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MRD:BaiqiuTang

2020-05-15T21:05:42Z

Bt3418: /* Reference list */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''. [1]

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive. [1]

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively. [2]

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguished. The most obvious difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.[2]

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision. No recrossing. Relatively high reaction rate.||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. The transition state is not reached. No collision. No recrossing.||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. No recrossing. The rate is slower and the oscillation is greater than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. The transition state is reached. Recrossing taken place for once and the product was formed and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The transition state is reached.Recrossing taken place for twice: the first time turned the products back to reactants, the second time formed the products again. Reaction rate is relatively slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the higher the reaction rate. [2] Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In the real circumstances, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by applying the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected. [2][3]

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.[2][4]

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice-Hall.

MRD:BaiqiuTang

2020-05-15T21:03:46Z

Bt3418: /* Reference list */

MRD:BaiqiuTang

2020-05-15T21:03:04Z

Bt3418: /* Reaction rate comparision */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''. [1]

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive. [1]

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively. [2]

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguished. The most obvious difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.[2]

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision. No recrossing. Relatively high reaction rate.||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. The transition state is not reached. No collision. No recrossing.||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. No recrossing. The rate is slower and the oscillation is greater than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. The transition state is reached. Recrossing taken place for once and the product was formed and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The transition state is reached.Recrossing taken place for twice: the first time turned the products back to reactants, the second time formed the products again. Reaction rate is relatively slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the higher the reaction rate. [2] Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In the real circumstances, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by applying the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected. [2][3]

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.[2][4]

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

MRD:BaiqiuTang

2020-05-15T20:59:42Z

Bt3418: /* Reactive and unreactive trajectories */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''. [1]

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive. [1]

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively. [2]

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguished. The most obvious difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.[2]

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision. No recrossing. Relatively high reaction rate.||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. The transition state is not reached. No collision. No recrossing.||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. No recrossing. The rate is slower and the oscillation is greater than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. The transition state is reached. Recrossing taken place for once and the product was formed and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The transition state is reached.Recrossing taken place for twice: the first time turned the products back to reactants, the second time formed the products again. Reaction rate is relatively slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the higher the reaction rate. [2] Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected. [2][3]

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.[2][4]

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

MRD:BaiqiuTang

2020-05-15T20:59:07Z

Bt3418: /* Reactive and unreactive trajectories */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''. [1]

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive. [1]

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively. [2]

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguished. The most obvious difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.[2]

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision. No recrossing. Relatively high reaction rate.||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. The transition state is not reached. No collision. No recrossing.||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. No recrossing. The rate is slower and the oscillation is greater than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. The transition state is reached. Recrossing taken place for once and the product was formed and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The transition state is reached.Recrossing taken place for twice: the first time turned the products back to reactants, the second time formed the products again. Reaction rate is relatively slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. [2] Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected. [2][3]

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.[2][4]

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

MRD:BaiqiuTang

2020-05-15T20:49:15Z

Bt3418:

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''. [1]

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive. [1]

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively. [2]

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguished. The most obvious difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.[2]

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. [2] Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected. [2][3]

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.[2][4]

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

File:Figure4.jpg

2020-05-15T20:44:37Z

Bt3418: Bt3418 uploaded a new version of File:Figure4.jpg

MRD:BaiqiuTang

2020-05-15T20:38:41Z

Bt3418: /* Reference list */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.

=Reference list=
1.Bostock, L.; Chandler, S.; Rourke, R. (1985) Further pure mathematics. Oxford: Oxford University Press.

2.Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

3. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

4.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

MRD:BaiqiuTang

2020-05-15T19:23:50Z

Bt3418: /* Reference list */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.

=Reference list=
1. Laidler, K. (1987) ''Chemical kinetics''. New York city: Harper & Row.

2. Atkins, P., Keeler, J. and Paula, J. (2018) ''Atkins' Physical Chemistry''. Oxford: Oxford University Press.

3.Steinfeld, J., Francisco, J. and Hase, W. (1999) ''Chemical Kinetics And Dynamics''. Upper Saddle River, N.J.: Prentice Hall.

MRD:BaiqiuTang

2020-05-15T19:12:08Z

Bt3418: /* Discussion of energy distribution between translational and vibrational modes */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.

=Reference list=

MRD:BaiqiuTang

2020-05-15T19:00:24Z

Bt3418: /* H + HF reaction */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.

MRD:BaiqiuTang

2020-05-15T18:56:17Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==
===F + H2 reaction===

In this reaction r1 and r2 no longer share the same value and must be adjusted one by one. The value for r2 was set to 74 pm, which is equal to the bond length of H-H bond, while the value for r1 at the transition state still need to be found out in the experiment. The momentum of the system was set to zero to avoid any initial kinetic interference. The transition state is found at approximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

===H + HF reaction===

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (transition state) || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied, the F-H bondlength was fixed at 92 pm and used the same transition state of the F-H-H system. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 (Transition state) || 74 || -433.945
|-
| 92 || 50 || -112.980
|-
| 92 || 74 || -424.096
|-
| 92 || 100 || -517.680
|-
| 92 || 200 || -559.348
|-
| 92 || 500 || -560.698
|-
| 92 || 1000 || -560.700
|-
| 92 || 2000 || -560.700
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. The state energy of the reactants is taken as -560.700 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''126.755 kJ.mol-1'''.

==Discussion of energy distribution between translational and vibrational modes==
The trajectories of the reactions with early transition state are mainly controlled by translational energies. In contrast, the trajectories of the reactions with late transition state are mainly controlled by vibrational energies. The enthalpic properties of the reaction, exothermic or endothermic, are not as critical as the position of the transition state, early or late, on the impact of the reaction trajectories.

MRD:BaiqiuTang

2020-05-15T18:32:04Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The interaction between the fluorine atom The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in Figure10, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The same method applied. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

MRD:BaiqiuTang

2020-05-15T18:31:27Z

Bt3418: /* Activation energies identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

{| class="wikitable" border=1
! F-H distance (r1) pm !!H-H distance (r2) pm !!Energytot kJ.mol-1
|-
| 182 || 74 || -433.945
|-
| 200 || 74 || -434.141
|-
| 400 || 74 || -435.087
|-
| 600 || 74 || -435.100
|-
| 800 || 74 || -435.100
|-
| 1000 || 74 || -435.100
|}

At the transition state, r1 = 182 pm and r2 = 74 pm，the total energy of the system is found to be -433.945 kJ.mol-1. When the distance between fluorine atom and the hydrogen molecule is increased, the interaction between is greatly decreased and a plateau of the total energy is found at -435.100 kJ.mol-1, which will be taken as the state energy of the reactants. The interaction between the fluorine atom The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in FigureX, the bond length of the product will periodically oscillate and the momentum of the product will also change periodically within a certain range.

===H + HF reaction===
The same method applied. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

MRD:BaiqiuTang

2020-05-15T18:15:43Z

Bt3418: /* Activation energies identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===

The activation energy of the reaction can be calculated from the difference between the transition state and the reactant state. The reactant state can be approximated by eliminating the interaction between the fluorine atom and the hydrogen molecule by expanding their distance greatly. The relationship between the distance of the fluorine atom and hydrogen molecule (r1) and the energy of the state is shown in the table below:

At the transition state, r1 = 182 pm and r2 = 74 pm， the total energy of the system is found to be -433.945 kJ.mol-1. When r1 = 1000 pm and r2 = 74 pm, the total energy is found to be -435.100 kJ.mol-1. The activation energy can be found as the difference of the state energies to be '''1.155 kJ.mol-1'''.

The release of the energy of this reaction will tend to end in the form of vibration. as shown in FigureX, the bond length of the product will periodically oscillate, and the momentum of the product will also change periodically within a certain range.

===H + HF reaction===
Same method applied. The relationship between the distance of the hydrogen atom and HF molecule (r2) and the energy of the state is shown in the table below:

MRD:BaiqiuTang

2020-05-15T17:36:43Z

Bt3418: /* Activation energies identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
At the transition state, the total energy of the system is found to be -433.945 kJ.mol-1. When r2 = 74 pm and r1 → +∞, the total evergy is foound to be -435.100 kJ.mol-1. The difference of the two state energies is the activation energy, which can be found to be '''1.155 kJ.mol-1'''.
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T17:23:19Z

Bt3418: /* Thermodynamic discussion of the reactions */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction. The F + H2 reaction is an early transition state reaction with a relatively low activation energy.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process. The F + H2 reaction is an late transition state reaction with a relatively high activation energy.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:22:24Z

Bt3418: /* Thermodynamic discussion of the reactions */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is '''exothermic'''. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is '''endothermic''' and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:21:57Z

Bt3418:

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:21:23Z

Bt3418: /* Transition state identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at aaroximately '''r1 = 182 pm and r2 = 74 pm'''.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

File:Figure12.jpg

2020-05-15T15:20:30Z

Bt3418: Bt3418 uploaded a new version of File:Figure12.jpg

MRD:BaiqiuTang

2020-05-15T15:19:35Z

Bt3418: /* Transition state identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|800px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

File:Figure12.jpg

2020-05-15T15:19:14Z

Bt3418: Bt3418 uploaded a new version of File:Figure12.jpg

MRD:BaiqiuTang

2020-05-15T15:14:11Z

Bt3418: /* Transition state identification */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|400px|center|Figure12: Illustration of the transition state of F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:13:00Z

Bt3418: /* Approximation of transition state position (rts) */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed. Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|400px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:12:30Z

Bt3418: /* Approximation of transition state position (rts) */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

The transition state position (rts)is found approximately at '''90.8 pm'''. Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|400px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:05:05Z

Bt3418: /* Defining the transition state */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|400px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:04:12Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|400px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|400px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T15:03:23Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|400px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|300px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

File:Figure12.jpg

2020-05-15T15:02:35Z

Bt3418:

File:Figure10.jpg

2020-05-15T15:02:15Z

Bt3418:

File:Figure9.jpg

2020-05-15T15:01:58Z

Bt3418:

MRD:BaiqiuTang

2020-05-15T14:54:17Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|500px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

<div id="figure12">
[[File:figure12.jpg|thumb|500px|center|Figure12: Illustration of the transition state F-H-H system]]
</div>

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T14:52:27Z

Bt3418: /* Thermodynamic discussion of the reactions */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of the F-H-H system]]
</div>

===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure11">
[[File:figure11.jpg|thumb|500px|center|Figure11: Illustration of potential energy change of H + HF reaction]]
</div>

==Transition state identification==

The transition state is found at r1 = 182 pm and r2 = 74 pm.

==Activation energies identification==
===F + H2 reaction===
===H + HF reaction===

MRD:BaiqiuTang

2020-05-15T14:45:01Z

Bt3418: /* H + HF reaction */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==
===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic and should identify an increase in the potential energy of the reaction. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of H + HF reaction]]
</div>

MRD:BaiqiuTang

2020-05-15T14:42:04Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==
===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 436 and 568 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 132 kJ of heat as the change of enthalpy of the reaction.

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic. Figure10 shows a possible reaction trajectory which identified an increase in the reaction potential energy. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 132 kJ of heat during the process.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of H + HF reaction]]
</div>

MRD:BaiqiuTang

2020-05-15T14:40:58Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==
===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential energy. From the angle of bond energies, giving the H-H and H-F bond energies at 432 and 565 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 133 kJ of heat as the change of enthalpy of the reaction.

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic. Figure10 shows a possible reaction trajectory which identified an increase in the reaction potential energy. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 133 kJ of heat during the process.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of H + HF reaction]]
</div>

MRD:BaiqiuTang

2020-05-15T14:40:27Z

Bt3418:

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==
===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential. From the angle of bond energies, giving the H-H and H-F bond energies at 432 and 565 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 133 kJ of heat as the change of enthalpy of the reaction.

<div id="figure9">
[[File:figure9.jpg|thumb|500px|center|Figure9: Illustration of potential energy change of F + H2 reaction]]
</div>

===H + HF reaction===
The reaction is endothermic. Figure10 shows a possible reaction trajectory which identified an increase in the reaction potential. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 133 kJ of heat during the process.

<div id="figure10">
[[File:figure10.jpg|thumb|500px|center|Figure10: Illustration of potential energy change of H + HF reaction]]
</div>

MRD:BaiqiuTang

2020-05-15T14:37:20Z

Bt3418: /* F-H-H system */

Physical lab term3: Molecular reaction dynamics

=H-H-H system=

==Defining the transition state==
The transition state is the structure of the reactant(s) with the highest energy in the reaction process. As shown in Figure1, the transition state is the global maximum point on the reaction pathway.

<div id="figure1">
[[File:figure1.jpg|thumb|400px|center|Figure1: Illustration of transition state via reaction progress]]
</div>

In the experiment, it is critical to identify the transition state from the potential energy surface to analyse the progress of the reaction based on the transition state theory. At the transition state, any changes in r1 or r2 will destroy the vulnerable structure and resulting in a decrease in the total potential of the system in either direction of product or reactants. The transition state itself is the saddle point of the curve and is the global maxima on one dimension and the global minima in the other. As a result, the transition state is reached only when the first derivatives of the potential on both dimensions equal to zero, giving: '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0'''.

<div id="figure8">
[[File:figure8.jpg|thumb|500px|center|Figure2: Demonstration of potential change with respect to dual dimentions of r1 and r2]]
</div>

However, there also exist local maximum and minimum points on the potential energy surface, which fulfil the first derivative condition perfectly. The second derivative is used to examine the actual property of the critical points from mathematics, a maximum, minimum or saddle point. The product of the second derivatives will be negative for the transition state since it is a combination of one maximum and one minimum, giving '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''. As for local maximum and minimum, these second derivatives will both share the same sign, being positive and negative at the same time and give a product which is always positive.

So, the transition state of the H-H-H system can be defined as '''∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0''' and '''∂2V(r1)/∂r12 × ∂V2(r2)/∂r22 < 0'''.

For other local minimum points on the potential energy surface, they may fulfill the derivative condion: ∂V(r1)/∂r1 = ∂V(r2)/∂r2 = 0, but the only point which will fulfill the distance condition, r1 = r2, is the transition state point of the system. By applying both conditions, the local minimum points will be filtered off and leaving only the transition state.

==Approximation of transition state position (rts)==

<div id="figure2">
[[File:figure2.jpg|thumb|500px|center|Figure3: Demonstration of H-H-H system]]
</div>

Since the system consists of three identical H atoms, there should be no difference in the transition state of Ha-Hb reacting with Hc or Ha reacting with Hb-Hc. From the symmetry of two reacting possibilities, r1 must be equal to r2 at the transition state, giving: r1 = r2, so r1 and r2 are changed simutaneously. The transition state position (rts) is found around 90.8 pm. At the transition state, r1 must be equal to r2. The initial momentum of the hydrogen atoms are set to zero, indicating no initial speed and thus the three atoms can only oscillate due to their internuclear attraction and repulsion. As r1 and r2 setting apporach the rts, the oscillation amplitude will decrease and the distance between the atoms will tend to be less changing. When r1 = r2 = rts under the conditiion of p1 = p2 = 0, it can be observed that all three atoms are fixed and no oscillatin property is existed.Figure3 shows the changing of internuclear distance with time under the condition of momentum set to zero and innitial intermolecular distance set to 70, 80 and 90.8 pm respectively.

<div id="figure3">
[[File:figure3.jpg|thumb|1000px|center|Figure4: Impact of rinitial on the oscillating situation of the H-H-H system]]
</div>

==Difference between MEP and Dynamics reaction trajectories==
Both simulations are conducted under the same condition of the positions r1 = 91.8 pm (rts + 1), r2 = 90.8 pm (rts) and the momenta p1 = p2 = 0 g.mol-1.pm.fs-1. As is shown in Figure4, these two trajectories share many similarities and can give the same potential energy range of the reaction. However, a few differences can still be distinguihed and the first difference is the length of the trajectories, indicating that the potential energy change of the H-H-H system has stopped when r1 and r2 are around 190 amd 75 pm respectively(∂V(ri)/∂ri = 0), while the plotting from the dynamics methods will still continue to record the changing trend of r1 and r2 no matter whether the systemetic potential energy has been fixed.

<div id="figure4">
[[File:figure4.jpg|thumb|800px|center|Figure5: Simulation results of reaction trajectory via MEP and Dynamics]]
</div>

The second difference is the fluctuation trend in the two plottings, the Dynamics plotting shows more fluctuating property than the MEP plotting. This is due to that the MEP algorithm ignores many factors, for example, the atomic mass, atomic inertia, atomic momenta, which can lead to the incorrect simulation of the motion of the atoms and resulting in a more idealistic reaction trajectory which lacks the participation of oscillation. As shown in Figure5, the momentum of the system was fixed at zero when the MEP algorithm applied, while the momentum appears normal when the Dynamics algorithm is applied, which clearly shows the limitation and inaccuracy of the MEP algorithm.

<div id="figure5">
[[File:figure5.jpg|thumb|800px|center|Figure6: Comparison of the momentum of the sysytem via MEP and Dynamics]]
</div>

==The effect of switching the testing values for r1 and r2==
The values for r1 and r2 are switched in this step, giving r1 = rts and r2 = rts + 1. The comparision between the internuclear distances are compared in Figure6. The pathways are generally the same with the plottings of A-B and B-C atoms reversed, which means that the reverse of r1 and r2 will only affect the appearance of the data but share the same result of the reaction.

<div id="figure6">
[[File:figure6.jpg|thumb|800px|center|Figure7: Diatance results of reaction trajectory via MEP and Dynamics]]
</div>

The comparison between the momentum is compared in Figure7 and the pathways are generally the same as the plottings of A-B and B-C atoms reversed. These two results both indicate that the reaction will repeat itself in the identical trajectory, will give the same product and movement of the molecules from the microscopic angle and is the same repetition of itself from the macroscopic angle.

<div id="figure7">
[[File:figure7.jpg|thumb|800px|center|Figure8: Momentum results of reaction trajectory via MEP and Dynamics]]
</div>

==Reactive and unreactive trajectories==
In this section of the experiment, r1 and r2 values are fixed at 74 and 200 pm respectively and the momentum of the reactants varied. The experimental conditin and result are shown in the table below:

{| class="wikitable" border=1
! p1 / g.mol-1.pm.fs-1 !! p2 / g.mol-1.pm.fs-1 !!p1 / p2!! Etot kJ.mol-1!! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||0.502|| -414.28 || Reactive || A direct and clean reaction. No collision in the reaction. ||[[File:0-1.jpg|300px]]
|-
| -3.1 || -4.1 ||0.756|| -420.077 || Unreactive || The single atom reactant lacks kinatic energy and is driven off by repulsion before reaction. ||[[File:0-2.jpg|300px]]
|-
| -3.1 || -5.1 ||0.608|| -413.977 || Reactive || A direct and clean reaction. No collision. The rate is slower and the oscillation is greateer than the first set.||[[File:0-3.jpg|300px]]
|-
| -5.1 || -10.1 ||0.505|| -357.277 || Unreactive || Reactants have high kinatic energy and collided. Product was formed once and turned back to the reactants. ||[[File:0-4.jpg|300px]]
|-
| -5.1 || -10.6 ||0.481|| -349.477 || Reactive || A very struggle reaction. The prodect was formed twice before the termination. Reaction rate could be slow.||[[File:0-5.jpg|300px]]
|}

It can be concluded that， firstly, reactants with a lower momentum can decrease the possibility of collisions in the reaction process, form the transition state much easier and avoid the reverse reaction. If there is no collision in the reaction process, the reaction trajectory will pass the saddle point of the momentum surface and take that as the transition state. If one or multiple collisions take place in the reaction, it is not likely that the reaction trajectory will pass the saddle point and the transition state under that reaction circumstance will be shifted to somewhere else. Secondly, the rate of the reaction is connected to the momentum in an inverse proportion relationship, the lower the momentum, the faster the reaction rate. Also, from the result that the reaction can take place under the condition of -3.1 < p1 < -1.6  g.mol-1.pm.fs-1 and p2 = -5.1 g.mol-1.pm.fs-1, a hypothesis can be established that if p1 / p2 falls in the region of [0.314, 0.608] without any collision of the reaction, the reaction will be reactive. The p1 / p2 values are calculated in the table and the first three sets which do not experience collision all fall in the region and underpin the hypothesis.

==Reaction rate comparision==

The prediction of the reaction rate from the transition state theory will overestimate the reaction rate.

The diagonal of the contour plot of the potential energy graph is called the barrier of the reaction. Based on the hypothesis of transition state theory, if the reaction is reactive then the reaction trajectory should cross the reaction only once. However, recrossing is likely to take place in this reaction to cross the barrier again and take the product in its unstable status back to the reactants state, which indicates the limitation of the transition state theory. In fact, not all of the reactants at their transition state will enter the product valley. The proportion of the reactants that are able to enter the product valley can be quantified by the transmission coefficient. Systems with higher energy at the transition state will be likely to have a smaller transmission coefficient, indicating fewer molecules are likely to give the product compared to the systems with fewer energies and a greater portion of the reactants will experience the recrossing effect at higher possibilities. As for the tunnelling effect, the mass of the H-H-H system is too large to adapt the theory and thus the tunnelling effect can be neglected.

=F-H-H system=
==Thermodynamic discussion of the reactions==
===F + H2 reaction===
The reaction is exothermic. Figure9 shows a possible reaction trajectory which identified a decrease in the reaction potential. From the angle of bond energies, giving the H-H and H-F bond energies at 432 and 565 kJ.mol-1 respectively, the reaction of one mole of the reactants will form one mole of the product, during which one mole of H-H bond will break and one mole of H-F bond will be formed, and releasing 133 kJ of heat as the change of enthalpy of the reaction.

pic

===H + HF reaction===
The reaction is endothermic. Figure10 shows a possible reaction trajectory which identified an increase in the reaction potential. From the angle of bond energies, the reaction of one mole of the reactants will form one mole of the product and consume 133 kJ of heat during the process.

pic

MRD:BaiqiuTang

2020-05-15T14:18:43Z

Bt3418: /* F-H-H system */

MRD:BaiqiuTang

2020-05-15T14:12:17Z

Bt3418:

File:Figure8.jpg

2020-05-15T14:10:12Z

Bt3418: Bt3418 uploaded a new version of File:Figure8.jpg

MRD:BaiqiuTang

2020-05-15T14:06:09Z

Bt3418: /* Reaction rate comparision */

MRD:BaiqiuTang

2020-05-15T13:47:43Z

Bt3418: /* Reactive and unreactive trajectories */

MRD:BaiqiuTang

2020-05-15T13:47:23Z

Bt3418: /* Reaction rate comparision */

MRD:BaiqiuTang

2020-05-15T13:44:35Z

Bt3418:

MRD:BaiqiuTang

2020-05-15T13:21:59Z

Bt3418: /* Reactive and unreactive trajectories */

MRD:BaiqiuTang

2020-05-15T11:41:15Z

Bt3418: /* Reactive and unreactive trajectories */

MRD:BaiqiuTang

2020-05-15T11:09:12Z

Bt3418: /* Reactive and unreactive trajectories */

MRD:BaiqiuTang

2020-05-15T11:08:05Z

Bt3418: /* Reactive and unreactive trajectories */