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== Molecular reaction dynamics of triatomic systems ==

=== Notation ===

In all the calculations below, r<sub>BC</sub> and p<sub>BC</sub> refer to the internuclear distance and momentum of the molecule respectively. r<sub>AB</sub> and p<sub>AB</sub> refer to the internuclear distance and momentum between the individual atom A and atom B which is part of the molecule.

=== H<sub>2</sub> + H system ===

==== Initial conditions ====

The initial system was ran with the atoms in H<sub>2</sub> labelled as B and C and the Hydrogen atom labelled as A. The internuclear distance of the hydrogen molecule, r<sub>BC</sub>, was set to 0.74Å and the momentum, p<sub>BC</sub>, was set to 0. The internuclear distance between the hydrogen molecule and the hydrogen atom, r<sub>AB</sub>, was set to 2.30Å and the momentum, p<sub>AB</sub>, was set to -2.70.

When the potential energy surface reaches a minimum the total gradient will be 0. Since the transition state is a maximum on the minimum energy path between reactants and products, a transition structure will also have a total gradient of 0. The two can be differentiated by taking the second derivatives and looking at their signs. A minimum point will have a positive second derivative while a maximum point will have a negative one.

{{fontcolor1|gray|(This is only for one coordinate (in the case here, the reaction coordinate). What about for an N-coordinate system? [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

==== Identifying the transition state ====

In order to identify the transition state, the internuclear distances r<sub>AB</sub> and r<sub>BC</sub> were set equal and the momenta p<sub>AB</sub> and p<sub>BC</sub> were set to 0. The internuclear distances were varied but kept equal for both AB and BC starting from a distance of 2Å and then decreasing the distance until the reaction pathway on the surface plot gave only a small dot, the atoms in the Animation mode appeared almost inanimate and the internuclear distance in the Internuclear Distances vs Time plot appeared to be almost completely flat. The best estimate obtained for the transition state position, r<sub>ts</sub> was 0.908Å.



[[File:Dc3415 mrd trajectory best estimate ts.png|600px|center|thumb|Figure 1: Internuclear Distance vs Time plot for r<sub>ts</sub>=0.908Å.]]



From the Internuclear Distance vs Time plot in Figure 1 it is seen that r<sub>AB</sub> and r<sub>BC</sub> remain almost constant with increasing time unit. Since a transition state occurs at an energy maximum and given that the atoms possess no momentum, the transition state is a point of infinite potential energy. This is demonstrated by the constant kinetic energy which is obtained at r<sub>ts</sub>=0.908Å

{{fontcolor1|gray|(This does not mean infinite potential energy! And infinite potential energy is not demonstrated by constant kinetic energy. The kinetic energy of your calculation should be zero throughout because the gradient is zero and you started with no kinetic energy. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

==== Comparing minimum energy path (mep) calculations to Dynamics calculations ====

The MEP for the reaction was calculated by keeping the same initial conditions as described above but changing r<sub>BC</sub> by +0.01Å to 0.918Å, increasing the number of steps to 2000 and the calculation type to MEP. By looking at the surface plot of the MEP calculation the trajectory appears to be flat along the floor of the plot. The MEP calculation resets the gradient of the calculation to zero at each step thus allowing the reactants to follow a minimum energy pathway that results in zero kinetic energy being observed in the Kinetic energy vs Time plot. In other words, the potential energy of the reactants is set to the maximum at the beginning of every step, ensuring that the path followed is the minimum energy path.

{{fontcolor1|gray|(No. The velocity is set to zero, not the gradient. The last sentence is also wrong. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

The Dynamics calculation shows the potential energy changing in an continuous manner since the reactants can have enough energy to go uphill after the minimum has been reached. Since the gradient does not reset at the end of every step, the potential energy is no longer infinite and can convert to kinetic energy allowing for the reactants to go uphill. 

{{fontcolor1|gray|(No. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

By reversing the bond distances so that r<sub>AB</sub>=0.918Å and r<sub>BC</sub>=0.908Å, the reverse reaction occurs. r<sub>AB</sub> increases and r<sub>BC</sub> oscilates close to 0.78Å. The final positions r<sub>AB</sub> and 
r<sub>BC</sub> after 10 seconds were found to be 0.733Å and 36.85Å respectively. The final momenta, p<sub>AB</sub> and p<sub>BC</sub> were 1.22 and 2.48 respectively. 

A calculation was set up so that the final position parameters were used as the initial parameters and the final negative momenta were used as the initial momenta. By inspection of the surface plot, it appears that the molecule and the atom collide but no reaction occurs after collision. The momenta appear to become positive after collision and reach the same magnitude in the positive direction. The internuclear distance r<sub>AB</sub> appears to remain the same and is slightly elevated at the moment when the internuclear distances between AB and BC become the same. After that, the internuclear distance AB keeps increasing.

{{fontcolor1|gray|(Any explanation for this? Did you give it enough steps? [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

==== Dynamic calculations to predict the reactivity of pathways ====

The reactivity of pathways with r<sub>BC</sub>=0.74Å and r<sub>AB</sub>=2.0Å and varying momenta was investigated. Table 1 below summarises the findings:


{| class="wikitable" border="1"
|+ Table 1:The reactivity of pathways with different momenta
! p<sub>BC</sub> !! p<sub>AB</sub> !! Reactivity
|-
| -1.25 || -2.5 || Reactive
|-
| -1.5 || -2.0 || Unreactive
|-
| -1.5 || -2.5 || Reactive
|-
| -2.5 || -5.0 || Unreactive
|-
| -2.5 || -5.2 || Reactive
|}



[[File:Dc3415 p12525.png|thumb|center|Surface plot of p<sub>BC</sub>=-1.25 and p<sub>AB</sub>=-2.5|450px]]


In this case it can be seen that as the hydrogen atom A approaches the hydrogen molecule, r<sub>AB</sub> decreases while r<sub>BC</sub> appears to be relatively constant at 0.74Å. There is very little vibrational energy at this point. Once the H<sub>A</sub> approaches at a distance of around 1.2Å, r<sub>BC</sub> begins to increase. After collision, the product appears to vibrate more compared to the reactants. r<sub>BC</sub> increases, indicating that the reaction trajectory is reactive resulting in the formation of H<sub>A</sub>-H<sub>B</sub> and H<sub>C</sub> as a single atom.



[[File:Dc3415 p1520.png|thumb|center|Surface plot of p<sub>BC</sub>=-1.5 and p<sub>AB</sub>=-2.0|450px]]



From the trajectory above it can be seen that r<sub>BC</sub> remains relatively constant, but with a higher vibrational energy compared to the case above. After the moment of collision r<sub>AB</sub> appears to increase again while r<sub>BC</sub> still remains fairly constant. This suggests that the collision does not result in reaction hence the reaction trajectory is unreactive. The potential energy of the vibrations is very similar before and after collision.


[[File:Dc3415 p1525.png|thumb|center|Surface plot of p<sub>BC</sub>=-1.5 and p<sub>AB</sub>=-2.5|450px]]



This set of momenta results in a reactive trajectory as it can be seen by the shape of the surface plot, which is similar to case 1. r<sub>AB</sub> decreases and when the two reactants come close to each other, reaction occurs resulting in H<sub>AB</sub> being formed which vibrates at a slightly higher energy than the reactants. r<sub>BC</sub> appears to increase which is consistent with the formation of H<sub>A</sub>-H<sub>B</sub> + H<sub>C</sub>.


[[File:Dc3415 p2550.png|thumb|center|Surface plot of p<sub>BC</sub>=-2.5 and p<sub>AB</sub>=-5.0|450px]]


This trajectory shows an initial reaction as r<sub>AB</sub> decreases and after collision, r<sub>BC</sub> increases. However the products are quickly reverted back to reactants as the products recross the activation barrier. This is due to the high kinetic energy of the system as a result of the large momenta of the molecules. The higher energy can be also seen by the large vibrations resulting in large potential energy wells. This return to the reactants results in this trajectory being essentially unreactive. After recrossing the barrier, the energy of the system is significantly reduced, as seen from the decrease in potential energy and the size of the wells. 


[[File:Dc3415 p2552.png|thumb|center|Surface plot of p<sub>BC</sub>=-2.5 and p<sub>AB</sub>=-5.2|450px]]



This trajectory is a reactive trajectory. A reaction takes place as H<sub>A</sub> approaches the hydrogen molecule. Initially there is very little vibrational energy however once the reaction occurs, the H<sub>C</sub> atom appears to be vibrationally excited as r<sub>BC</sub> increases.

{{fontcolor1|gray|(No, the single atom cannot be vibrationally excited. It's the AB atoms that are vibrating. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

==== Transition state theory ====

Transition state theory assumes an equilibrium between the reactants and the transition state complex. It uses the Eyring equation<ref>Pechukas, P. Transition State Theory Ann. Rev. Phys. Chern. 1981.32:159-77</ref> to allow the calculation of the rate constant of a reaction using statistical thermodynamics. The calculation of a partition function for an activated complex or a transition state can be difficult especially in the case where there is limited information available for the reaction or its mechanism. Transition state theory relies on multiple assumptions such that the atoms and molecules involved behave according to classical mechanics<ref>Atkins, P.; De Paula, J. Physical Chemistry, 9th ed.; W. H. Freeman and Company, 2010</ref> and are treated as hard spheres. Transition state theory assumes that a specific amount of energy is required before a collision can lead to the formation of a transition state and then products, otherwise the collision will not have enough energy to lead to product formation. In reality, quantum mechanical processes such as barrier crossing can cause product formation between particles with energy lower than the activation energy. While it is known that there are multiple activated complexes that can occur with different energies, transition state theory states that only the lowest energy activated complex (the transition state) will lead to product formation and all other pathways will not. While this is true at low temperatures, at higher temperatures much more energy is available in the system and the reaction rate might no longer be in direct control of the transition state pathway<ref>Pechukas, P. Transition State Theory Ann. Rev. Phys. Chern. 1981.32:159-77</ref>. Transition state theory would not have predicted the correct outcome for the 4th case and the 5th case which involve barrier recrossing since it assumes that once the transition state turns to products, it would stay as products.

=== F - H - H system ===

==== Thermodynamics and PES inspection ====

There are 2 possible reactions in the H - H - F system. One is the reaction between H<sub>2</sub> and F and the reverse reaction is between HF and H. By inspection of the H<sub>2</sub>+F surface plot it can be seen that this reaction is exothermic since the overall potential energy of HF + H is lower than the reactant H<sub>2</sub> + F. This can be seen in Figure X. Conversely, the reverse reaction is endothermic, since the overall potential energy of H<sub>2</sub> and F is greater than the energy of HF + H as seen in figure Y. The enthalpy of a reaction depends on the bonds that are being broken (endothermic process) and the bonds being formed (exothermic process). Since the enthalpy change of the reaction, ΔH, is the difference between the bonds being broken and the bonds being formed, in the case of the exothermic reaction, the energy required to break the H<sub>2</sub> bond is less than the energy released during the formation of the H-F bond hence the overall sign of ΔH will be negative. Thus it can be concluded that H-F has a stronger covalent bond than H<sub>2</sub>.


[[File:Dc3415 sp exotherm.png|thumb|left|Figure X: Surface plot of H<sub>2</sub>+F|450px]]
[[File:Dc3415 sp endotherm.png|thumb|right|Figure Y: Surface plot of HF+H|450px]]




























With reference to the Hammond postulate, the transition state is much closer in energy to the reactants, therefore it will resemble closely to the reactants. The aproximate position of the transition state for the exothermic reaction was found to be at r<sub>BC</sub>=0.74Å and r<sub>AB</sub>= 1.813Å.


The activation energies for each reaction were calculated for both reactions by subtracting the energy of the reactants from the energy of the transition state. The activation energies (E<sub>a</sub>) of each reaction are reported below:

H<sub>2</sub> + F: -103.7-(-)103.9= 0.2kcal/mol = 0.8368 kJ/mol

HF + H: -103.7-(-)133.9= 30.2kcal/mol = 126.3568 kJ/mol


[[File:Dc3415 H2plusF reactants energy.png|thumb|center|Reactant energy of exothermic reaction|450px]]
[[File:Dc3415 HHF ts position.png|thumb|center|Transition state energy|450px]]
[[File:Dc3415 HFplusH reactants energy.png|thumb|center|Reactant energy of endothermic reaction|450px]]

==== Reaction dynamics: F + H<sub>2</sub>====

A set of initial parameters that led to a reaction was identified with r<sub>BC</sub>=0.74Å, r<sub>AB</sub>=1.8Å, p<sub>BC</sub>=-2.0 and p<sub>AB</sub>=0. The surface plot and internuclear momenta vs time plot of the initial set of conditions is shown below:


[[File:Dc3415 reactive path H2plusF.png|thumb|center|Surface plot of initial conditions set|450px]]
[[File:Dc3415 reactive path H2plusF int p vs t.png|thumb|center|Internuclear momenta vs Time plot of initial conditions set|450px]]



The law of conservation of energy states that the total energy of a system remains unchanged i.e. the individual forms of energy may change however the total energy will remain constant. By observation of the internuclear momenta vs time plot, it is seen that p<sub>AB</sub> which is initially -2, increases after collision with the hydrogen molecule and is now changing between -5.56 and 9.35. In contrast, the momentum of the molecule, which was originally changing between -2 and 0 as a result of vibrations of the bond is now at a constant momentum of 2. This demonstrates the change between the potential energy, which is a result of the internuclear distances and attractions, and the kinetic energy that is due to the momentum of the molecule. Since the momenta change after the collision, so does the kinetic energy of the molecules. The "excess" energy of the reaction is converted to heat energy and is released to the surrounding environment of the reaction. A calorimetric experiment using a bomb calorimeter can be conducted to measure the change in temperature of the reaction as it occurs. The change in temperature can then be related to ΔH using the fundamental equation dH=dU+PdV+VdP and adapting it to include the temperature change term. 


When p<sub>BC</sub> was changed in the range of -3 to 3, at p<sub>AB</sub>=-0.5, r<sub>BC</sub>=0.74Å and r<sub>AB</sub>=1.8Å most of the momenta resulted in a reactive trajectory. The majority of the reactive trajectories at high momenta resulted in vibrational excitation of the product molecule, demonstrated by the increase in the height of the peaks along the r<sub>BC</sub> axis after the products were formed. 

When p<sub>AB</sub> was increased to -0.8 and p<sub>BC</sub> to -0.1 a reactive trajectory was obtained, however the vibrational excitation of the product molecules was significantly reduced.

==== Reaction dynamics: HF + H ====

A set of initial conditions was established starting from the bottom of the channel (r<sub>BC</sub>=0.91Å, p<sub>BC</sub>=0 and r<sub>AB</sub>=2.5Å, p<sub>AB</sub>=-10.0) to simulate a collision between a molecule of HF and a high kinetic energy H atom. The reaction appears to be reactive and at the moment of collision, there is initial a decrease in r<sub>AB</sub> to 0.56Å due to the high kinetic energy of the colliding molecule. The potential energy then spikes producing a potential energy maximum as r<sub>AB</sub> increases to 1.02Å. After this maximum, the reaction proceeds to product formation with r<sub>BC</sub> and the potential energy following the path floor.

The p<sub>AB</sub> was then decreased slowly, so that the kinetic energy of the H atom decreases and r<sub>BC</sub> was decreased, to increase the potential energy of the HF molecule until a reactive pathway was obtained. The first reactive pathway obtained is shown below:

[[File:Dc3415 react path HFplusH high p.png|thumb|center|surface plot of initial conditions at high momentum (side view)|450px]]
[[File:Dc3415 react path HFplusH lower p.png|thumb|center|surface plot of decreased p<sub>AB</sub> and r<sub>BC</sub> (top view)|450px]]

==== Polyani's empirical rules discussion ====

Polyani's empirical rules are used to classify early barriers or transition states that resemble the reactants and late barriers or transition states which resemble the products<ref>Jiang, B; Guo, H; The Journal of Chemical Physics 138, 234104, '''2013'''</ref>. In each case, the reactivity and efficacy of reaction trajectory will be different depending on the amount of kinetic energy/translational motion which depends on the momenta of the reactants and the potential energy/vibrational motion which is governed by the internuclear distances in the reactant molecules. 

Polyani's rules state that for an early transition state, a higher momentum or translational motion<ref>Jiang, B; Guo, H; The Journal of Chemical Physics 138, 234104, '''2013'''</ref> is preferred in order to overcome the barrier whereas in a late transition state a higher potential energy is needed to result in an efficient reaction. The reaction between H<sub>2</sub> and F is an example of a reaction which results in an early transition state, therefore a high energy in the form of translational motion would result in a reactive pathway whereas supplying a large amount of vibrational energy and a low translational energy would less likely result in a reactive trajectory despite the fact that the total energy is unchanged and higher than the activation energy.

HF+H has a late transition state, so a high amount of energy supplied as vibrational energy would would increase the efficiency of the reaction. The same effect as explained above would result in the case where most of the energy was supplied in the form of translational energy and the reaction pathway would most likely be unreactive.

[[File:Dc3415 final hi p.png|thumb|center|A F + H<sub>2</sub> trajectory at high momentum|450px]]
[[File:Dc3415 final hi v.png|thumb|center|A F + H<sub>2</sub> trajectory at high vibrational energy|450px]]

By observation of the two trajectories it is seen that while both of the trajectories are reactive, there is significant recrossing of the activation barrier in the case of the higher vibrational energy which makes the reaction less efficient.

{{fontcolor1|gray|(Why did you change the initial conditions here? It doesn't help that you are already on the product side of the TS [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:35, 31 May 2017 (BST))}}

=== References ===