=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F → H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. The location of the transition state in H2+F system]]

[[File:Ykx2_2_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

With Hammond's postulate, it is found that, when '''rAB''' = 0.7444 a.u. and '''rBC''' = 1.8110, the H2+F system is at the transition state (Figure 11.1 and 11.2). For the reverse reaction, the coordinator is reversed as well.

====The activation energy====

[[File:Ykx2_3_1.jpeg|thumb|400px|none|Figure 11.1. The activation energy for the H2+F system]]

Figure 11.1 shows that, in the H2+F system, the energy decreased from -103.751 kJ/mol (potential energy at transition state) to -104.02 kJ/mol (potential energy at transition state), starting as the BC distance was 0.001 a.u. larger than that of the transition state. The total energy change is ~ -0.3 kJ/mol and hence the activation energy was 0.3 kJ/mol. This is a relatively small value, meaning that this reaction needs very little external energy to become spontaneous.

[[File:Ykx2_3_2.jpeg|thumb|400px|none|Figure 11.2. The activation energy for the HF+H system]]

The energy change in the HF+H system, as given in Figure 11.2, was (-133.749) - (-103.755)= -30.0 kJ/mol when the AB distance was set 0.1 a.u. smaller than that at the transition state. Thus, the activation energy is 30.0 kJ/mol, shich means it indeed needs more energy than H2+F system to be reactive.

===Reaction dynamics===

====F + H2 system====

[[File:Ykx2_4_1.jpeg|thumb|400px|none|Figure 12.1. The momentum plot when '''rHH'''=0.74, '''rHF'''=2.3, '''pHH'''=-2 and '''pHF'''=-1.5.]]

[[File:Ykx2_4_2.jpeg|thumb|400px|none|Figure 12.2. The momentum plot when '''rHH'''=0.74, '''rHF'''=2.0, '''pHH'''=-3 and '''pHF'''=-3.]]

[[File:Ykx2_4_3.jpeg|thumb|400px|none|Figure 12.3. The momentum plot when '''rHH'''=0.74, '''rHF'''=2.3, '''pHH'''=-0 and '''pHF'''=-2.]]

From the momentum plot, it can be observed the product HF vibrates more intensively than the reactant H2. This means a increase in vibrational energy. Also, the momentum between leaving atom and the molecule after the reaction is greater than the approaching atom and the molecule before collision. This shows a increase in translational energy. Overall, the momentum increased along the reaction coordinator. Some of the potential energy transferred to kinetic energy, resulting in these increase.

Experimentally, IR spectra can be used to visualize this change. The vibration frequency remained constant, but the intensity increased. As a result, there can be seen a peak at the same position on both IR spectra for the reactant and the product, but the peak for the product should be higher than that for the reactant. Temperature is a more intuitive property that can also manifest the increase in the kinetic energy since the temperature rises.

====H + HF system====

H2 + F → H + HF is a exothermic reaction. By Hammond's postulate, it has a early transition state. The translational energy that push the reagent together gives more contribution to the reaction efficiency than the vibrational energy which though can lead the system across the transition state, it is very likely to go back if the vibrational energy is too high.

H + HF → H2 + F system, on the other hand, is a endothermic reaction with a late transitions state. Vibrational energy is more important for the reactivity energy in light of overcome the H-F bond. Translational energy does not give much merit in the formation of H-H bond.

==References==

2018-05-21T09:30:14Z

Yq816: Yq816 uploaded a new version of File:Ykx2 4 3.jpeg

File:Ykx2 4 3.jpeg

2018-05-21T09:28:41Z

Yq816:

File:Ykx2 4 2.jpeg

2018-05-21T09:25:10Z

Yq816:

MRD:YQ816

2018-05-21T09:22:07Z

Yq816: /* Molecular Reaction Dynamics: Applications to Triatomic systems */

=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F ==> H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. The location of the transition state in H2+F system]]

[[File:Ykx2_2_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

With Hammond's postulate, it is found that, when '''rAB''' = 0.7444 a.u. and '''rBC''' = 1.8110, the H2+F system is at the transition state (Figure 11.1 and 11.2). For the reverse reaction, the coordinator is reversed as well.

====The activation energy====

[[File:Ykx2_3_1.jpeg|thumb|400px|none|Figure 11.1. The activation energy for the H2+F system]]

Figure 11.1 shows that, in the H2+F system, the energy decreased from -103.751 kJ/mol (potential energy at transition state) to -104.02 kJ/mol (potential energy at transition state), starting as the BC distance was 0.001 a.u. larger than that of the transition state. The total energy change is ~ -0.3 kJ/mol and hence the activation energy was 0.3 kJ/mol. This is a relatively small value, meaning that this reaction needs very little external energy to become spontaneous.

[[File:Ykx2_3_2.jpeg|thumb|400px|none|Figure 11.2. The activation energy for the HF+H system]]

The energy change in the HF+H system, as given in Figure 11.2, was (-133.749) - (-103.755)= -30.0 kJ/mol when the AB distance was set 0.1 a.u. smaller than that at the transition state. Thus, the activation energy is 30.0 kJ/mol, shich means it indeed needs more energy than H2+F system to be reactive.

===Reaction dynamics===

[[File:Ykx2_4_1.jpeg|thumb|400px|none|Figure 12.1. The momentum plot when '''rHH'''=0.74, '''rHF'''=2.3, '''pHH'''=-2 and '''pHF'''=-1.5.]]

==References==

File:Ykx2 4 1.jpeg

2018-05-21T09:19:29Z

Yq816:

MRD:YQ816

2018-05-20T22:50:07Z

Yq816: /* The activation energy */

=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F ==> H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. The location of the transition state in H2+F system]]

[[File:Ykx2_2_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

With Hammond's postulate, it is found that, when '''rAB''' = 0.7444 a.u. and '''rBC''' = 1.8110, the H2+F system is at the transition state (Figure 11.1 and 11.2). For the reverse reaction, the coordinator is reversed as well.

====The activation energy====

[[File:Ykx2_3_1.jpeg|thumb|400px|none|Figure 11.1. The activation energy for the H2+F system]]

Figure 11.1 shows that, in the H2+F system, the energy decreased from -103.751 kJ/mol (potential energy at transition state) to -104.02 kJ/mol (potential energy at transition state), starting as the BC distance was 0.001 a.u. larger than that of the transition state. The total energy change is ~ -0.3 kJ/mol and hence the activation energy was 0.3 kJ/mol. This is a relatively small value, meaning that this reaction needs very little external energy to become spontaneous.

[[File:Ykx2_3_2.jpeg|thumb|400px|none|Figure 11.2. The activation energy for the HF+H system]]

The energy change in the HF+H system, as given in Figure 11.2, was (-133.749) - (-103.755)= -30.0 kJ/mol when the AB distance was set 0.1 a.u. smaller than that at the transition state. Thus, the activation energy is 30.0 kJ/mol, shich means it indeed needs more energy than H2+F system to be reactive.

===Reaction dynamics===

==References==

MRD:YQ816

2018-05-20T22:48:12Z

Yq816: /* The activation energy */

=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F ==> H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. The location of the transition state in H2+F system]]

[[File:Ykx2_2_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

With Hammond's postulate, it is found that, when '''rAB''' = 0.7444 a.u. and '''rBC''' = 1.8110, the H2+F system is at the transition state (Figure 11.1 and 11.2). For the reverse reaction, the coordinator is reversed as well.

====The activation energy====

[[File:Ykx2_3_1.jpeg|thumb|400px|none|Figure 11.1. The activation energy for the H2+F system]]

Figure 11.1 shows that, in the H2+F system, the energy decreased from -103.751 kJ/mol (potential energy at transition state) to -104.02 kJ/mol (potential energy at transition state), starting as the BC distance was 0.001 a.u. larger than that of the transition state. The total energy change is ~ -0.3 kJ/mol and hence the activation energy was 0.3 kJ/mol. This is a relatively small value, meaning that this reaction needs very little external energy to become spontaneous.

[[File:Ykx2_3_2.jpeg|thumb|400px|none|Figure 11.2. The activation energy for the HF+H system]]

The energy change in the HF+H system, as given in Figure 11.2, was (-133.749) - (-103.755)= -30.0 kJ/mol when the AB distance was set 0.1 a.u. smaller than that at the transition state. Thus, the activation energy is 30.0 kJ/mol, shich means it indeed needs more energy than H2+F system to be reactive.

==References==

MRD:YQ816

2018-05-20T22:14:48Z

Yq816: /* The activation energy */

=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F ==> H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. The location of the transition state in H2+F system]]

[[File:Ykx2_2_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

With Hammond's postulate, it is found that, when '''rAB''' = 0.7444 a.u. and '''rBC''' = 1.8110, the H2+F system is at the transition state (Figure 11.1 and 11.2). For the reverse reaction, the coordinator is reversed as well.

====The activation energy====

[[File:Ykx2_3_1.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in H2+F system]]

[[File:Ykx2_3_2.jpeg|thumb|400px|none|Figure 11.2. The location of the transition state in HF+H system]]

HF + H system: -103.755 -133.749

==References==

File:Ykx2 3 1.jpeg

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2018-05-20T16:29:01Z

Yq816: /* Reaction type and the bond strength */

=Molecular Reaction Dynamics: Applications to Triatomic systems=

==H + H2 system==

[[File:Ykx1_1_0.jpg|thumb|500px|none|Figure 1]]

In this model, three hydrogen named H1, H2 and H3 (or A, B and C in short correspondingly)

[[File:Ykx1.1.JPG|thumb|400px|none|Figure 2. The contour trajectory diagram of the molecules when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and the momenta '''pAB'''(0)=0 ''' pBC'''(0) = -2.7.]]

===Dynamics from the transition state region===

Before the collision, the first derivative of the potential energy against '''rAB''' is zero because they need minimum potential energy to stay bonded; the first derivative of energy '''rBC''' is greater than one because the potential energy increases during approaching. After the collision, the property is the same but in opposite way. At the transition state, both the first derivatives becomes zero because both the interactions between AB and BC changes in order to form or break a bond. However, the second derivatives are not zero. Instead, the second derivative is positive for potential energy between AB and negative between BC during the transition state. This can also be proved by the diagram. These properties can help distinguish the minimum energy and the transition states by measuring the first derivatives against both distance between A and B and distance between B and C.

===Trajectories from r1 = r2: locating the transition state===

[[File:Ykx1_2_3.JPG|thumb|400px|none|Figure 3.1]]
[[File:Ykx1_2_4.JPG|thumb|400px|none|Figure 3.2]]

The diagrams (Figure 3.1 and 3.2) when '''rAB'''(0) = 0.74 '''rBC'''(0) = 2.30 and '''pAB'''(0)=0 ''' pBC'''(0) = -2.7 was inspected. The interception of A-B and B-C line indicates that A and C are as far away from B,so the transition state position is near this point which is approximately (3.8, 9.2).

[[File:Ykx1_2_2.JPG|thumb|400px|none|Figure 4]]

Therefore, '''rAB'''='''rBC'''=0.90 was first tried as shown in Figure 4. The wave-like diagram indicates the vibration of A and C around the transition state position. The starting part shows that the molecules first went apart, meaning that the attempt is smaller than the real '''rts'''

[[File:Ykx1_2_1.JPG|thumb|400px|none|Figure 5]]

'''rAB'''='''rBC'''=0.91 is then tried (Figure 5). There is still vibrations shown although with lower amplitude. However, This number is slightly larger since the lines start with decrease. Therefore, the real value should lie between 0.90 and 0.91. A number between 0.905 and 0.910 is tried.

[[File:Ykx1_2_5.JPG|thumb|400px|none|Figure 6]]

More attempts with higher accuracy was tried and the generated diagrams were judged, until the lines become flat as shown in Figure 6, which means the atoms fix in the location where it is initially. The value under this situation (0.90775 ±0.00005 a.u.) is then the transition state position. (As the diagrams when '''rAB''' = '''rBC''' is equal to 0.9077 and 0.9078 both show slight vibrations, but any values between gives almost as flat lines, the uncertainty of ±0.00005 a.u. is deduced here.)

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

'''rBC''' was set to be '''rts''' (0.90775 a.u.), and '''rAB''' was set to be rts+0.01 (0.91775 a.u.), while the initial momenta among the atoms are all 0. The results were calculated using both minimum energy pathway (mep) and dynamic (500 steps).

[[File:Ykx1_3_1.jpeg|thumb|400px|none|Figure 7.1 The contour plot of the trajectory calculated by mep]]

[[File:Ykx1_3_2.jpeg|thumb|400px|none|Figure 7.2 The contour plot of the trajectory calculated by dynamic]]

[[File:Ykx1_3_3.jpeg|thumb|400px|none|Figure 7.3 The energy plot of the trajectory calculated by mep]]

[[File:Ykx1_3_4.jpeg|thumb|400px|none|Figure 7.4 The energy plot of the trajectory calculated by dynamic]]

Both diagrams (Figure 7.1 and Figure 7.2) tell us that the reaction had passed the transition state, and the H2-H3 (the product) bond was formed.

In the dynamic calculation The potential energy was transferred to kinetic energy which was 0 initially. (Figure 7.3) The kinetic energy was contributed by the vibrational energy of the H2-H3 molecule and the translational energy. The total energy is always reserved. This result is closer to the reality.

In the mep calculation, however, the potential energy decreased as shown in Figure 7.4, but this calculation doesn't have to take account of the kinetic energy as time elapse and the initial momenta are all 0 so that the kinetic energy remains 0. This makes the total energy not concstant. This calculation cannot give the real reaction dynamic but a clearer presentation of how the potential energy can reach and progress for every coordinators so that the activation energy can be found.

===Reactive and unreactive trajectories===

For the same initial positions '''rAB''' = 0.74 and '''rBC''' = 2.0, different results from different initial momenta were obtained:

{| class="wikitable" border=1
! pAB !! pBC !! Total Energy !! Reactivity !! Contour plot !! Description
|-
| -1.25 || -2.5 || -99.018 || Reactive || [[File:Ykx1_4_1.jpeg|thumb|350px|none|Figure 8.1]]
| The particles possessed the momenta so that initially the H2 molecule had little vibrational energy but the total translational energy was strong enough to lead the particles to pass the transition state. After this point, some of the translational energy turned to the vibrational energy.
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:Ykx1_4_2.jpeg|thumb|350px|none|Figure 8.2]]
| The relative velocity between the H2 molecule and the H atom was too small and hence the kinetic energy of the system was too small as well. The system did not possess enough energy for crossing the transition state. The reaction was hence not able to happen.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:Ykx1_4_3.jpeg|thumb|350px|none|Figure 8.3]]
|This is similar to the first situation, but as the initial relative speed between A and B in the molecule was higher, the molecule started with vibrations.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:Ykx1_4_4.jpeg|thumb|350px|none|Figure 8.4]]
|The initial translational energy here was too high so that when it pass the transition state, the vibration of the product was so strong that exceeded the speed of the departing atom A. Consequently, A and B fell together again while particle C was far away enough to leave the molecule. This allowed the system re-cross the transition point back to the product. However, after the collision, particle C lost its kinetic energy which turned into the vibrational energy between A and B.
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:Ykx1_4_5.jpeg|thumb|350px|none|Figure 8.5]]
|This is similar to the fourth situation. But because of the higher kinetic energy, the system was able to re-cross the transition state twice. This gave rise to large vibrational energy after collision.
|}

===Assumptions of Transition State Theory===

In the transition state theory (TST), the analysis on the kinetics of the hydrogen particles is mostly based on classical mechanics. It does not take account of the quantum-tunneling effect, and the relative effect of the electrons on the nucleus is neglected (in other words, Born-Oppenheimer approximation is applied).<ref>T. Bligaard, J.K. Nørskov, in Chemical Bonding at Surfaces and Interfaces, 2008</ref> This greatly simplifies the calculation while the errors that may occur are negligible.

TST also assumes the equilibrium to be maintained during the reaction, which means that the ideal Boltzmann distribution keeps applied along the reaction coordinator. <ref>6 – Reactivity in Thermalized Systems, Luis Arnaut, Sebastiao Formosinho, Hugh Burrows, [https://www.elsevier.com/books/chemical-kinetics/arnaut/978-0-444-52186-6 "in Chemical Kinetics: From Molecular Structure to Chemical Reactivity"], 2007</ref>. If the equilibrium is disturbed towards the reactants, the TST would predict a higher activation energy, and vice versa.

The other postulate is that the reactants can not re-cross the transition state and go back to the products.<ref>[https://courses.lumenlearning.com/boundless-chemistry/chapter/activation-energy-and-temperature-dependence/ "Activation Energy and Temperature Dependence, Boundless Chemistry"]</ref> However, inreality, as we can see in the calculation above, the results can be greatly biased if the system is under some extreme conditions such as high temperature when the reactants can re-cross the transition state several times, so that it is more complex to predict the precise reaction rate.

==F - H - H system==

[[File:Ykx2_0_1.jpg|thumb|350px|none|Figure 9.1. H2+F system]]

[[File:Ykx2_0_2.jpg|thumb|350px|none|Figure 9.2. HF+H system]]

===PES inspection===
====Reaction type and the bond strength====

[[File:Ykx2_1_1.jpeg|thumb|400px|none|Figure 10.1. Surface plot of the trajectory of H2+F system]]

[[File:Ykx2_1_2.jpeg|thumb|400px|none|Figure 10.2. Surface plot of the trajectory of HF+H system]]

[[File:Ykx2_1_3.jpeg|thumb|400px|none|Figure 10.3. Energy plot of the trajectory of H2+F system]]

[[File:Ykx2_1_4.jpeg|thumb|400px|none|Figure 10.4. Energy plot of the trajectory of HF+H system]]

From Figure 10.1 and 10.3, it can be seen that the minimum potential of the reaction decreases along the reaction coordinators. During the reaction the potential energy transfers to the kinetic energy reflected as heat macroscopically. This is hence a exothermic reaction.

Figure 10.2 and 10.4 shows that the minimum potential energy increases during the reaction. Energy (i.e. heat) is required to enable the successful reaction. Thus, the reaction type is endothermic.

According to the literature value, the bond strength of H-H is 436 kJ/mol and that of H-F is 568 kJ/mol.<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html"Bond Lengths and Energies"] Chung Chieh</ref> Fot the reaction H2 + F ==> H + HF, the H-H bond needs 436 kJ/mol of energy to be broken while the formation of the H-F bond releases 569 kJ/mol of energy. This manifests that this is a exothothermic reaction. For the reverse reaction, the process and the result are opposite.

====The transition state====

[[File:Ykx2_2_1.jpeg|thumb|400px|none|Figure 11.1. Contour plot of the trajectory of H2+F near transition state]]

==References==

File:Ykx2 2 2.jpeg

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