Minimum Energy Pathway and Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics Calculations===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:35:56Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time plot, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:34:23Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the two reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:32:11Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r_1}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:28:35Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]] || [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time || Figure 3 - Plot of Energy VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal. This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:25:36Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction (eg. F atom in F + H2 reaction). However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:WXY0119

2018-05-18T15:17:33Z

Nw716: /* Reaction Dynamics */

= Molecular Reaction Dynamics Report =

== EXERCISE 1ː H + H2 SYSTEM ==

[[File:HandH2wxy0119.png]]

In the H + H2 system, the distance between the first two H atoms is r1 and the distance (bond length) between the two H atoms in the H2 molecule is r2.

=== Dynamics from the transition state region ===

{{fontcolor|red|Question 1 ː What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.}}

{| class="wikitable" border=1
|[[File:Q1TStransitionwxy0119.png|390px]]
|[[File:Q1TS2wxy0119.png|390px]]
|[[File:Q1TS3wxy0119.png|390px]]
|-
|Figure.1 Surface plot of reaction trajectory
|Figure.2 Transition state surface plot
|Figure.3 Transition state surface plot (different angle)
|}

In the reaction trajectory surface plot (Fig.1), AB is the distance r1 and BC is the distance r2. The gradient of the potential energy with regard to r1 and r2 are ∂V(r1)/∂r1 and ∂V(r2)/∂r2 respectively. The second derivatives of the potential energy with regard to r1 and r2 are ∂2V(r1)/∂r12 and ∂2V(r2)/∂r22 respectively.

At the two minimum structures which are at the two ends of the reaction pathway, ∂V(r1)/∂r1 = 0 and ∂V(r2)/∂r2 = 0 respectively. Since the two minimum structures are at two local minimum point , the second derivatives r2 are ∂2V(r1)/∂r12 and ∂2V(r2)/∂r22 are both greater than zero.

On the other hand, the components at the transition state structure (Fig.2) are different from that at minimum points. The gradient ∂V(r1)/∂r1 and ∂V(r2)/∂r2 are both equal to zero at the transition structure. However, r2 are ∂2V(r1)/∂r12 >0 and ∂2V(r2)/∂r22 <0. The transition state point is actually a saddle point which is observed more clearly in Fig.3. The Transition state point is thus distinguished from the minimum points as the potential energy surface curves inwards and downwards at the minimum points.

=== Locating the Transition State ===

{{fontcolor|red|Question 2ː Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.}}

{| class="wikitable" border=1
|[[File:TScontourwxy0119.png|390px]]
|[[File:TSsurfaceplwxy0119.png|390px]]
|[[File:TSestimatewxy0119.png|390px]]
|-
|Figure.4 Transition state contour plot
|Figure.5 Transition state surface plot
|Figure.6 Internuclear distance against time plot
|}

The best estimate of the transition state position is rts = 0.9078 Å.

When r1 = r2 = rts and momenta are set to zero, a graph of intermolecular distance against time is plotted. (Fig.6) In this graph, AB and BC lines are the same and overlap, the two lines on the graph are flat and horizontal, indicating that the atomic distances are the same at the position and with no momenta the state is at equilibrium, validating the position is the transition state position. The transition state is shown as a cross on the counter plot (Fig.4) and a dot on the surface plot (Fig.5).

=== Calculating and comparing the reaction path and trajectory ===

{{fontcolor1|red|Question 3ː Comment on how the ''mep'' and the trajectory you just calculated differ.}}

{| class="wikitable" border=1
|[[File:Mep1wxy0119.png|390px]]
|[[File:Mep2wxy0119.png|390px]]
|[[File:Mep3wxy0119.png|390px]]
|-
|Figure.7 mep calculation contour plot
|Figure.8 mep calculation surface plot
|Figure.9 mep calculation internuclear distance against time plot
|-
|[[File:Dynamic1wxy0119.png|390px]]
|[[File:Dynamic2wxy0119.png|390px]]
|[[File:Dynamic3wxy0119.png|390px]]
|-
|Figure.10 Dynamics calculation contour plot
|Figure.11 Dynamics calculation surface plot
|Figure.12 Dynamics calculation internuclear distance against time plot
|}
{| class="wikitable" border=1
|[[File:dynamics4wxy0119.png|600px]]
|[[File:Mep5wxy0119.png|600px]]
|-
|Figure.13 Dynamics calculation internuclear momenta against time plot
|Figure.14 mep calculation internuclear momenta against time plotsurface plot
|}

The mep (minimum energy path) trajectory is a smooth line on contour and surface plot (Fig.7 Fig.8) while the trajectory under dynamics calculation is oscillating along the pathway. (Fig.10 Fig.11) The mep trajectory takes more steps and longer time than the dynamics trajectory as shown in internuclear distance against time plots. (Fig. 9 Fig.12) The momentum is zero all the time under mep calculation (Fig.13) but under dynamics calculation, the momenta lines are oscillating against time.

The differences arise from mep and dynamics calculations can be explained by the the natures of the two calculations. Under mep calculation, the atoms are in extremely slow motion and the momentum is zero for each step, i.e. the atoms are "stop" after every step. The pathway under mep calculation is formed by connecting all the minimum energy points of each step and appears as a smooth line. On the other hand, atoms are in continuous motion and the momentum is accumulated after each step, atoms are then oscillating on the potential energy surface along the pathway. The steps are set to 5000 in the mep calculation but are only 500 in the dynamics calculatoin. Since the steps in mep are small, more time is needed to complete the same length on the trajectories.

=== Reactive and unreactive trajectories ===

{{fontcolor1|red|Question 4ː Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.}}

For the initial positions '''r1''' = 0.74 and '''r2''' = 2.0, run trajectories with the following momenta p1 and p2 combinations:
{| class="wikitable" border=1
| Condition
| p1
| p2
| Total Energy/ kcal mol -1
| Reactivity
|-
|1.
| -1.25
| -2.5
| -99.018
| reactive
|-
|2.
| -1.5
| -2.0
| -100.456
| unreactive
|-
|3.
| -1.5
| -2.5
| -98.956
| reactive
|-
|4.
| -2.5
| -5.0
| -84.956
| unreactive
|-
|5.
| -2.5
| -5.2
| -83.416
| reactive
|}

{| class="wikitable" border=1
| Condition
| Surface Plot
| Contour Plot
| Description
|-
|1.
| [[File:condition1wxy0119.png|350px]]
| [[File:firstwxy0119.png|350px]]
| r2 (BC) decreases when HC approaches bonded HA and HB. The energy processed by the system is sufficient to overcome the activation barrier and crosses the transition state structure to break the HA and HB bond and form a new HB and HC bond. The new bond oscillate as r1 (AB) increases.
|-
|2.
| [[File:condition2wxy0119.png|350px]]
| [[File:secondwxy0119.png|350px]]
| HC approaches HB but the energy is insufficient to reach the transition state point, HC then moves further away from the bonded H2 molecule and no new bond is formed. The oscillation along r2 is a result of an increase in momentum p1.
|-
|3.
| [[File:condition3wxy0119.png|350px]]
| [[File:thirdwxy0119.png|350px]]
| Similarly to condition 1 but with a more negative p1, HA - HB bond oscillates more when r2 decreases. The reaction proceeds and crosses the transition state point to form the products.
|-
|4.
| [[File:condition4wxy0119.png|350px]]
| [[File:fourthwxy0119.png|350px]]
| Both p1 and p2 are more negative than the previous conditions. The HA - HB bond oscillates vigorously with larger amplitudes. The barrier recrossing happens after the system crosses the transition state region but eventually reverts back to the reactants.
|-
|5.
| [[File:condition5wxy0119.png|350px]]
| [[File:fifthwxy0119.png|350px]]
| With a slight more negative value of p1 as compared to that in condition 4, the system now processes appropriate energy to proceed to forming products. The vigorous oscillations are resulting from high momenta and barrier recrossing occurs.
|}

=== Transition State Theory ===

{{fontcolor|red|Question 5ː State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?}}

The main assumptions of Transition State Theory (TST) are<ref name="TS" />ː

1. The behaviours of the atomic nuclei follow the classical mechanics rules, i.e. the transition structure is formed as long as atoms or molecules collide with enough energy.

2. The intermediates have a long enough lifetime for the energies to be Boltzmann distributed preceding the next step.

3. The lowest energy saddle point on the potential energy surface is passed over in the reaction system.

The TST does not take into account of the quantum tunnelling effect which allows reactions to happen even when the energy carried by the system is lower than the activation energy barrier, especially for reactions with low energy barriers. Nor does the theory consider the intermediates with short lifetimes. When the energies are not fully distributed, the momentum of the reaction trajectory from the reactants to the intermediate can have effects on product selectivity. Moreover, the theory fails at high temperature because higher vibrational energy modes can be populated for a molecule at high temperature. The molecules can have complex motion and their collisions may result in a higher energy transition structure. The experimental results may thus deviate from that predicted by the TST. For example in condition 4, the kinetic energy of the reactants is much higher than the activation energy required. However, the momenta between atoms are very high and higher vibrational modes are populated that result in higher energy transition structures formed and the system does not pass though the lowest transition saddle point as shown in the diagram. Barrier recrossing occurs but the products are not formed as opposed to the predictions by the TST.

== EXERCISE 2: F - H - H SYSTEM ==

=== PES inspection ===

=== Reaction Energetics ===

{{fontcolor1|red|Question 6ː Classify the F + H2 and H + HF reactions according to their energetics (endothermic or exothermic). How does this relate to the bond strength of the chemical species involved?}}

{| class="wikitable" border=1
|[[File:Q6FH2wxy0119.png|600px]]
|[[File:Q6HFHwxy0119.png|600px]]
|-
|Figure.15 F + H2 position on surface plot
|Figure.16 HF and H position on surface plot
|}

The surface plots of the potential energy surface of H + HF → F + H2 reaction is shown (Fig.15 Fig.16). Distance AB is the distance between H and F atoms and distance BC is the distance between two H atoms. The two minimum points shown in the graph are positions of F + H2 (Fig.15) and H + HF (Fig.16) respectively. If forward reaction is H + HF → F + H2, then the backward reaction is F + H2 → H + HF. As clearly shown in the diagram F + H2 are at a minimum point of higher potential energy than that of H + HF. Thus, by comparing the potential energy of reactants and products, the signs of changes in enthalpy (ΔH) and the energetics are determined. H + HF → F + H2 is endothermic and F + H2 → H + HF is exothermic.

The energetics of the two reactions are in accordance to the predictions by comparing bond strengths of chemical species. The bond energy of H-F bond (565 kJ/mol) is higher than the bond energy of H-H bond (432 kJ/mol). Thus, energy released by forming the H-H bond is not enough to compensate for energy required for breaking the H-F bond and the reaction H + HF → F + H2 is endothermic. On the contrary, the reaction F + H2 → H + HF is exothermic.

=== Transition State Approximation ===

{{fontcolor1|red|Question 7ː Locate the approximate position of the transition state.}}

{| class="wikitable" border=1
|[[File:Q7TStransitionwxy0119.png|390px]]
|[[File:Q7TS2wxy0119.png|390px]]
|[[File:Q7TS3wxy0119.png|390px]]
|-
|Figure.17 Transition state surface plot
|Figure.18 Transition state contour plot
|Figure.19 Internuclear distance against time plot (at TS point)
|}

The transition state (TS) is shown as a black dot on the potential energy surface plot (Fig.17) and a red cross on the contour plot. (Fig,18) The position of the TS is at the point where the distance between F and H (rFH) is 1.810Å and the distance between two H atoms (rHH) is 0.746Å. At the transition state position, the distance between F and H, between two H atoms are constant with no momentum as shown in the internuclear distance against time plot as horizontal and flat lines, validating the position is the transition state point (Fig.19)

=== Activation Energies ===

{{fontcolor1|red|Question 8ː Report the activation energy for both reactions.}}

{| class="wikitable" border=1
|[[File:Q8Ea2wxy0119.png|600px]]
|[[File:Q8Ea1wxy0119.png|600px]]
|-
|Figure.21 Energy against time plot (HF + H)
|Figure.20 Energy against time plot (F + H2)
|}

By performing MEP calculation with slight increase and decrease of the rFH to 1.820Å to perform F + H2 → H + HF reaction and to 1.800Å to perform H + HF → F + H2 reaction, the activation energies (Ea) of both reactions can be calculated from the differences in their potential energies (Fig.20 Fig.21)ː

F + H2 → H + HFː Ea = -103.751 - (-133.624) = +29.873 kcal mol-1

H + HF → F + H2ː Ea = -103.751 - (-103.972) = + 0.221 kcal mol-1

=== Reaction Dynamics ===

{{fontcolor1|red|Question 9ː In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?}}

Reactionː F + H2 → H + HF

Initial condition setː rHF = 2Å rHH = 0.74Å pHF = -0.5 pHH = 0.1

{| class="wikitable" border=1
|[[File:Q9plot1wxy0119.png|390px]]
|[[File:Q9plot2wxy0119.png|390px]]
|[[File:Q9plot3wxy0119.png|390px]]
|-
|Figure.23 Contour plot
|Figure.24 Surface plot
|Figure.25 Internuclear momentum against time plot
|-
|[[File:Q9ani1wxy0119.png|390px]]
|[[File:Q9ani2wxy0119.png|390px]]
|-
|Figure.26 Animation figure (at the start of the reaction)
|Figure.27 Animation figure (at the end of the reaction)
|}

As shown in the plots and animation figures, the H-H bond in the reactants is broken and H-F bond in the products is formed. There are small momenta between both H and F and H-H bond that provide the initial kinetic energy of the reactants and the initial vibrational energy of the H-H bond in the reactants. The initial energy is required to break the H-H bond and reduce the distance of H and F for bond forming and potential energy is released when H-F bond is formed. Since energy is conserved, the reaction energy released is converted to the vibrational energy of the H-F bond and the translational kinetic energies of the products. This is shown in the internuclear momentum against time graph as the initial vibrations of B-C (H-H bond) becomes flat as two H atoms are separated in the product and the A-B (H-F bond) formed in the product vibrates with a large amplitude. (Fig.25) The contour plot and the surface plot also clearly show there is a significant increase in the amplitude of the oscillation of H-F bond which is an indication of the large vibrational energy of H-F bond. (Fig.23 Fig.24)

The predictions can be proved experimentally. The vibrational energy of H-F bond can be determined by measuring frequency and the intensity of the absorption band of the H-F bond with infrared spectroscopy. The conversion to the kinetic energies of the products can be observed by measuring the temperature of the reaction as the kinetic energy can be converted to thermal energy as products collide with solvent molecules. The temperature is expected to increase as this is an exothermic reaction.

hahahahahahahahahahahahahaha

=== Energy Distribution and Reactivity ===

{{fontcolor|red|Question 10ː Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.}}

for investigation, 1st set condition pFH = -0.5 pHH = -3 unreactive; 2nd pHH = -2.45 reactive; 3rd pHH = -1.1 unreactive; 4th pHH = -0.9 reactive; 5th pHH = 1.1 unreactive ; 6th pHH = 1.9 reactive ; 7th pHH = 2.4 to 3 unreactive
; pHH = 2.1 no reaction

The cases studied are an illustration of Polanyi's empirical rules.

For H and HF

failed conditionsː 091, 2, 0.05, -20

initial condition of H HF systemː
HF distance = 0.91 HH distance = 2 pHF = 0.8 pHH = -7.5

== Rreferences ==

<references>
<ref name="TS">Eyring, H. (1935). "The Activated Complex in Chemical Reactions". J. Chem. Phys. 3 (2): 107–115.</ref>
</references>

MRD:NW716

2018-05-18T15:15:41Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two bonded atoms (eg. HF in H + HF reaction) in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:11:42Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:11:29Z

Nw716: /* Minimum Energy Pathway and Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:11:20Z

Nw716: /* Reaction with Different Momenta */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:11:11Z

Nw716: /* Transition State Theory */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:10:49Z

Nw716: /* Reactive Conditions */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

'''Qn: In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?'''

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:10:14Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==
'''Qn: Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.'''

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T15:08:02Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above ( pFH more effectively affects the reactivity of the reaction.) and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:54:30Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:54:10Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH can affect the reactivity of the reaction even with just a small change at constant pFH but there is no trend observed for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:52:15Z

Nw716: /* Reactivity with Variation of pHH */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:51:41Z

Nw716: /* Reactive Conditions */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the initial oscillation of the trajectory is extremely small and the final HF molecule contains great vibrational energy, as seen from the large amplitude of the oscillation once the reaction completes. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation".<ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, more effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:46:56Z

Nw716: /* Activation Energy */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. These distances are chosen so that the trajectory is towards the reactants and the change in potential energy is therefore the activation energy. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:44:02Z

Nw716: /* H + HF */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction as this is the reverse reaction. The transition state in this endothermic reaction should resemble more closely to the products, which are H2 and a separate F atom, based on Hammond postulate. Hence, the F-H distance is 1.8107 Å and H-H distance is 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:41:01Z

Nw716: /* F + H2 */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances stay constant, shown in Fig 23.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:40:36Z

Nw716: /* F + H2 */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction should compose of longer H-H and H-F bond distances. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants, H2 and a separate F atom. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:38:38Z

Nw716: /* Potential Energy Surface */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surrounding. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:37:40Z

Nw716: /* Potential Energy Surface */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need energy input. These results reflect that H-F bond is stronger than H-H bond, which agrees with a higher H-F bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:33:27Z

Nw716: /* Potential Energy Surface */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2, reactants on the left || Figure 21 - Surface Plot of H + HF, reactants on the left
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:28:12Z

Nw716: /* Transition State Theory */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy ('''lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme''') would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:27:42Z

Nw716: /* Transition State Theory */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a higher momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy of -99.305 - (-103.869) = 4.564 kcal mol-1, calculated from the programme) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:24:17Z

Nw716: /* Potential Energy Surface */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the reactants to be 2.3 Å away from each other, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:23:23Z

Nw716: /* Transition State Theory */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn 5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:22:46Z

Nw716: /* Reaction 3: p1 = -1.5, p2 = -2.5 */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system. The amplitude of the oscillation is greater after the reaction. This indicates that the release of vibrational energy from the reaction.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:19:42Z

Nw716: /* Reaction 1: p1 = -1.25, p2 = -2.5 */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because part of the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:17:25Z

Nw716: /* Reaction with Different Momenta */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn 4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:12:48Z

Nw716: /* Minimum Energy Pathway and Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn 3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T14:12:38Z

Nw716: /* Transition State and Minima */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn 1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn 2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:58:10Z

Nw716: /* Reactive Conditions */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Contour Plot'''
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:57:22Z

Nw716: /* H + HF */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distances for this reaction should be the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å as this is just the reverse reaction of the above reaction. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:56:09Z

Nw716: /* Activation Energy */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distance for this reaction are the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:55:31Z

Nw716: /* Activation Energy */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distance for this reaction are the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.869 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.869) = 0.117 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:51:56Z

Nw716: /* Activation Energy */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distance for this reaction are the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.624 kcal mol-1 and that in H + HF reaction is -103.790 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-103.790) = 0.038 kcal mol-1

H + HF : Ea = -103.752 - (-133.624) = 29.872 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:45:30Z

Nw716: /* Activation Energy */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distance for this reaction are the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.807 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-133.807) = 30.055 kcal mol-1

H + HF : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-026.png|500px]] || [[File:NW716-MRD-027.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

<references>
<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>

MRD:NW716

2018-05-18T13:42:20Z

Nw716: /* Conclusion from Reaction Dynamics */

=H + H2 System=

==Potential Energy Surface==

===Transition State and Minima===

'''Qn1: What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-PES.png|600px|left]]
|-
| Figure 1 - Transition State and Minimum <ref name="TS"/>
|}

The gradients of the potential energy surface at a minimum and at a transition structure are both 0. At a local minimum, the distance between two bonded atoms is a constant, hence, one of the component, eg. <math>{ \partial V\over \partial r}</math>, is zero and <math>{ \partial V^2\over \partial^2 r_1}</math> > 0. The other component, eg. <math>{ \partial V^2\over \partial^2 r_2}</math>, is increasing as r2 decreases, i.e. the single atom approaches the diatomic molecule. Transition state linking the two minima represents a maximum along the direction of the reaction coordinate, but along all other directions, it is a minimum. At the transition state, which is the saddle point of the graph, both <math>{ \partial V\over \partial q_1}</math> and <math>{ \partial V\over \partial q_2}</math> are zero. However, for the reaction coordinates, one of the second derivatives is negative and the other is positive. Hence, if the point is a minimum in one direction and does not decrease in the orthogonal direction, it is a minimum. However, if the point is a minimum in one direction but a maximum in the orthogonal direction, it is a saddle point, which corresponds to the transition state.

'''Qn2: Report your best estimate of the transition state position ('''rts''') and explain your reasoning illustrating it with a “Internuclear Distances vs Time” plot for a relevant trajectory.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-001.png|500px|left]]
|-
| Figure 2 - Plot of Internuclear Distance VS Time
|}

rts is estimated to be 0.9077 Å. Since r1 = r2 and there is no momentum, the distances of A-B and B-C are the same and should not vary. Hence, only two lines are observed in the Internuclear Distances vs Time plot as two lines overlap and the lines are perfectly horizontal.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-002.png|500px|left]]
|-
| Figure 3 - Plot of Energy VS Time
|}

This can also be confirmed using the Energy vs Time graph, Fig 3. At the transition state position, the kinetic energy is zero and potential energy should be a constant.

==Reaction Trajectories==

===Minimum Energy Pathway and Dynamics===

'''Qn3: Comment on how the ''mep'' and the trajectory you just calculated differ.'''

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-003.png|500px|left]] || [[File:NW716-MRD-004.png|500px|left]]
|-
| Figure 4 - Minimum Energy Path (MEP) Calculation || Figure 5 - Dynamics Calculation
|}

The MEP trajectory is a smooth line without oscillation. The trajectory calculated using Dynamics shows the vibration of HB-Hc bond. This difference is owing to the different calculation methods, MEP corresponds to an infinitely slow motion. Each step is extremely small and the velocity is set to zero after each step. Hence, the motion of the molecule at each step is independent of the previous step and is a trajectory connecting all of the lowest energy points for each step. Therefore, MEP is a smooth and non-oscillatory line. On the contrary, Dynamics calculation corresponds to a continuous motion and every step is dependent upon the previous step (motion of atoms is inertial). Hence, the molecule possesses a velocity to climb up the potential energy surface and results in the oscillation. Moreover, to obtain the length of MEP shown in Fig 4 above, the Steps set for calculation is 50000. However, the Steps set for Dynamics calculation is only 500. Since each step for MEP is extremely small, more steps are required to obtain the same length of the trajectory with the same amount of time.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-005.png|500px|left]] || [[File:NW716-MRD-006.png|500px|left]]
|-
| Figure 6 - Dynamics Calculation: Plot of Internuclear Momenta VS Time || Figure 7 - Dynamics Calculation: Plot of Internuclear Distance VS Time
|}

Using the Dynamics calculation, at large time, r1 increases linearly as HB and HC separates and r2 decreases slightly (to 0.74) as HA-HB bond forms, seen in Fig 7. At large time, p1 increases to 2.5 and p2 increases to 1.25 (on average). These values indicate that once the reactants surpass the transition state, even slightly, the reaction proceeds readily.

When final position is used as the initial conditions and the signs of the momentum are reversed, the reaction will proceed and end at the transition state, as seen in Fig 8 and 9 below. The final position is the red cross at the transition state. This is confirmed by the plot of Internuclear Distance VS Time as r1 is equal to r2 at the end of the reaction.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-007.png|500px|left]] || [[File:NW716-MRD-008.png|500px|left]]
|-
| Figure 8 - Contour Plot || Figure 9 - Plot of Internuclear Distance VS Time
|}

===Reaction with Different Momenta===

'''Qn4: Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.'''

{| class="wikitable" border="1"
|+ Table 1 - Trajectories with Various Momenta Combination
|-
| '''Reaction No.''' || '''p1''' || '''p2''' || '''Total Energy / kcal mol-1''' ||''' Kinetic Energy / kcal mol-1''' || '''Reactivity'''
|-
| 1 || - 1.25 || - 2.5 || - 99.018 || + 4.687 || Reactive
|-
| 2 || - 1.5 || - 2.0 || - 100.456 || + 3.250 || Unreactive
|-
| 3 || - 1.5 || - 2.5 || - 98.956 || + 4.750 || Reactive
|-
| 4 || - 2.5 || - 5.0 || - 84.956 || + 18.750 || Unreactive
|-
| 5 || - 2.5 || - 5.2 || - 83.416 || + 20.290 || Reactive
|}

====Reaction 1: p1 = -1.25, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-009.png|500px|left]] || [[File:NW716-MRD-014.png|500px|left]] || HA moves towards HB with HB-HC distance increases slightly. The reactants then reach the transition state structure and the reaction proceeds with HA-HB bond formation and HC moves away. The initial reaction path is smooth without oscillation as p1 is much smaller than p2. The kinetic energy in the system is dominantly in the AB coordinate. Once the reaction completes, the oscillation in AB coordinate indicates the vibration of HA-HB bond. This is because the kinetic energy and convert to vibrational energy of the bond.
|-
| Figure 10 - Surface Plot || Figure 11 - Contour Plot || Description
|}

====Reaction 2: p1 = -1.5, p2 = -2.0 ====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-010.png|500px|left]] || [[File:NW716-MRD-015.png|500px|left]] || An increase in p1 with a decrease in p2 from the conditions in Reaction 1 result in the initial system with relatively more kinetic energy in BC coordinate which can be observed from the oscillation of HB-HC bond. However, the atoms do not possess sufficient kinetic energy to climb up the energy surface and reach the transition state, this is owing to the decrease in p2, which leads to a decrease in translational energy in AB coordinate. Hence, HA moves away from HB with HB-HC bond retains. No reaction takes place.
|-
| Figure 12 - Surface Plot || Figure 13 - Contour Plot || Description
|}

====Reaction 3: p1 = -1.5, p2 = -2.5====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-011.png|500px|left]] || [[File:NW716-MRD-016.png|500px|left]] || An increase of p2 from Reaction 2 enables the atoms to climb up the energy surface to allow the reaction to proceed. The vibration of HB-HC bond is smaller compared to Reaction 2 as p2 increases which cancels out some of the kinetic energy in the BC coordinate. However, the vibration is greater compared to Reaction 1 with an increase of p1 only. This illustrates that the relative values of p1 and p2 affect the initial shape of the trajectory by altering the kinetic energy in the two coordinates of the system.
|-
| Figure 14 - Surface Plot || Figure 15 - Contour Plot || Description
|}

====Reaction 4: p1 = -2.5, p2 = -5.0====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-012.png|500px|left]] || [[File:NW716-MRD-017.png|500px|left]] || There is a huge increase in both p1 and p2. The system does reach the transition state region but barrier recrossing takes place. The initial trajectory is smooth without oscillation as p1 is significantly lower than p2 (half of p2, similar to Reaction 1). Hence, the initial kinetic energy is mainly along the AB coordinate. With a high energy content, after collision, HB-HC bond vibrates more rigorously as shown with a large amplitude of oscillation owing to energy transfer from kinetic to vibrational.

|-
| Figure 16 - Surface Plot || Figure 17 - Contour Plot || Description
|}

====Reaction 5: p1 = -2.5, p2 = -5.2====

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-013.png|500px|left]] || [[File:NW716-MRD-018.png|500px|left]] || p2 increases slightly with p1 unchanged from conditions for Reaction 4. Barrier recrossing takes place but the reaction proceeds in this case. This means a small change in p2 will change the reactivity of the reaction. Initial trajectory is smooth without oscillation as p1 is relatively small comparing to p2. HA-HB bond formed vibrates more rigorously owing to energy transfer from kinetic to vibrational.
|-
| Figure 18 - Surface Plot || Figure 19 - Contour Plot || Description
|}

===Transition State Theory===

'''Qn5: State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?'''

Transition State Theory (TST) assumes that the motion of the atoms obeys classic mechanics and with higher initial momenta, trajectories starting with the same position would be more likely to
form the transition state and react as the system contains more kinetic energy to overcome the activation energy barrier. Hence, the reaction should be more likely to proceed with a highest momenta combination as the system would possess a higher kinetic energy. However, from the experimental values above, it can be concluded that the system with a lower kinetic energy (lower than the activation energy) would not be reactive, for instance, a kinetic energy of 3.25 kcal mol-1 in Reaction 2. However, a system would be unreactive despite a very high kinetic energy which is higher than the activation energy, for instance, Reaction 4 above. This is because as the momenta increases, the atoms populate higher energy vibrational modes. Their motion becomes more complex and collisions might cause the transition state to deviate from the lowest energy saddle point. Hence, even with sufficient energy, a reaction does not occur. This means TST will fail at high temperatures when more reactant molecules occupy higher energy vibrational modes.

TST also assumes that transitions from a reactant state to a product state occur without barrier recrossings.<ref name='BR'/> However, Reaction 4 and 5 above show barrier recrossing which does not agree with TST.

Moreover, since the TST is based on the assumption that nuclei behave according to classic mechanics, it does not describe the quantum effects, specifically, quantum tunnelling. There is always a possibility that the reactants will react even if they do not collide to form the transition state and cross the activation barrier. Quantum tunnelling is significant if the activation barrier is low as the tunnelling probability increases with decreasing barrier height.

TST also assumes that the transition state is long-lived so that the reaction continues. It fails if the transition state is short-lived and could affect product selectivity.<ref name="TST"/>
Therefore, whether a reaction will take place does not solely depend on the initial momenta of the reactants, which illustrates the kinetic energy the system possessed. The Transition State Theory is not accurate to predict the reaction rate with a low activation barrier, a short-lived transition state and at high temperatures.

=F - H - H System=
==Potential Energy Surface==

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-019.png|500px]] || [[File:NW716-MRD-020.png|500px]]
|-
| Figure 20 - Surface Plot of F + H2 || Figure 21 - Surface Plot of H + HF
|}

Setting the separate atom to be 2.3 Å away from the molecule, B-C distance to be H-H and H-F bond length respectively and both momenta to be 0, the above two surface plots are obtained. From Fig 20, by observing the two minima, the reactants, H2 and F are at a higher minimum comparing to the product. Hence, the products are lower in potential energy which means that the reaction is exothermic.

Similarly, from Fig 21, the products are higher in potential energy and the reaction between H and HF is endothermic.

Formation of H-F bond and breaking of H-H bond releases energy to the surroundings. Formation of H-H bond and breaking of H-F bond need external energy input.These results reflect that H-F bond is stronger than H-H bond, which agrees with the bond energies. Bond energy of H-F is 565 kJ mol-1 and that of H-H is 432 kJ mol-1.

==Transition State Approximation==

===F + H2===
Transition state of F + H2 reaction composes of longer H-H and H-F bond lengths. Based on observation of Fig 20, distance AB is around 1.8 and distance BC is around 0.75. This agrees with the Hammond postulate as the transition state of an exothermic reaction should resemble more closely to the reactants. Hence, the H-H bond is only stretched a bit from the bond length of 0.74 Å. Using trial and error, F-H distance is estimated to be 1.8107 Å and H-H distance is 0.7450 Å for the transition state structure. From the contour plot, Fig 22, the reactants do not move along the PES and inter-atomic distances shown in Fig 23 stay constant.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-021.png|500px]] || [[File:NW716-MRD-022.png|500px]]
|-
| Figure 22 - Surface Plot of Transition State || Figure 23 - Internuclear Distance VS Time of Transition State
|}

===H + HF===

The H-H and H-F distance for this reaction are the same as the above reaction with F-H distance as 1.8107 Å and H-H distance as 0.7450 Å. The transition state is illustrated with Fig 24 an 25 below.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-023.png|500px]] || [[File:NW716-MRD-024.png|500px]]
|-
| Figure 24 - Surface Plot of Transition State || Figure 25 - Internuclear Distance VS Time of Transition State
|}

===Activation Energy===

A MEP calculation from a structure neighbouring the transition state, H-F length used in F + H2 reaction 1.8207 Å and is 1.8007 Å in H + HF reaction. The potential energy of the transition state was determined to be -103.752 kcal mol-1, which is the same for both reactions.

The potential energy of reactants in F + H2 reaction is -133.807 kcal mol-1 and that in H + HF reaction is -103.886 kcal mol-1. Hence the activation energies for the two reactions are:

F + H2 : Ea = -103.752 - (-133.807) = 30.055 kcal mol-1

H + HF : Ea = -103.752 - (-103.886) = 0.134 kcal mol-1

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-027.png|500px]] || [[File:NW716-MRD-026.png|500px]]
|-
| Figure 26 - Energy VS Time (F + H2) || Figure 27 - Energy VS Time (H + HF)
|}

==Reaction Dynamics==

===F + H2===

====Reactive Conditions====

{| class="wikitable" border="1"
|-
| Reaction No. || p1 || p2 || Contour Plot
|-
| 1 || -1.5 || 0 || [[File:NW716-MRD-028.png|500px|thumb|Figure 28]]
|-
| 2 || -1.5 || -0.25 || [[File:NW716-MRD-029.png|500px|thumb|Figure 29]]
|-
| 3 || -2.0 || -1.25 || [[File:NW716-MRD-030.png|500px|thumb|Figure 30]]
|-
| 4 || -2.2 || -1.25 || [[File:NW716-MRD-031.png|500px|thumb|Figure 31]]
|-
| 5 || -2.2 || -1.5 || [[File:NW716-MRD-032.png|500px|thumb|Figure 32]]
|}

From all of the five conditions above, it can be observed that the final HF molecule contains great vibrational energy, as seen from the oscillation of the reaction paths. Reaction energy released as F approaches H2 becomes the motion in HF, the product vibration, whereas energy released as HF separates from H becomes the motion along the BC distance coordinate, the product translation. The HF vibrational energy can be determined using IR and analyse the frequency of the vibrational band. Translational energy of H atom can be confirmed by measuring the scatter of the products. The energy distribution can be measured by recording the infrared chemiluminescence of the reaction under "arrested relaxation". <ref name="PER"/>

The above five conditions illustrate that a higher p1 (pFH) is always required for the reaction to be reactive, especially in Reaction 1 when p2 (pHH) is zero. Hence, one can assume that pFH, which corresponds to a high translational energy of the reactants, effectively affects the reactivity of the reaction.

====Reactivity with Variation of pHH====

When rHH = 0.74 and momentum pFH = -0.5, values of pHH in the range from -3 to 3 were used for calculation. As pHH gets closer to -3 or 3, barrier recrossing takes place and the reaction conditions render the reaction unreactive. Starting from pHH = -3, when pHH is greater than approximately -2.5, the reaction is reactive. When pHH reaches 1, or extremely close to 1, the reaction is unreactive and remains unreactive until pHH is above 1.5 and below approximately 2.4, although the reaction may be unreactive with pHH in between 1.5 and 2.4 (eg. 1.7 and 2.1). This shows that change in pHH substantially affect the reactivity of the reaction even with just a small change at constant pFH and there is no trend for whether the reaction is reactive or not.

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-033.png|500px]]
|-
| Figure 33 - Contour Plot with pFH = -0.8 and pHH = 0.1
|}

However, increasing pFH slightly to -0.8, and reduce the overall energy of the system by reducing pHH to 0.1, the reaction is now reactive as shown in Fig 34. This means that a higher overall energy of the system does not mean that the reaction will be reactive, but a high pFH, which corresponds to the translational energy, is required.

The reaction between F and H2 is highly exothermic with an extremely low activation barrier. The transition state is in the entrance valley, hence, an early barrier. Using the assumption from the five reactive conditions above and observations from various pHH values, pFH does have a significant effect on the reactivity of the reaction.

===H + HF===

{| class="wikitable" border="1"
|-
| [[File:NW716-MRD-034.png|500px]] || [[File:NW716-MRD-035.png|500px]]
|-
| Figure 34 - Contour Plot with pHH = -50.0 and pHF = 0.01 || Figure 35 - Contour Plot with pHH = -8.0 and pHF = 0.5
|}

pHH = -8.0 is still very large. This complies with the high activation energy of this reaction as the products formed are thermodynamically less stable than the reactants. It can be concluded that an increase in pHF, which corresponds to a high vibrational energy in H-F coordinate, is necessary for the reaction to take place. pHH, which corresponds to the translational energy in the H-H coordinate that is always very large, does not contribute too much to affect the reactivity of the reaction.

The transition state is present in the exit valley, hence, a late barrier for this endothermic reaction. The reactivity of endothermic reactions is thus dominated by the vibrational energy of the system from the results above.

==Conclusion from Reaction Dynamics==

For substantial exothermic reactions, such as F + H2, the transition state is located in the entrance valley, corresponds to an early barrier of the PES. For substantially endothermic reactions, such as H + HF, the transition state is in the exit valley, corresponds to a late barrier. The favoured degree of freedom for barrier crossing in exothermic reactions would be translation. This means that the momentum of the approaching atom and the atom which it is going to collide with has a greater impact on the rate of the reaction. However, reagent vibration, which is related to the momentum of the two atoms in the colliding molecule, would be most effective in enabling endothermic reactions to take place.<ref name="PER"/>

=References=

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<ref name="TS">E. G. Lewars, Computational Chemistry (Springer Netherlands, Dordrecht, 2011; http://link.springer.com/10.1007/978-90-481-3862-3).</ref>
<ref name="TST">D. Dyson, Advanced Chemical Kinetics, World Technologies, 2012.</ref>
<ref name="PER">J. C. Polanyi, Some Concepts in Reaction Dynamics. Accounts of Chemical Research. 5, 161–168 (1972).</ref>
<ref name="BR">T. Komatsuzaki, M. Nagaoka, Study on “regularity” of barrier recrossing motion. Journal of Chemical Physics. 105, 10838–10848 (1996).</ref>
</references>