ChemWiki - User contributions [en]

MRD:jc2916

2018-05-18T16:11:39Z

Jc2916: /* References */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot type !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| In dynamic plot, the momentum between AB is constant at high t, because the A and B atoms are dar part and there is no vibrational energy between them, while BC is bonded and there is interaction and thus fluctuation in the momentum. In MEP plot, because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all time steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in MEP calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

== References ==

1.Zhang Z, Zhou Y, Zhang DH, Czakó G, Bowman JM. Theoretical study of the validity of the polanyi rules for the late-barrier Cl + CHD3 reaction. J Phys Chem Lett. 2012;3(23):3416–9.

2.Masel, R. (1996). Principles of Adsorption and Reactions on Solid Surfaces. New York: Wiley

3.Pineda, J. R.; Schwartz, S. D. (2006). "Protein dynamics and catalysis: The problems of transition state theory and the subtlety of dynamic control". Phil. Trans. R. Soc. B. 361 (1472): 1433–1438. doi:10.1098/rstb.2006.1877. PMC 1647311 Freely accessible. PMID 16873129

MRD:jc2916

2018-05-18T16:11:28Z

Jc2916: /* References */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot type !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| In dynamic plot, the momentum between AB is constant at high t, because the A and B atoms are dar part and there is no vibrational energy between them, while BC is bonded and there is interaction and thus fluctuation in the momentum. In MEP plot, because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all time steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in MEP calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

== References ==

1. Zhang Z, Zhou Y, Zhang DH, Czakó G, Bowman JM. Theoretical study of the validity of the polanyi rules for the late-barrier Cl + CHD3 reaction. J Phys Chem Lett. 2012;3(23):3416–9.

2.Masel, R. (1996). Principles of Adsorption and Reactions on Solid Surfaces. New York: Wiley

3.Pineda, J. R.; Schwartz, S. D. (2006). "Protein dynamics and catalysis: The problems of transition state theory and the subtlety of dynamic control". Phil. Trans. R. Soc. B. 361 (1472): 1433–1438. doi:10.1098/rstb.2006.1877. PMC 1647311 Freely accessible. PMID 16873129

MRD:jc2916

2018-05-18T16:11:05Z

Jc2916: /* References */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot type !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| In dynamic plot, the momentum between AB is constant at high t, because the A and B atoms are dar part and there is no vibrational energy between them, while BC is bonded and there is interaction and thus fluctuation in the momentum. In MEP plot, because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all time steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in MEP calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

== References ==
1. Zhang Z, Zhou Y, Zhang DH, Czakó G, Bowman JM. Theoretical study of the validity of the polanyi rules for the late-barrier Cl + CHD3 reaction. J Phys Chem Lett. 2012;3(23):3416–9.

2.Masel, R. (1996). Principles of Adsorption and Reactions on Solid Surfaces. New York: Wiley

3.Pineda, J. R.; Schwartz, S. D. (2006). "Protein dynamics and catalysis: The problems of transition state theory and the subtlety of dynamic control". Phil. Trans. R. Soc. B. 361 (1472): 1433–1438. doi:10.1098/rstb.2006.1877. PMC 1647311 Freely accessible. PMID 16873129

MRD:jc2916

2018-05-18T16:03:16Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

MRD:jc2916

2018-05-18T15:49:44Z

Jc2916:

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot type !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| Because mep path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all tie steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in mep calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

== References ==

MRD:jc2916

2018-05-18T15:49:04Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot type !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| Because mep path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all tie steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in mep calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:48:14Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the reaction rolls toward the product and the bond between A-B broke, and the bond between B-C was formed. The following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| Because mep path takes the infinite slowly motion, the velocity will be reset to zero at each time step, so the momentum is zero at all tie steps.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in mep calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.
|}

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:44:33Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

{| class="wikitable" border="1"
|+
! Plot !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| In dynamic plot, the bond between A-B broke, and the bond between B-C wad formed, and there is no interaction between AB at high t when the two atoms are to far apart to interact, so the momentum stays constant at high t. Because mep path takes the infinite slowly motion, the velocity will be reset to zero at each time step.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| Because in mep calculation, the velocity will be reset to zero in each time step, so there is no kinetic energy and thus no fluctuation of bond distance in MEP trajectory plot.

|}

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:36:57Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====

{| class="wikitable" border="1"
|+
! Plot !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: dynamic2.png |300px]] || [[File: Mep2jc2916.png |300px]]|| -99.018 .
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| -100.456

|}

[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:35:46Z

Jc2916: /* Distance vs Time */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====

{| class="wikitable" border="1"
|+
! Plot !! Dynamic !! MEP !! Description
|-
| Momentum vs time ||[[File: Plot1jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| Contour plot || [[File: Contour2jc2916.png |300px]] || [[File: Contourjc2916.png |300px]]|| -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.

|}

[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

File:Contour2jc2916.png

2018-05-18T15:35:24Z

Jc2916:

File:Contourjc2916.png

2018-05-18T15:34:12Z

Jc2916:

MRD:jc2916

2018-05-18T15:25:22Z

Jc2916: /* H + HF System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not. The vibrational energy for the reactants is high enough, so that the reaction is active.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:24:13Z

Jc2916: /* H + HF System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the this H atom bounce off with also high kinetic energy, so the other H atom cannot react with it to form H-H bond. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive. Because this reaction has a late transition state reaction, so the vibrational energy is the dominating factor to determine whether the reaction is active or not.

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:11:23Z

Jc2916: /* Trajectories from r1 = rts, r2 = rts+δ, */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable and the energy profiles for the reactants and products are the same in this reaction.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:06:52Z

Jc2916: /* Inverse Plot */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |450px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T15:05:43Z

Jc2916: /* F + HH System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive. And according to first law of thermodynamics, the energy is conserved. Therefore, the energy possessed by the reactants has been transformed to the kinetic energy for the leaving H atom and the vibrational energy for the new HF bond. On the graph, the vibrational energy of HF bond has larger amplitude than vibrational energy of HH bond, which means the translational energy between H and F has been transformed to the vibrational energy between HF.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:41:32Z

Jc2916: /* PES Inspection */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The transition state for this reaction is at the minimum point of V( rHH ), but the maximum point of V( rFH). Therefore, if we set rFH (r1) larger than 1.8115, it will roll back towards the reactants, and if we set rFH to be slightly smaller than 1.8115, it will roll towards the products.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:29:30Z

Jc2916: /* Transition State Theory */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can be fixed by multiplying rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:28:02Z

Jc2916: /* Reactive and Unreactive Trajectories */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and the reaction is unreactive.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:26:41Z

Jc2916: /* Inverse Plot */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state to form the products.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:23:43Z

Jc2916: /* Trajectories from r1 = rts, r2 = rts+δ, */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part because the three H atoms are indistinguishable.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:19:19Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly, so the time scale for the distance between B-C to decrease to a point where the energy is at its minimum is much larger and need more steps.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:14:33Z

Jc2916: /* Trajectories from r1 = rts+δ, r2 = rts */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:12:48Z

Jc2916: /* Calculating the Reaction Path */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:12:11Z

Jc2916:

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, and ∂V1(q1)/∂q11>0. So transition state is the cross point of two curves, and if we adjust r2 by a small amount to in the direction of the products, it will roll towards the product.

===Trajectories from r1 = r2===
The transition state distance is estimated to be: rts = 0.9077500

Because the trajectory starts at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will break, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===

When r1 is 0.01 higher than rts, the following plots will be obtained.

==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T14:06:38Z

Jc2916: /* Dynamics from Transition State Region */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===

When r1 is changing , the potential energy varies. At a certain r1 value, the potential energy is at its minimum, so the gradient vanishes and ∂V(r1)/∂r1=0. Because it is the minimum point of the V(r1), so the secondary derivative ∂V2(q2)/∂q22>0. The same can be applied to r2.

However, the transition state is the local maximum on the minimum energy pathway, which means it is the energy minima of V(q1), but energy maxima of V(q2). Therefore, ∂V2(q1)/∂q12<0, ∂V1(q1)/∂q11>0.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T13:48:32Z

Jc2916: /* H + HF System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive, and three separate atoms are hanging around.

[[File: -0.110.png |500px]]

If pFH = 10, and pHH = -0.1, the reaction is reactive because this reaction is a late transition state reaction, so the vibrational energy, which is the
pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T13:46:33Z

Jc2916: /* H + HF System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

If pFH = -0.1, and pHH = 10, which is very high, H atom will hit the HF bond with very high kinetic energy so that it breaks the HF bond. However, the H atom bounce off with also high kinetic energy, so the other H atom from the HF bond cannot reach it. So the overall reaction is not reactive.

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T13:43:26Z

Jc2916: /* F-H-H system */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the pFH=-0.5, and the pHH=3, the reaction trajectory is unreactive although in this case the vibrational energy is very high.

[[File: 3jc2916.png |500px]]

If the pFH is increased to -0.8, and pHH is decreased to only 0.1, the reaction trajectory is reactive although in this case the vibrational energy is very low. This is because for the reaction with early transition state, the translational energy is the dominate factor to determine whether or not the reaction trajectory is reactive.

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

Setting the rFH = 0.92, which is the normal F-H bond distance. And rHH = 2.3, so that two H atoms are far apart .

pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T13:35:20Z

Jc2916: /* Reaction dynamics */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

Setting the distance between F and H to be 2.3, so they are apart and energy between them are translational energy. The H-H distance is set to be 0.74, which is the bond distance between normal H-H bond. Energy between them is dominated by vibrational energy.

If the p=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-18T13:27:40Z

Jc2916: /* F-H-H system */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kcal/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the reverse reaction.

In this experiment, r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and r2 = 0.7437.

The activation energy for the F + H-H reaction is 0.262 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly loner than its distance at transition state. The least negative energy is -103.752 kcal-1 , and the most negative energy is -104.014 kcal-1 .

[[File: activationenergyjc2916.png |500px]]

The activation energy for the H + F-H reaction is 30.247 kcal-1 , which is obtained from the energy vs time plot below. In this calculation, the H-F distance is set to be slightly shorter than its distance at transition state. The least negative energy is -103.751 kcal-1 , and the most negative energy is -133.998 kcal-1 .

[[File: Ac2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T21:22:27Z

Jc2916: /* PES Inspection */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: activationenergyjc2916.png |500px]]
F+HH ---- H+FH
-103.752
-104.014

[[File: Ac2jc2916.png |500px]]
H+FH --- F+HH
-103.751
-133.998

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

File:Ac2jc2916.png

2018-05-17T21:21:58Z

Jc2916:

MRD:jc2916

2018-05-17T21:21:06Z

Jc2916: /* PES Inspection */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: activationenergyjc2916.png |500px]]
F+HH ---- H+FH
-103.752
-104.014

[[File: Acjc2916.png |500px]]
H+FH --- F+HH
-103.751
-133.998

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T21:20:49Z

Jc2916: /* PES Inspection */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: activationenergyjc2916.png |500px]]
F+HH ---- H+FH
-103.752
-104.014

[[File: Acjc2916.PNG |500px]]
H+FH --- F+HH
-103.751
-133.998

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

File:Acjc2916.png

2018-05-17T21:19:52Z

Jc2916:

MRD:jc2916

2018-05-17T21:10:26Z

Jc2916: /* PES Inspection */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: activationenergyjc2916.png |500px]]
F+HH ---- H+FH
-103.752
-104.014

[[File: TS4jc2916.PNG |500px]]
H+FH --- F+HH
-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

File:Activationenergyjc2916.png

2018-05-17T21:08:41Z

Jc2916:

MRD:jc2916

2018-05-17T20:49:53Z

Jc2916: /* F + HH System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T20:49:20Z

Jc2916: /* F + HH System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

rab=2.3 rbc=0.74
pab=-0.5 pbc=-3

[[File: 3jc2916.png |500px]]

pab=-0.8 pbc=0.1

[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T20:47:04Z

Jc2916: /* H + HF System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

[[File: 3jc2916.png |500px]]
[[File: jc2916.png |500px]]

====H + HF System ====

Polanyi's empirical rules state that vibrational energy is more efficient to promote a reaction with late transition state.

rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T20:46:38Z

Jc2916: /* F + HH System */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====

Polanyi's empirical rules state that transnational energy is more efficient to promote a reaction with early transition state.

[[File: 3jc2916.png |500px]]
[[File: jc2916.png |500px]]

====H + HF System ====
rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

MRD:jc2916

2018-05-17T20:44:25Z

Jc2916: /* Reaction dynamics */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
====F + HH System====
[[File: 3jc2916.png |500px]]
[[File: jc2916.png |500px]]

====H + HF System ====
rab=0.92 rbc=2.3
pab:-0.1 pbc:10

[[File: -0.110.png |500px]]

pab:10 pbc:-0.1

[[File: 10-0.1.png |500px]]

File:-0.110.png

2018-05-17T20:42:00Z

Jc2916:

File:10-0.1.png

2018-05-17T20:41:40Z

Jc2916:

MRD:jc2916

2018-05-17T20:26:19Z

Jc2916:

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===

[[File: jc2916.png |500px]]
[[File: 3jc2916.png |500px]]

File:3jc2916.png

2018-05-17T20:25:45Z

Jc2916:

File:Jc2916.png

2018-05-17T20:23:46Z

Jc2916:

MRD:jc2916

2018-05-17T19:55:07Z

Jc2916: /* Transition State Theory */

== H + H2 System ==

Three H atoms system. The distance between A-B is r1, and distance between B-C is r2.

===Dynamics from Transition State Region ===
Transition state is the local maximum on the minimum energy path, so ∂V(r1)/∂r1=0, and ∂V(r2)/∂r2=0, which means transition state is at the energy minimum. However, the trajectory at the transition state is the local maximum on the minimum energy pathway, which enables the transition structure to roll towards reactants and products easily. Therefore, minima and transition structure can be distinguished by taking the secondary derivative. For minimum point, the secondary derivative should be higher than 0. However, at transition state, it is the maximum point, where ∂V2(q1)/∂q12<0, on the minimum energy path, where ∂V2(q2)/∂q22>0. So this point is the cross point of two curves with two secondary derivative values.

===Trajectories from r1 = r2===
rts = 0.9077500

Because the trajectory stars at transition state, there is no initial momentum. The distance between three atoms, r1 and r2 are the same, and will stay the same without rolling to any other directions. Therefore, on the ''Internuclear Distances vs Time'' plot, distances between A-B, r1, and B-C, r2, stay constant at 0.9077500, and distance between A-C, which is double the length of rts, stays constant at 1.8155.

[[File: distance.jpeg |500px]]

===Calculating the Reaction Path===
By setting r1 slightly higher than rts, the geometry was changed in the direction of the product and it rolls toward the product, that the distance between A-B increasing overtime, and the bond between them will be broken, and the bond between B-C will be formed.

[[File: change.png |500px]]

===Trajectories from r1 = rts+δ, r2 = rts ===
When r1 is 0.01 higher than rts, the following plots will be obtained.
==== Distance vs Time ====
[[File: dynamic1.png |400px]]

This is the distance vs time plot obtained from the dynamic calculation. The distance between A-B increases rapidly. The bond between B-C vibrates, so the distance between B-C also fluctuates.

[[File: Mep1jc2916.png |400px]]

This is the distance vs time plot obtained from the mep calculation. Because in each time step, the velocity will be reset to zero, the distance between A-B rises slowly. The distance between B-C decreases to a point where the energy is at its minimum.

==== Momentum vs Time ====
[[File: dynamic2.png |400px]]

This is the momentum vs time plot obtained from the dynamic calculation. Because we set the r1 to be greater than rts , atom A will move further away from the other two atoms over time, while B-C atoms are linked by bond. Therefore the distance between A-B is large enough that there is no interaction between them, and the momentum stays constant at high t.

[[File: Mep2jc2916.png |400px]]

This is the momentum vs time plot obtained from the mep calculation. Because this path takes the infinite slowly motion, the velocity will be reset to zero at each time step. Therefore, the momentum is zero.

===Trajectories from r1 = rts, r2 = rts+δ, ===
[[File: distancejc2916.png |400px]][[File: momentumjc2916.png |400px]]

By setting r2 slightly higher than rts while keeping r1 = rts, atom C will move further away from the other two atoms to generate the reactant, A-B. The distance vs time plot and the momentum vs time plot obtained from dynamic calculation is exactly the same as the former part, while we just need to swap lines of A-B with that of B-C.

==== Inverse Plot ====
[[File: Inversejc2916.png |400px]][[File: Inverseplot2.png |400px]]

When the initial positions are setup to be the final positions of the trajectory from the internuclear distance vs time plot above, which are 0.810753 and 9.06604, and the momenta are setup to be the negative value of the average momenta at high t on the internuclear momentum vs time graph, which are -1.24756 and -2.4931, new calculations are generated. On the distance vs time plot, r1 and r2 line meet each other at around time 2.13. Therefore, it demonstrates that theses atoms have enough kinetic energy to overcome the activation barrier, so they pass the transition state.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1"
|+
! Experiment !! P1 !! P2 !! Total Energy !! Reactivity !! Plot of trajectory !! Distance vs time !! Descriptions
|-
| 1 ||-1.25 || -2.5 || -99.018 || reactive || [[File: Plot1jc2916.png |300px]] || [[File: Plot11jc2916.png |300px]] || Under this circumstance, three atoms have enough kinetic energy to overcome the activation barrier. On the contour plot, A-B start from the initial position and pass through the transition stage. Finally, the bond between A-B breaks and the product is formed between B-C.
|-
| 2 || -1.5 || -2.0 || -100.456 || unreactive || [[File: Pot2jc2916.png |300px]] || [[File: Plot22.png |300px]] || In this scenario, the momentum P2 is not high enough in negative direction, therefore there is not enough energy to overcome the activation barrier. Thus no product is formed, and A-B bond is intact.
|-
| 3 || -1.5 || -2.5 || -99.119 || reactive || [[File: Plot3jc2916.png |300px]] || [[File: Plot33jc2916.png |300px]] || Very similar to circumstance 1.
|-
| 4 || -2.5 || -5.0 || -84.956 || unreactive || [[File: Plot4jc2916.png |300px]] || [[File: Plot44jc2916.png |300px]] || In this scenario, the kinetic energy possessed by these three atoms are higher than activation energy. But the reactants did not pass the TS region. This trajectory is defined as unreactive because it crosses the saddle point and then turns around and comes back.
|-
| 5 || -2.5 || -5.2 || -83.416 || reactive || [[File: Plot6jc2916.png |300px]] || [[File: Plot66jc2916.png |300px]] || Under this circumstance, three atoms have even higher kinetic energy. In this case, the reactants pass the TS region and the products are formed.
|}

===Transition State Theory===
Assumption in transition state theory:

1. each intermediate is long-lived enough to reach a Boltzmann distribution of energies before continuing to the next step.

2. unless atoms or molecules collide with enough energy to form the transition structure, then the reaction does not occur.

3.the reaction system will pass over the lowest energy saddle point on the potential energy surface.

Second assumption stated that when the system possesses energy greater than or equal to the activation energy, the reaction will proceed to form the products from the reactants. However, the experiment 4 and 5 both have enough energy to overcome the activation energy, but experiment 4 is unreactive while 5 is reactive. This is because in experiment 4, the transition state theory overestimates the rate constant. This can sometimes be fixed by multiplying transition-state rate constants by a transmission coefficient.

== '''F-H-H system''' ==
===PES Inspection===
The bond energy between H-H is 432 kJ/mol, and the bond energy between H-F is 565 kJ/mol. Therefore H-H bond is weaker and higher in energy. For F + H2 system, bond between H-H is broken, which absorbs energy, and bond between H-F is formed, which releases energy. Energy that has been released overwrites the energy that has been absorbed, so this is an exothermic reaction. Vice versa for the other reaction.

r1 is the distance between F and H atoms, and r2 is the distance between two H atoms. According to Hammond postulate, when the reaction is an exothermic reaction, the transition state is in close proximity to the reactant, which is H-H and F in this case, so r2 at the transition state should be shorter and more like H-H bond distance, and r1 is longer. The approximate positions of the transition state are r1 = 1.8115, and 22 = 0.7437.

[[File: FHHjc2916.png |500px]]

-103.766
-103.943

[[File: TS4jc2916.PNG |500px]]

-103.770
-133.981

[[File: TS2jc2916.png |500px]]

===Reaction dynamics===
PA-B: -0.5818 -2.58194