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MRD:01546737

2020-05-08T22:13:49Z

Jb4318: /* Distribution of Energy in the System */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

====Distribution of Energy in the System====
For a system starting with the reactants F + H2, with the initial conditions rHH= 74 pm, rHF=200 pFH= -1.0 g.mol-1.pm.fs-1, a range of values for pHH were investigated, between -6.1 to 6.1 g.mol-1.pm.fs-1. It was observed that the negative pHH had overall less vibrational kinetic energy to begin with than a positive value of the same magnitude. It was also observed that because the system had a lot greater energy in the form of initial vibrational energy, system recrossing could sometimes occur. When system recrossing occured, the H2 molecule was observed to travel away from F with less vibrational and translational kinetic energy than when approaching F. However when the reaction was successful, HF had much greater vibrational energy than the H2 reactant, and both products had more translational energy. This shows that if system recrossing occurs, then there is a net gain in potential energy, due to some kinetic energy being used to overcome the small initial activation energy in the first crossing.

When the overall energy of the system is significantly reduced by changing pFH to -1.6 g.mol-1.pm.fs-1, and pHH to 0.2 g.mol-1.pm.fs-1, the system still has enough energy to overcome the initial activation energy barrier, however there is not sufficient energy for system recrossing to happen, meaning that overall there is a formation of F-H, and again a release of kinetic energy to the system, primarily in the form of translational and vibrational energy.

Though the inversion of momentum procedure was used, a reactive trajectory that successfully achieved the formation of H2 from HF + H was not found. However in the process of trying to find such a reactive trajectory, it was observed that a higher vibrational energy of the reactants is much more important to the success and efficiency of a reaction than high translational energy. This is because the trajectory of the reactants is already oscillating about the potential well, so therefore less external energy (in the form of translational energy) is required to overcome the saddle point than if the molecules were residing at the bottom of the potential well. If the position of the transition state were more central between the products and the reactants, then overall for an endothermic reaction, as in this example, it would resemble the products less and would in turn require less potential energy to be overcome. This would mean that the reaction rate would be less dependent on the vibrational energy possessed by the reactants, and translational energy of reactants would have more of an effect in determining whether a reaction is successful.

MRD:01546737

2020-05-08T21:52:57Z

Jb4318: /* Distribution of Energy in the System */

MRD:01546737

2020-05-08T21:10:32Z

Jb4318: /* Distribution of Energy in the System */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

====Distribution of Energy in the System====
For a system starting with the reactants F + H2, with the initial conditions rHH= 74 pm, rHFM=200 pFH= -1.0 g.mol-1.pm.fs-1, a range of values for pHH were investigated, between -6.1 to 6.1 g.mol-1.pm.fs-1. It was observed that the negative pHH had overall less vibrational kinetic energy to begin with than a positive value of the same magnitude. It was also observed that because the system had a lot greater energy in the form of initial vibrational energy, system recrossing could sometimes occur. When system recrossing occured, the H2 molecule was observed to travel away from F with less vibrational and translational kinetic energy than when approaching F. However when the reaction was successful, HF had much greater vibrational energy than the H2 reactant, and both products had more translational energy. This shows that if system recrossing occurs, then there is a net gain in potential energy, due to some kinetic energy being used to overcome the small initial activation energy in the first crossing.

When the overall energy of the system is significantly reduced by changing pFH to -1.6 g.mol-1.pm.fs-1, and pHH to 0.2 g.mol-1.pm.fs-1, the system still has enough energy to overcome the initial activation energy barrier, however there is not sufficient energy for system recrossing to happen, meaning that overall there is a formation of F-H, and again a release of kinetic energy to the system, primarily in the form of translational and vibrational energy.

MRD:01546737

2020-05-08T21:05:02Z

Jb4318: /* Distribution of Energy in the System */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

====Distribution of Energy in the System====
For a system starting with the reactants F + H2, with the initial conditions rHH= 74 pm, rHFM=200 pFH= -1.0 g.mol-1.pm.fs-1, a range of values for pHH were investigated, between -6.1 to 6.1 g.mol-1.pm.fs-1. It was observed that the negative pHH had overall less vibrational kinetic energy to begin with than a positive value of the same magnitude. It was also observed that because the system had a lot greater energy in the form of initial vibrational energy, system recrossing could sometimes occur. When system recrossing occured, the H2 molecule was observed to travel away from F with less vibrational and translational kinetic energy than when approaching F. However when the reaction was successful, HF had much greater vibrational energy than the H2 reactant, and both products had more translational energy. This shows that if system recrossing occurs, then there is a net gain in potential energy, due to some kinetic energy being used to overcome the small initial activation energy in the first crossing.

MRD:01546737

2020-05-08T21:04:31Z

Jb4318: /* Distribution of Energy in the System */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

====Distribution of Energy in the System====
For a system starting with the reactants F + H2, with the initial conditions rHH= 74 pm, rHFM/sub>=200 pFH= -1.0 g.mol-1.pm.fs-1, a range of values for pHH were investigated, between -6.1 to 6.1 g.mol-1.pm.fs-1. It was observed that the negative pHH had overall less vibrational kinetic energy to begin with than a positive value of the same magnitude. It was also observed that because the system had a lot greater energy in the form of initial vibrational energy, system recrossing could sometimes occur. When system recrossing occured, the H2 molecule was observed to travel away from F with less vibrational and translational kinetic energy than when approaching F. However when the reaction was successful, HF had much greater vibrational energy than the H2 reactant, and both products had more translational energy. This shows that if system recrossing occurs, then there is a net gain in potential energy, due to some kinetic energy being used to overcome the small initial activation energy in the first crossing.

MRD:01546737

2020-05-08T20:47:52Z

Jb4318: /* Distribution of Energy in the System */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

====Distribution of Energy in the System====
For a system starting with the reactants F + H2, with the initial conditions rHH= 74 pm, rHFM/sub>=200 pFH= -1.0 g.mol-1.pm.fs-1, a range of values for pHH were investigated, between -6.1 to 6.1 g.mol-1.pm.fs-1. It was observed that

MRD:01546737

2020-05-08T20:42:54Z

Jb4318: /* Reaction Dynamics */

MRD:01546737

2020-05-08T20:42:34Z

Jb4318: /* Reaction Dynamics */

MRD:01546737

2020-05-08T20:42:16Z

Jb4318: /* Reaction Dynamics */

MRD:01546737

2020-05-08T20:35:41Z

Jb4318: /* The Release of Reaction Energy */

MRD:01546737

2020-05-08T20:35:00Z

Jb4318: /* The Release of Reaction Energy */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between the saddle point energy and energies of reactants for the system were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.
[[File:momenta-time energy release 01546737.png|250px|thumb|right|A plot of momenta against time for a trajectory that leads to the successful formation of the H-F product, illustrating the increase of kinetic vibrational energy released into the system upon a successful exothermic reaction]]

File:Momenta time energy release 01546737.png

2020-05-08T20:32:20Z

Jb4318:

MRD:01546737

2020-05-08T20:28:43Z

Jb4318: /* EXERCISE 2: F-H-H system */

MRD:01546737

2020-05-08T20:28:17Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

MRD:01546737

2020-05-08T20:18:03Z

Jb4318: /* The Release of Reaction Energy */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====
Upon inspection of a momenta-time graph for a successful reaction for the formation of H-F from H2, upon the formation of the product, there is greater vibrational energy in the H-F bond, in comparison to the H-H bond of the H2 reactant. This is because there is a large release of potential energy from the reactants, which is in turn converted to kinetic energy, primarily in the form of vibrational energy in the products. This vibrational energy in turn increases the overall heat energy present in the system. There is therefore an increase in temperature of the system. This change in temperature can be measured experimentally using a calorimeter. The heat released from the exothermic reaction is transferred to the solution in the calorimeter, with the change in temperature of solution being calculated by the equipment.

MRD:01546737

2020-05-08T19:22:57Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:22:46Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:22:32Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:21:02Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HH formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:20:04Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the formation of H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for the formation of H-F was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-F formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:15:24Z

Jb4318: /* Reaction Dynamics */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===
====The Release of Reaction Energy====

MRD:01546737

2020-05-08T19:14:58Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:14:48Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:14:37Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:14:21Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:14:00Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:13:45Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:13:29Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:13:02Z

Jb4318: /* Energetics of the F + H2 and H + HF reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:12:45Z

Jb4318: /* PES Inspection */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:12:21Z

Jb4318: /* The Approximate Position of the Transition State */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:11:57Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:11:44Z

Jb4318: /* EXERCISE 2: F-H-H system */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

===Reaction Dynamics===

MRD:01546737

2020-05-08T19:09:46Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

MRD:01546737

2020-05-08T19:07:57Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be -126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be -0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|right|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

MRD:01546737

2020-05-08T19:07:44Z

Jb4318: /* PES Inspection */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-H + F reaction, to form HF was calculated to be -126.624 KJ.mol-1, whilst the activation energy for H-F + H, to form H2 was found to be -0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.
[[File:energy-time HH formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-F formation]]
[[File:energy-time HF formation 01546737.png|250px|thumb|left|The energy time plot used to determine the activation energy of H-H formation. Whilst the energy of the plot looks to be constant, there is a slight negative gradient.]]

MRD:01546737

2020-05-08T19:02:43Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-F + H reaction, to form H2 was calculated to be -126.624 KJ.mol-1, whilst the activation energy for H-H + F, to form HF was found to be -0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.

File:Energy-time HF formation 01546737.png

2020-05-08T19:01:34Z

Jb4318:

File:Energy-time HH formation 01546737.png

2020-05-08T18:55:45Z

Jb4318:

MRD:01546737

2020-05-08T18:48:19Z

Jb4318: /* Activation Energy for H2 + F and HF + H reactions */

==EXERCISE 1: H + H2 system==
=== Dynamics from the transition state region ===
==== Defining the Transition State ====
The transition state is mathematically defined on a potential energy surface as the point where the gradient of the potential is equivalent to zero, i.e. ∂V(ri)/∂ri=0. It is a saddle point, and lies on the minimum energy path between reactants and products. It can be easily identified if trajectories are started near the transition state. If the transition state is present, then the trajectory will 'roll' in the direction of either the reactants or the products, whereas if a local energy minima is present, then the trajectory may become stuck in a potential well and will not 'roll' towards the reactants or products.
[[File:Rts_internuclear_distance_vs_time.png|250px|thumb|right|A graph of internuclear distance against time for the transition state position of the H+H2 system]]
The transition state and local potential minima can be further distinguished by looking at the Hessian matrix for the point. For an energy minima, then it would be expected that overall the dot product of the eigenvalues and orthogonal eigenvectors would lead to both vectors being positive, i.e. going up the 'slope' of the potential surface, whereas at the transition state, on the saddle point, one of the vectors would be positive, going 'up the slope', and the other would be negative, going 'down the slope'.

====Locating the Transition State Using r1=r2====
Using r1=r2 and p1=p2=0, the transition state position, rts was found to be located at 90.77pm. This is clearly shown in the internuclear distances vs time graph for this position, as the gradients for A-B, B-C and A-C are all equivalent to 0, showing that at this location the atoms exhibit no oscillation along the ridge or r1=r2. This will occur when there is no force acting upon the atoms. One definition of force is the variation of momentum, or mathematically ∂Pi/∂t . This partial derivative is equal to -∂V(r1,r2,...)/∂ri, and as we have previously discussed at the transition state this is equal to zero. As such at the transition state, no force will be acting on the atoms, meaning that they will not oscillate.

==== Trajectories from r1+δ, r2 ====
Through a comparison of the trajectory from contour plots created from the MEP and Dynamics calculations, it is clear that the trajectory follows the minimum energy path in the MEP calculation, whilst oscillation occurs in the Dynamics trajectory. This is because in the MEP calculation, there is an infinite 'friction' acting on the trajectory, meaning that as it 'rolls', it is not able to pick up any momentum, and therefore no net force acts on the system. In contrast, the Dynamics calculation is an accurate representation of the system, and therefore means that momentum is allowed to build up as it 'rolls' towards the products, applying a force on the atoms in the system, in turn causing them to oscillate.
[[File:MEP slight displacement 01546737.png|250px|thumb|left|The MEP trajectory calculated from a slight displacement from the transition state position of the H+H2 system]] [[File:Dynamics slight displacement 01546737.png|250px|thumb|right|The Dynamics trajectory calculated from a slight displacement from the transition state position of the H+H2 system]]

The following plots are for the compare the 'Internuclear Distances vs Time' plots and 'Momenta vs Time' plots for the system previously described and the system r1=rts, r2=rts+δ, using the Dynamics calculation for both. It is clear that both graphs look identical, however in the second system the lines representing A-B and B-C have been swapped. This is because the change in displacement in the second system means a change in trajectory as the system 'rolls' towards a different set of products, HA + HB-HC.
[[File:1st system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:1st system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts+δ, r2=rts trajectory of the H+H2 system]]
[[File:2nd system IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]
[[File:2nd system momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for r1=rts, r2=rts+δ trajectory of the H+H2 system]]

The following graphs correspond to a system where the initial positions are the final positions of the r1=rts, r2=rts+δ, and the initial momenta is the negative of the final momenta from the previous system. It is observed that the trajectory of the system is reversed, and the system returns to the initial conditions of r1=rts+δ, r2=rts.
[[File:reverse momenta-time 01546737.png|250px|thumb|right|Momenta vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]
[[File:reverse IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the trajectory to return r1=rts, r2=rts+δ to it's initial positions and momenta for the H+H2 system]]

=== Reactive and Unreactive Trajectories ===
====Table of Varying Trajectories====
{| class="wikitable" border="1"
|+ A summary of trajectories run at varying values of p1 and p2, with the same initial conditions r1= 74 pm, r2= 200 pm
! p1/g.mol-1.pm.fs-1 !! p2/g.mol-1.pm.fs-1 !! Etot/KJ.mol-1 !! Reactive? !! Description of Dynamics !! Illustration of the Trajectory
|-
| -2.56 || -5.1 || -414.280 || yes || The system has sufficient energy to overcome the saddle point, after which the atoms continue along the trajectory with some oscillation. || [[File: table 1 01546737.png|200px|none]]
|-
| -3.1 || -4.1 || -420.077 || no || This system has a lower total energy than the previous one, and therefore does not have sufficient energy to overcome the saddle point, meaning that it is unreactive, and the trajectory travels back towards the reactants. || [[File: table 2 01546737.png|200px|none]]
|-
| -3.1 || -5.1 || -413.977 || yes || Similarly to the first trajectory, the system is able to overcome the saddle point, however the fact that there is more energy in the system means that when 'rolling' towards the products, there is greater oscillation. || [[File: table 3 01546737.png|200px|none]]
|-
| -5.1 || -10.1 || -357.277 || yes || This trajectory is an example of barrier recrossing. Due to the sufficient energy in the system, after the products are formed, the trajectory 'rolls' back over the saddle point towards the reactants || [[File: table 4 01546737.png|200px|none]]
|-
| -5.1 || -10.6 || -349.477 || yes || This trajectory has the greatest total energy. After some oscillation of the atoms about the saddle point, the trajectory crosses over and 'rolls' towards the products with the greatest oscillation of any of the trajectories due to the greatest total energy. || [[File: table 5 01546737.png|200px|none]]
|}

====Transition State Theory====
Transition State Theory will overestimate the rate of reaction in comparison to experimental values. One of the conditions for Transition State Theory is that once the system has crossed over the potential energy barrier and is 'rolling' towards the products, it is not able to cross the saddle point again and therefore the system cannot revert back to the products. In reality however, barrier recrossing can occur, as shown in the fourth trajectory of the previous table, meaning that experimentally not all trajectories that cross over the transition state saddle point will lead to the formation of the products, slowing down the overall rate of reaction.
== EXERCISE 2: F-H-H system==
===PES Inspection===
====Energetics of the F + H2 and H + HF reactions ====
The F+ H2 reaction is exothermic as, according to Hammond's postulate, for an exothermic reaction, the transition state will lie close to the reactants and will therefore resemble said reactants. In contrast, the H + HF reaction is endothermic as the transition state lies close to the products, and will in turn resemble the products. These reaction energetics directly relate to the H-F and H-H bond strengths. The F-H bond is stronger than the H-H bond, and as such the formation of the H-F bond is overall exothermic as there is a net energy release caused by the formation of a strong bond. The H-H bond formation from H-F + H is endothermic as there is a net energy loss, because the strong H-F bond must be broken.

====The Approximate Position of the Transition State====
The transition state of the H-H-F system lies approximately at H-F = 181.28 pm and H-H = 74.15 pm. A change of δ= ±1 pm on either of the bond distances changes the trajectory in the direction of either the reactants or the product. In addition, the internuclear distances - time graph shows that the bond lengths exhibit only slight oscillation when the trajectory starts at this point, and are practically constant. This in turns means that very little force is acting on the atoms at these positions, suggesting that they are very close to the transition state position. Furthermore, the orthogonal vectors formed from the Hessian matrix of this point show one 'going up' the wall of the potential energy slope, whilst the other is 'pointing down' the slope towards the products, suggesting that it lies very close to the saddle point.
[[File:H-H-F IN-time 01546737.png|250px|thumb|left|Internuclear Distances vs Time for the H-H-F system starting at the approximate transition state position.]]
[[File:H-H-F contour 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position, with the vectors calculated from the Hessian matrix.]]
[[File:H-H-F F-H change 01546737.png|250px|thumb|left|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for H-H, and F-H + δ]]
[[File:H-H-F H-H change 01546737.png|250px|thumb|right|The contour plot for the trajectory of the H-H-F system starting at the approximate transition state position for F-H, and H-H - δ]]

==== Activation Energy for H2 + F and HF + H reactions====
The activation energy for the H-F + H reaction, to form H2 was calculated to be 126.624 KJ.mol-1, whilst the activation energy for H-H + F, to form HF was found to be 0.938 KJ.mol-1. These were found by performing an MEP reaction that started from a slight deviation from the transition state position, and the difference between start and final energies of the systems were taken using energy vs time plots.

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2020-05-08T14:54:36Z

Jb4318: /* EXERCISE 2: F-H-H system */

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Jb4318: /* EXERCISE 2: F-H-H system */

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Jb4318: /* The Approximate Position of the Transition State */

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Jb4318: /* EXERCISE 2: F-H-H system */

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Jb4318: /* EXERCISE 2: F-H-H system */

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2020-05-08T14:52:36Z

Jb4318: /* EXERCISE 2: F-H-H system */

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Jb4318: /* The Approximate Position of the Transition State */

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Jb4318: /* EXERCISE 2: F-H-H system */

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Jb4318: /* EXERCISE 2: F-H-H system */

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2020-05-08T14:51:19Z

Jb4318: /* EXERCISE 2: F-H-H system */