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MRD:01566447

2020-05-22T19:47:20Z

Nwt18: /* Polanyi's Empirical Rules */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.<ref name="Polanyi" /> For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

Exothermic FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 g.mol-1.pm.fs-1 and pBC=-6.1 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 g.mol-1.pm.fs-1 and pBC=0.2 g.mol-1.pm.fs-1]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

Endothermic FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 g.mol-1.pm.fs-1 and pBC=-10 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 g.mol-1.pm.fs-1 and pBC=-1 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 g.mol-1.pm.fs-1 and pBC=-1 g.mol-1.pm.fs-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polanyi's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
<ref name="Polanyi">K. J. Laidler, in Chemical Kinetics, Harper-Collins, 3rd ed, 1951, ch.12, pp. 460-471.</ref>
</references>

MRD:01566447

2020-05-22T19:46:04Z

Nwt18: /* Polanyi's Empirical Rules */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.<ref name="Polanyi" /> For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 g.mol-1.pm.fs-1 and pBC=-6.1 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 g.mol-1.pm.fs-1 and pBC=0.2 g.mol-1.pm.fs-1]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 g.mol-1.pm.fs-1 and pBC=-10 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 g.mol-1.pm.fs-1 and pBC=-1 g.mol-1.pm.fs-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 g.mol-1.pm.fs-1 and pBC=-1 g.mol-1.pm.fs-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polanyi's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
<ref name="Polanyi">K. J. Laidler, in Chemical Kinetics, Harper-Collins, 3rd ed, 1951, ch.12, pp. 460-471.</ref>
</references>

MRD:01566447

2020-05-22T19:36:23Z

Nwt18: /* References */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.<ref name="Polanyi" /> For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 and pBC=0.2]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 and pBC=-10]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 and pBC=-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 and pBC=-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polanyi's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
<ref name="Polanyi">K. J. Laidler, in Chemical Kinetics, Harper-Collins, 3rd ed, 1951, ch.12, pp. 460-471.</ref>
</references>

MRD:01566447

2020-05-22T19:35:57Z

Nwt18: /* References */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.<ref name="Polanyi" /> For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 and pBC=0.2]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 and pBC=-10]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 and pBC=-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 and pBC=-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polanyi's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
<ref name="Polanyi">K. J. Laidler, in Chemical Kinetics, Harper-Collins, 3rdedn, 1951, ch.12, pp. 460-471.</ref>
</references>

MRD:01566447

2020-05-22T19:32:01Z

Nwt18: /* Polanyi's Empirical Rules */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.<ref name="Polanyi" /> For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 and pBC=0.2]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 and pBC=-10]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 and pBC=-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 and pBC=-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polanyi's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T19:28:57Z

Nwt18: /* Mechanism of Energy release */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====
[[File:01566447_Plot2_momenta.png|500px|thumb|center|Momenta-Time graph of a reactive trajectory for the reaction of F + H2]]
From the momenta graph above, we can see the oscillations of the product molecule is much larger than that of the reactant molecules. This implies that during the reaction, the decrease in potential energy releases energy in the form of kinetic energy, more specifically vibrational energy as shown by the large oscillations of product molecule after TS is crossed. This gain in vibrational energy can be experimentally measured using Infrared Chemiluminescence. The intensities of the IR absorption peaks can be used to calculate the relative populations of the product at each vibrational energy level. Products with more vibrational energy will populate higher energy levels, giving rise to overtones which has a smaller wavenumber compared to the fundamental peak. Over time, the overtones will become smaller while the fundamental peak becomes larger. This is because product molecules in the excited state release kinetic energy as heat and fall to the ground state, less product is at higher vibrational states.

This provides an alternative way to experimentally confirm the mechanism of energy release. The heat released, converted from kinetic energy can be measured using calorimetry. The increase in temperature due to kinetic energy is measured and used to calculate the amount of kinetic energy released. However, we are unable to differentiate between vibrational kinetic energy and translational kinetic energy using this method. Therefor, infrared chemiluminescence is preferred over calorimetry.

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC. For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 and pBC=0.2]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 and pBC=-10]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 and pBC=-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 and pBC=-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polyani's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T19:15:24Z

Nwt18: /* Polanyi's Empirical Rules */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC. For both cases, there must be enough kinetic energy(vibrational and translational) to overcome the activation barrier for reaction to proceed.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|400px|thumb|center|Plot 1: Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|400px|thumb|center|Plot 2: Reactive trajectory, pAB=-1.6 and pBC=0.2]]

The momenta along AB represents translational energy of F, whereas momenta along BC represents vibrational energy of H2. In Plot 1, momenta along BC is larger, which means more vibrational energy over translational energy was used to drive the reaction, but it did not succeed. On the other hand, in Plot 2, momenta of AB is larger which means more translational energy instead of vibrational energy was used to drive the reaction. The reaction was successful as seen from the reactive trajectory. This is in line with Polanyi's Empirical rules, where an exothermic reaction with early TS is more effectively driven by translational energy.

FA-HB + HC:
For positions rAB=92pm and rBC=225pm,
[[File: 01566447_Plot3.png|400px|thumb|center|Plot 3: Unreactive trajectory, pAB=0 and pBC=-10]]
[[File: 01566447_Plot4.png|400px|thumb|center|Plot 4: Unreactive trajectory, pAB=-20 and pBC=-1]]
[[File: 01566447_Plot5.png|400px|thumb|center|Plot 5: Reactive trajectory, pAB=-29 and pBC=-1]]

The momenta along AB represents vibrational energy of F-H, while the momenta along BC represents the translational energy of the lone H atom. For Plot 3, momenta along BC is quite high, which means the system has a lot of translational energy which is ineffective in driving the reaction, resulting in an unreactive trajectory. For Plot 4, momenta along BC is reduced whereas momenta along AB increases, which raises the amount of vibrational energy compared to translational energy. However, there is still not enough vibrational energy to yield a reactive trajectory. Lastly, in Plot 5, momenta along AB is large enough to effectively drive the reaction and produce a reactive trajectory. This is consistent with Polyani's empirical rules which states endothermic reactions with late TS are more effectively driven by vibrational energy.

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T18:59:24Z

Nwt18: /* Polanyi's Empirical Rules */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====

====Polanyi's Empirical Rules====
For exothermic reactions with early TS, more translational energy over vibrational energy tends to lead to more effective reactions. This can be illustrated by the cases studied using the reaction of FA + HB-HC. Whereas for endothermic reactions with late TS, more vibrational energy over translational energy usually leads to more effective reactions and is shown by the cases studied for reaction of FA-HB + HC.

FA + HB-HC:
For positions rAB=190pm and rBC=74pm,
[[File: 01566447_Plot1.png|600px|thumb|Unreactive trajectory, pAB=-1 and pBC=-6.1]]
[[File: 01566447_Plot2.png|600px|thumb|Reactive trajectory, pAB=-1.6 and pBC=0.2]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

File:01566447 Plot2.png

2020-05-22T18:46:24Z

Nwt18:

File:01566447 Plot5.png

2020-05-22T18:45:28Z

Nwt18:

File:01566447 Plot4.png

2020-05-22T18:45:12Z

Nwt18:

File:01566447 Plot3.png

2020-05-22T18:44:54Z

Nwt18:

File:01566447 Plot2 momenta.png

2020-05-22T18:44:39Z

Nwt18:

File:01566447 Plot1.png

2020-05-22T18:42:49Z

Nwt18:

MRD:01566447

2020-05-22T18:31:32Z

Nwt18: /* Reaction Dynamics */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
====Mechanism of Energy release====

====Polanyi's Empirical Rules====

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T18:18:26Z

Nwt18: /* Activation energy */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
The energies of the TS, reactants and products were calculated using an mep trajectory and shown in the figure below:
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is 1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is 126.564 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T18:15:39Z

Nwt18: /* Activation energy */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: 01566447_activationE.png|600px|thumb|center|Energy against reaction coordinate graph and calculation of activation energies]]

The approximate activation energy for the exothermic reaction(F + H2) is +1.118 kJ mol-1.

The approximate activation energy for the endothermic reaction(FH + H) is +126.564 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

File:01566447 activationE.png

2020-05-22T18:12:42Z

Nwt18:

MRD:01566447

2020-05-22T17:47:13Z

Nwt18: /* Transition State Theory (TST) */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:46:16Z

Nwt18: /* Dynamics from Transition State region */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one Hessian eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:44:41Z

Nwt18: /* Locating TS position */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:43:41Z

Nwt18: /* PES inspection */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:42:40Z

Nwt18: /* Transition state */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state. rAB=180.6pm and rBC=74.5pm.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]

====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:40:33Z

Nwt18: /* PES inspection */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below(left to right). As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447_endoexo.png|600px|thumb|center|PES of F + H2 and FH + H system]]

Whereas the reaction in the opposite direction(right to left): FA-HB + HC -> FA + HB-HC, shows an increase in energy, indicating the reaction is endothermic.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located with the help of Hammond's postulate. Below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this is a saddle point and the transition state.

[[File: 01566447TS_parameters.PNG|500px|thumb|center|Parameters and calculated values for the TS of F-H-H system]]

A summary of the reaction showing the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px|center]]
====Activation energy====
[[File: ]]
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

File:01566447 endoexo.png

2020-05-22T17:32:57Z

Nwt18:

MRD:01566447

2020-05-22T17:28:57Z

Nwt18: /* How bond strength affects energetics of reaction */

== H + H2 system ==
===Dynamics from Transition State region ===
HA + HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px|thumb|center|Potential energy surface for reaction of HA + HB-HC.]]

On the PES, the transition state (TS) is a first order saddle point. It is the maxima on the minimum energy path, and the minima in the directions orthogonal to this path.

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the TS of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. Since Force=∂V(r)/∂r, zero gradient means zero force along AB and BC. To distinguish the TS from a local minima on the PES, we can use the Hessian matrix to compute the Hessian eigenvalues. For local energy minima, both Hessian eigenvalues will be positive. For a TS, one eigenvalue is positive while the other is negative.

====Locating TS position====
If initial rAB and rBC distances are set to the distances corresponding to the TS, with no initial momentum, the force acting on the atoms is 0 thus would remain there indefinitely. Since the H + H2 system is symmetrical, rAB=rBC at the TS.

[[File:01566447r_ts_2.png|500px|thumb|center|Internuclear distance-time graph of rAB=rBC=90.8pm at zero momenta, internuclear distances at TS]]

The TS position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Trajectories from rAB = rts, rBC = rts+1pm===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below. Products formed are HA-HB and HC.

====Trajectory from 'Dynamics':====
{| class="wikitable" border="1"
| [[File:01566447dynamics_contour.png|thumb|Contour plot]] || [[File:01566447dynamics_energy.png|thumb|Energy-Time plot]]
|}

The trajectory calculated using the 'Dynamic' option shows greater oscillation, which means the product molecules vibrate and possess kinetic energy. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
{| class="wikitable" border="1"
| [[File:01566447mep_contour.png|thumb|Contour plot]] || [[File:01566447mep_energy.png|thumb|Energy-Time plot]]
|}

Whereas in the 'mep' contour plot, the trajectory is smooth and follows the valley floor of the PES, kinetic energy throughout the reaction path is constantly 0 as shown in the Energy-Time graph. Atoms possess no kinetic energy to vibrate as a result of removing the inertial effect, thus no oscillation is observed from the trajectory on the contour plot. This is not representative of reality because atoms have mass and thus have inertia.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.

====Changing initial conditions====
If we used the initial conditions rAB = rts+1pm, rBC = rts, reaction would proceed in the opposite direction instead, products formed are HA and HB-HC.

After setting the initial parameters with the final positions and reversing the signs of the momenta, trajectories from the Dynamics calculation differs from that of mep. For the dynamics calculation, reaction proceeds back towards the initial position. For the mep calculation, the pathway proceeds in the opposite direction and continues to follow the valley floor of the PES.

===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || HA and HB-HC has sufficient energy to overcome the energy barrier to form the TS and form the products HA-HB and HC. The products possess a lot of vibrational energy as shown by the large oscillations after TS is crossed.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once). Atoms must possess the right amount of vibrational energy and possess the correct orientation for a reaction to proceed successfully.

===Transition State Theory (TST)===
TST has certain assumptions which may lead to deviations in reaction rate values from experimental values.

Assumptions<ref name="TSTassumptions" />:
1)Born-Oppenheimer Approximation where electronic and nuclear motions can be separated.

2)Reactant molecules are distributed among their states according to the Maxwell-Boltzmann distribution.

3)Molecular systems that have crossed the TS in the direction of the products cannot recross the TS again to reform reactants

4)In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as translation, ignoring any quantum tunneling effects.

5)Even in the absence of equilibrum between the reactant and product molecules, the TS that are becoming products are distributed among their states according to Maxwell-Boltzmann laws.

Assumption 4 which states that quantum effects are ignored in the TST does not hold in reality, because quantum tunneling effects do occur, which means reactant molecules do not need to overcome the activation energy and cross the TS to form products. Hence in reality, experimental reaction rates would be larger than those predicted using the TST.

Assumption 3 is also another assumption that does not represent what happens in reality. As seen in the table of reactive and unreactive trajectories, the 4th reaction(p1=-5.1, p2=-10.1) is an example of how the TS can be crossed again even after formation of products. Leading to lower experimental reaction rates compared to the predicted values.

Effect of recrossing of TS is likely to be larger than that of quantum tunneling. Hence the overall effect is that the predicted rate constants using TST is an overestimation of the experimental rate constants.

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}

As shown in the table above, the bond strength of H-H is weaker than F-H. This means that for the reaction of F + H2, energy released from forming the F-H bond is greater than the energy required to break the H-H bond. The reaction releases energy thus it is exothermic.

On the other hand, the reaction of FH + H is endothermic because energy required to break the F-H bond is greater than the energy released from the cleavage of the H-H bond. Overall, energy is required for the reaction to proceed successfully.

====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="TSTassumptions">J. I. Steinfeld, J. S. Francisco, W. L. Hase, in Chemical Kinetic and Dynamics, Prentice-Hall, 2nd ed., 1998, ch. 10, pp. 287-321.</ref>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T17:19:49Z

Nwt18: /* References */

MRD:01566447

2020-05-22T17:13:18Z

Nwt18: /* Transition State Theory (TST) */

MRD:01566447

2020-05-22T16:52:14Z

Nwt18: /* Reactive and Unreactive Trajectories */

MRD:01566447

2020-05-22T16:46:07Z

Nwt18: /* Trajectories from rAB = rts+δ, rBC = rts */

MRD:01566447

2020-05-22T16:37:59Z

Nwt18: /* Trajectories from r1 = rts+δ, r2 = rts */

MRD:01566447

2020-05-22T16:36:56Z

Nwt18: /* Trajectories from r1 = rts+δ, r2 = rts */

MRD:01566447

2020-05-22T16:31:31Z

Nwt18: /* Trajectory from 'mep': */

MRD:01566447

2020-05-22T16:28:47Z

Nwt18: /* Trajectory from 'mep': */

MRD:01566447

2020-05-22T16:27:16Z

Nwt18: /* Trajectory from 'Dynamics': */

MRD:01566447

2020-05-22T16:23:54Z

Nwt18: /* Trajectory from 'Dynamics': */

MRD:01566447

2020-05-22T16:20:59Z

Nwt18: /* Calculation of reaction path and trajectories */

MRD:01566447

2020-05-22T16:17:32Z

Nwt18: /* Locating TS position */

MRD:01566447

2020-05-22T16:16:27Z

Nwt18: /* Dynamics from Transition State region */

MRD:01566447

2020-05-22T16:14:44Z

Nwt18: /* Dynamics from Transition State region */

MRD:01566447

2020-05-22T16:12:10Z

Nwt18: /* Dynamics from Transition State region */

MRD:01566447

2020-05-22T16:11:27Z

Nwt18: /* H + H2 system */

MRD:01566447

2020-05-22T09:59:08Z

Nwt18: /* Reaction Dynamics */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. It can be identified by looking at forceAB and forceBC(F=∂V(r)/∂r) which must be at 0(or very close to 0). The Hessian eigenvalues w2 must have opposing signs, i.e. one is positive and the other is negative, this distinguishes the TS from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===
Transition State Theory operates on a Classical model and does not take into account quantum effects. Some of the key assumptions are:

1)

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy. By varying the initial energy of system, we can measure how much of the excess energy is converted to vibrational and how much is translational.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T09:55:22Z

Nwt18: /* Transition State Theory (TST) */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. It can be identified by looking at forceAB and forceBC(F=∂V(r)/∂r) which must be at 0(or very close to 0). The Hessian eigenvalues w2 must have opposing signs, i.e. one is positive and the other is negative, this distinguishes the TS from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===
Transition State Theory operates on a Classical model and does not take into account quantum effects. Some of the key assumptions are:

1)

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T06:20:02Z

Nwt18: /* The Transition State */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES). The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0, where r is distance and p is momenta).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(rAB)/∂rAB=∂V(rBC)/∂rBC=0. It can be identified by looking at forceAB and forceBC(F=∂V(r)/∂r) which must be at 0(or very close to 0). The Hessian eigenvalues w2 must have opposing signs, i.e. one is positive and the other is negative, this distinguishes the TS from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

File:01566447F HH contour.png

2020-05-22T06:11:19Z

Nwt18:

MRD:01566447

2020-05-22T06:10:56Z

Nwt18:

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES), where the potential energy is plotted as a function of both rAB and rBC. The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(r)/∂r=0. It can be identified by looking at calculated Forces which must be at 0(or very close to 0), and the Hessian eigenvalues w2 where one must be positive and the other must be negative. If the Hessian is indefinite, it is a saddle point. This distinguishes it from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===
In light of the fact that energy is conserved, the energy released via the exothermic reaction contributes to the vibrational energy of F-H molecule as seen by the wavy trajectory line after the TS. To test this, we can conduct experiments and measure the vibrational energy of the product using Infra-red spectroscopy.

[[File:01566447F_HH_contour.png]]

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T06:01:40Z

Nwt18: /* F-H-H system */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES), where the potential energy is plotted as a function of both rAB and rBC. The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(r)/∂r=0. It can be identified by looking at calculated Forces which must be at 0(or very close to 0), and the Hessian eigenvalues w2 where one must be positive and the other must be negative. If the Hessian is indefinite, it is a saddle point. This distinguishes it from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

===Reaction Dynamics===

==Distribution of Energy==

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T04:28:45Z

Nwt18: /* Transition state and activation energy */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES), where the potential energy is plotted as a function of both rAB and rBC. The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(r)/∂r=0. It can be identified by looking at calculated Forces which must be at 0(or very close to 0), and the Hessian eigenvalues w2 where one must be positive and the other must be negative. If the Hessian is indefinite, it is a saddle point. This distinguishes it from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]
====Activation energy====
The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1 while the activation energy for the endothermic reaction(FH + H) is +126.717 kJ mol-1.

==Distribution of Energy==

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>

MRD:01566447

2020-05-22T04:27:39Z

Nwt18: /* Transition state and activation energy */

== H + H2 system ==
===The Transition State ===
HA +HB-HC -> HA-HB + HC

The progress of the reaction above can be mapped on a potential energy surface (PES), where the potential energy is plotted as a function of both rAB and rBC. The trajectory shown on the potential energy surface represents the change in potential energy, rAB and rBC as reaction proceeds given an initial set of parameters (rAB, rBC,pAB, pBC at time=0).

[[File:01566447HH2Surface_Plot1_1.PNG|900px]]

Assuming a set of initial parameters that result in the successful formation of the products HA-HB and HC, the trajectory shown passes though a saddle point which represents the transition state of the reaction. It is mathematically defined as ∂V(r)/∂r=0. It can be identified by looking at calculated Forces which must be at 0(or very close to 0), and the Hessian eigenvalues w2 where one must be positive and the other must be negative. If the Hessian is indefinite, it is a saddle point. This distinguishes it from a local minima on the PES.

[[File:01566447r_ts_2.png|500px]]

The transition state position rts is approximately 90.8 pm, where rts=rAB=rBC. The parameters set for the plotted trajectory rAB=rBC=90.8 pm and pAB=pBC=0 g.mol-1.pm.fs-1. Oscillation is at a minimum as shown by the horizontal lines on the Internuclear distance-time graph above, which means the system is at or minutely oscillating at the midpoint on the ridge, the atoms are stationary and system is at the transition state: rts=(rAC-rAB)=(rAC-rBC)=90.8 pm as illustrated in the graph.

===Calculation of reaction path and trajectories===
There are two ways to calculate the trajectory, 'Dynamics' and 'Minimum energy path(mep)', the latter of which resets the momenta and velocity to 0 at every time step. Results obtained from using the 2 different methods are shown below.

====Trajectory from 'Dynamics':====
[[File:01566447dynamics_contour.png]] [[File:01566447dynamics_energy.png]]

The trajectory calculated using the 'Dynamic' option are more zig-zag which means the atoms vibration. Potential energy is converted to kinetic energy as shown in the Energy-Time graph where the kinetic energy increases whilst potential energy decreases.

====Trajectory from 'mep':====
[[File:01566447mep_contour.png]] [[File:01566447mep_energy.png]]

Whereas in the 'mep' contour plot, the trajectory is smooth and atoms do not vibrate, kinetic energy throughout the reaction path is constant as shown in the Energy-Time graph.

The mep connects the 2 energy minima on the PES and the energy maxima along the MES corresponds to the saddle point that represents the transition state of the reaction.
===Reactive and Unreactive Trajectories===
{| class="wikitable" border="1" align=”center”
|+ Table of calculated trajectories from different combinations of momenta, using 'Dynamics' calculation type:
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ mol-1 !! Reactive? !! Description of Dynamics !! Illustration of trajectory
|-
| -2.56 || -5.1 || -414.280 || Yes || As HA approaches HB-HC, AB distance decreases with very small vibrations due to the low momenta, eventually collides with HB-HC with enough energy to cross the transition state(TS) which breaks the HB-HC bond and forms the HA-HB bond, thus BC distance increases. The trajectory after the TS oscillates more due to the greater momenta along BC. || [[File:01566447Surface_Plot_1.png|300px]]
|-
| -3.1 || -4.1 || -420.077 || No || As HA approaches HB-HC, AB distance decreases. The trajectory is wavy due to the momenta along AB. However, system does not have enough energy to cross reach the TS before HA starts moving away from HB-HC without any H-H bond formation or cleavage. || [[File:01566447Surface_Plot_2.png|300px]]
|-
| -3.1 || -5.1 || -413.977 || Yes || Dynamics of this system is very similar to the 1st system. The difference is the momenta along AB is larger than that of the 1st system, resulting in a more wavy trajectory before the TS which means more vibrations along AB. Reaction crosses the TS once and forms the products HA-HB and HC. || [[File:01566447Surface_Plot_3.png|300px]]
|-
| -5.1 || -10.1 || -357.277 || No || As HA approaches HB-HC, due to the larger momenta, there are larger vibrations along AB, enough energy to reach the TS and form HA-HB and HC. However, momenta along BC is too large, resulting in very large vibrations, which is sufficient for system to cross the TS again and break the HA-HB bond to reform the reactants, HA and HB-HC. || [[File:01566447Surface_Plot_4.png|300px]]
|-
| -5.1 || -10.6 || -349.477 || Yes || Dynamics of this system is very similar to the 4th system above. The difference is that after the TS is recrossed to form the reactants, HA and HB-HC has sufficient energy to overcome the energy barrier again to form the TS and form the products HA-HB and HC.|| [[File:01566447Surface_Plot_5.png|300px]]
|}

From the table, we can conclude that the hypothesis that trajectories starting with the same positions but with higher values of momenta (higher kinetic energy) would be reactive is false. Comparing the 1st and the 4th graph, we can see the 4th has greater momenta thus more kinetic energy, but the trajectory is unreactive, while the 1st graph shows a reactive trajectory. This means that whether a trajectory is reactive or not is not dependent only on the size of its values, because the TS can be recrossed. The momentas must be of the right combination such that the resulting products are formed after crossing the TS(once or more than once).

===Transition State Theory (TST)===

== F-H-H system ==
===PES inspection===
The reaction of FA + HB-HC -> FA-HB + HC is as shown in the PES below. As FAHB distance decreases, the potential energy decreases, indicating reaction in this direction is exothermic.

[[File:01566447F_HH_Surface_Plot.png]]

Whereas the reaction in the opposite direction, FA-HB + HC -> FA + HB-HC is endothermic. This is because as FAHB distance increases, energy increases.

====How bond strength affects energetics of reaction====
{| class="wikitable" border="1"
|+ Bond strengths
! Bond !! Bond energy (kJ mol-1)<ref name="BondStrength" />
|-
| H-H || 436
|-
| F-H || 569
|}
====Transition state and activation energy====
The approximate transition state was located, below are the parameters, force, energy and Hessian eigenvalues for the transition state. The forces are close to or at zero, the Hessian eigenvalues have opposing signs which indicates this saddle point is the transition state.

[[File: 01566447TS_parameters.PNG|500px]]

A summary of the structure of the reactants, products and transition state is shown below:

[[File:01566447FHH_Reaction.jpg|700px]]

The approximate activation energy for the exothermic reaction(F + H2) is +1.119 kJ mol-1-1.

==Distribution of Energy==

==References==
<references>
<ref name="BondStrength">Chemistry LibreTexts, https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_202_-_General_Chemistry_II/Unit_5%3A_Molecular_Shape/5.1%3A_Bond_Strength, (accessed May 2020).</ref>
</references>