MRD:aj3318

2020-05-08T22:46:08Z

Aj3318: /* Exercise 2: F + H2 System */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 is an exothermic and HF + H is an endothermic reaction. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

The approximate position of TS is at '''r1'''=181.1 and '''r2'''=74.5 pm, which represent the intermolecular distance between F-H and H-H respectively. The eigenvalues of the Hessian matrix were -0.002 and +0.332 for the eigenvectors (+1.000, -0.023), (-0.023, +1.000) respectively. The potential energy at the TS is -433.980 kJ mol-1.

The activation energy of the system was estimated by comparing the potential energy of the TS to the potential energy of the products when the system is slightly perturbed from TS. The following demonstrates how the calculation is performed.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Figure
|-
|MEP
|1000
|0.5
|181.1
|74.5
|0
|0
|15
|-
|MEP
|1000
|0.5
|181.2
|74.5
|0
|0
|16
|}
[[File:Figure 15 Ea of H HF to HH F.PNG|thumb|700px|none|Figure 15. (a) Potential energy surface of H+HF to H2+F. (b) Energy/Time graph. ]]

[[File:Figure 16 Ea of HH F to H HF.PNG|thumb|700px|none|Figure 16. (a) Potential energy surface of H2+F to H+HF. (b) Energy/Time graph. ]]

By iterating the calculations, the potential energy was estimated to the 6 significant figures. This involved finding the trajectory down the MEP from an initial condition, inserting the last geometry as the initial condition, and repeating until the potential energy stabilises.
{| class="wikitable"
!System
!Ep/kJmol-1
|-
|HF + H
|<nowiki>-560.699</nowiki>
|-
|F + H2
|<nowiki>-435.099</nowiki>
|-
|F-H-H
|<nowiki>-433.980</nowiki>
|}
The activation energy of HF + H -> H + H2 is +126.719 kJ mol-1, and F + H2 -> HF + H is +1.119 kJ mol-1. 

=== Reaction Dynamics ===
The following reaction condition resulted in the reaction from H2 + F -> HF + H.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
|-
|Dynamics
|1000
|0.5
|200
|75
|<nowiki>-1.0</nowiki>
|0.3
|}
[[File:Figure 17a EXO example.PNG|thumb|700px|none|Figure 17. (a) Potential energy surface of an exothermic reaction. (b) Distance/Time graph]]

<nowiki>Following an exothermic reaction, the loss in potential energy becomes replaced by the increase in kinetic energy. The kinetic energy can be in translational, rotational or vibrational form. However, since we are working on a 1D system, the rotational motion can be neglected. </nowiki>

The overall gain in the kinetic energy, including both translational and vibrational energy, can be measured using a bomb calorimeter. The collision of products with high translational and vibrational energy collides to the wall of the calorimeter, transferring the energy as heat. The heat produced from the collision can subsequently be measured using a thermal sensor.

The vibrational energy can specifically be measured using an IR spectroscopy. At the room temperature, most bonds occupy the lowest-energy vibrational levels. However, during an exothermic reaction, higher-level vibrational modes can become occupied as the potential energy converts into the vibrational kinetic energy. In an IR spectroscopy, the higher-level vibrational modes appear as an overtone signal as 1->2 and 2->3 etc. transitions become detected. As the vibrational modes relax back to the ground state, the overtone signal disappears.<nowiki></nowiki>

=== Polanyi's empirical rules ===
The Polanyi rules state for reactants given momentum along the direction of the reaction path, translational energy, as opposed to vibrational energy, is preferred to overcome the early transition state in an exothermic reaction. To investigate this, series of trajectories with a same initial kinetic energy but with different vibrational/translational components are simulated. This is done by projecting the triatomic system on the potential energy surface toward TS at an different angle as shown below.

[figure 18]

In these series of trajectories, the horizontal components represent the vibrational momentum, and then the vertical components represent the translational momentum.

From the previous section, H2 + F -> H + HF is an exothermic reaction, with the activation energy of +1.119 kJ/mol. The series of trajectories at ~1.5kJ/mol are summarised below.
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|<nowiki>-1.69</nowiki>
|0.00
|0.0
|1.503
|Yes
|-
|<nowiki>-1.49</nowiki>
|0.20
|7.6
|1.506
|Yes
|-
|<nowiki>-1.26</nowiki>
|0.40
|17.6
|1.500
|Yes
|-
|<nowiki>-1.01</nowiki>
|0.60
|30.7
|1.503
|Yes
|-
|<nowiki>-0.73</nowiki>
|0.80
|47.6
|1.504
|Yes
|-
|<nowiki>-0.41</nowiki>
|1.00
|67.7
|1.498
|No
|-
|<nowiki>-0.05</nowiki>
|1.20
|87.6
|1.501
|No
|}
At a low angle of projection, the trajectory is directed straight at the TS along the MEP. The higher angle indicates the trajectory with lower velocity toward the TS, but with greater oscillations. <nowiki>Hence, the exothermic reaction favours the low angle of projection (i.e. a system with greater distribution of translational energy compared to vibrational energy) is preferred to overcome the TS.</nowiki>
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|0.0
|<nowiki>-13.1</nowiki>
|0.0
|171.161
|No
|-
|3.0
|<nowiki>-11.5</nowiki>
|14.6
|171.487
|No
|-
|6.0
|<nowiki>-9.7</nowiki>
|31.7
|171.237
|No
|-
|9.0
|<nowiki>-7.7</nowiki>
|49.5
|171.122
|No
|-
|12.0
|<nowiki>-5.4</nowiki>
|65.8
|169.749
|No
|-
|15.0
|<nowiki>-2.9</nowiki>
|79.1
|170.331
|Yes
|-
|18.0
|0.0
|90.0
|170.526
|Yes
|}
<nowiki>On the other hand, the endothermic reaction favours the "horizontal" projection with a large vibrational kinetic energy and low translational energy for overcoming the barrier.</nowiki>

MRD:aj3318

2020-05-08T22:26:47Z

Aj3318: /* Polanyi's empirical rules */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 is an exothermic and HF + H is an endothermic reaction. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

The approximate position of TS is at '''r1'''=181.1 and '''r2'''=74.5 pm, which represent the intermolecular distance between F-H and H-H respectively. The eigenvalues of the Hessian matrix were -0.002 and +0.332 for the eigenvectors (+1.000, -0.023), (-0.023, +1.000) respectively. The potential energy at the TS is -433.980 kJ mol-1.

The activation energy of the system was estimated by comparing the potential energy of the TS to the potential energy of the products when the system is slightly perturbed from TS. The following demonstrates how the calculation is performed.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Figure
|-
|MEP
|1000
|0.5
|181.1
|74.5
|0
|0
|15
|-
|MEP
|1000
|0.5
|181.2
|74.5
|0
|0
|16
|}
[[File:Figure 15 Ea of H HF to HH F.PNG|thumb|700px|none|Figure 15. (a) Potential energy surface of H+HF to H2+F. (b) Energy/Time graph. ]]

[[File:Figure 16 Ea of HH F to H HF.PNG|thumb|700px|none|Figure 16. (a) Potential energy surface of H2+F to H+HF. (b) Energy/Time graph. ]]

By iterating the calculations, the potential energy was estimated to the 6 significant figures. This involved finding the trajectory down the MEP from an initial condition, inserting the last geometry as the initial condition, and repeating until the potential energy stabilises.
{| class="wikitable"
!System
!Ep/kJmol-1
|-
|HF + H
|<nowiki>-560.699</nowiki>
|-
|F + H2
|<nowiki>-435.099</nowiki>
|-
|F-H-H
|<nowiki>-433.980</nowiki>
|}
The activation energy of HF + H -> H + H2 is +126.719 kJ mol-1, and F + H2 -> HF + H is +1.119 kJ mol-1. 

=== Reaction Dynamics ===
The following reaction condition resulted in the reaction from H2 + F -> HF + H.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
|-
|Dynamics
|1000
|0.5
|200
|75
|<nowiki>-1.0</nowiki>
|0.3
|}
[[File:Figure 17a EXO example.PNG|thumb|700px|none|Figure 17. (a) Potential energy surface of an exothermic reaction. (b) Distance/Time graph]]

Following an exothermic reaction, the loss in potential energy becomes replaced by the increase in kinetic energy. The kinetic energy can be in translational, rotational or vibrational form. However, since we are working on a 1D system, the rotational motion can be neglected.

The overall gain in the kinetic energy, including both translational and vibrational energy, can be measured using a bomb calorimeter. The collision of products with high translational and vibrational energy collides to the wall of the calorimeter, transferring the energy as heat. The heat produced from the reaction can subsequently be measured using a thermal sensor.

The vibrational energy can specifically be measured using an IR spectroscopy. At the room temperature, most bonds occupy the lowest-energy vibrational levels. However, during an exothermic reaction, higher-level vibrational modes can become occupied as the potential energy converts into the vibrational kinetic energy. In an IR spectroscopy, the higher-level vibrational modes appear as an overtone signal as 1->2 and 2->3 etc. transitions become detected. As the vibrational modes relax back to the ground state, the overtone signal disappears.

=== Polanyi's empirical rules ===
The Polanyi rules state for reactants given momentum along the direction of the reaction path, translational energy, as opposed to vibrational energy, is preferred to overcome the early transition state in an exothermic reaction.

From the previous section, H2 + F -> H + HF is an exothermic reaction, with the activation energy of +1.119 kJ/mol. The series of trajectories at ~1.5kJ/mol but with different angle of projection was explored.
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|<nowiki>-1.69</nowiki>
|0.00
|0.0
|1.503
|Yes
|-
|<nowiki>-1.49</nowiki>
|0.20
|7.6
|1.506
|Yes
|-
|<nowiki>-1.26</nowiki>
|0.40
|17.6
|1.500
|Yes
|-
|<nowiki>-1.01</nowiki>
|0.60
|30.7
|1.503
|Yes
|-
|<nowiki>-0.73</nowiki>
|0.80
|47.6
|1.504
|Yes
|-
|<nowiki>-0.41</nowiki>
|1.00
|67.7
|1.498
|No
|-
|<nowiki>-0.05</nowiki>
|1.20
|87.6
|1.501
|No
|}
At a low angle of projection, the trajectory is directed straight at the TS along the MEP. The higher angle indicates the trajectory with lower velocity toward the TS, but with greater oscillations. The reaction favours the low angle of projection (i.e. a system with greater distribution of translational energy compared to vibrational energy) is preferred to overcome the TS.
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|0.0
|<nowiki>-13.1</nowiki>
|0.0
|171.161
|No
|-
|3.0
|<nowiki>-11.5</nowiki>
|14.6
|171.487
|No
|-
|6.0
|<nowiki>-9.7</nowiki>
|31.7
|171.237
|No
|-
|9.0
|<nowiki>-7.7</nowiki>
|49.5
|171.122
|No
|-
|12.0
|<nowiki>-5.4</nowiki>
|65.8
|169.749
|No
|-
|15.0
|<nowiki>-2.9</nowiki>
|79.1
|170.331
|Yes
|-
|18.0
|0.0
|90.0
|170.526
|Yes
|}
On the other hand, the endothermic reaction favours the "horizontal" projection with a large vibrational kinetic energy and low translational energy for overcoming the barrier.

MRD:aj3318

2020-05-08T21:53:27Z

Aj3318: /* Exercise 2: F + H2 System */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 is an exothermic and HF + H is an endothermic reaction. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

The approximate position of TS is at '''r1'''=181.1 and '''r2'''=74.5 pm, which represent the intermolecular distance between F-H and H-H respectively. The eigenvalues of the Hessian matrix were -0.002 and +0.332 for the eigenvectors (+1.000, -0.023), (-0.023, +1.000) respectively. The potential energy at the TS is -433.980 kJ mol-1.

The activation energy of the system was estimated by comparing the potential energy of the TS to the potential energy of the products when the system is slightly perturbed from TS. The following demonstrates how the calculation is performed.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Figure
|-
|MEP
|1000
|0.5
|181.1
|74.5
|0
|0
|15
|-
|MEP
|1000
|0.5
|181.2
|74.5
|0
|0
|16
|}
[[File:Figure 15 Ea of H HF to HH F.PNG|thumb|700px|none|Figure 15. (a) Potential energy surface of H+HF to H2+F. (b) Energy/Time graph. ]]

[[File:Figure 16 Ea of HH F to H HF.PNG|thumb|700px|none|Figure 16. (a) Potential energy surface of H2+F to H+HF. (b) Energy/Time graph. ]]

By iterating the calculations, the potential energy was estimated to the 6 significant figures. This involved finding the trajectory down the MEP from an initial condition, inserting the last geometry as the initial condition, and repeating until the potential energy stabilises.
{| class="wikitable"
!System
!Ep/kJmol-1
|-
|HF + H
|<nowiki>-560.699</nowiki>
|-
|F + H2
|<nowiki>-435.099</nowiki>
|-
|F-H-H
|<nowiki>-433.980</nowiki>
|}
The activation energy of HF + H -> H + H2 is +126.719 kJ mol-1, and F + H2 -> HF + H is +1.119 kJ mol-1. 

=== Reaction Dynamics ===
The following reaction condition resulted in the reaction from H2 + F -> HF + H.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
|-
|Dynamics
|1000
|0.5
|200
|75
|<nowiki>-1.0</nowiki>
|0.3
|}
[[File:Figure 17a EXO example.PNG|thumb|700px|none|Figure 17. (a) Potential energy surface of an exothermic reaction. (b) Distance/Time graph]]

Following an exothermic reaction, the loss in potential energy becomes replaced by the increase in kinetic energy. The kinetic energy can be in translational, rotational or vibrational form. However, since we are working on a 1D system, the rotational motion can be neglected.

The overall gain in the kinetic energy, including both translational and vibrational energy, can be measured using a bomb calorimeter. The collision of products with high translational and vibrational energy collides to the wall of the calorimeter, transferring the energy as heat. The heat produced from the reaction can subsequently be measured using a thermal sensor.

The vibrational energy can specifically be measured using an IR spectroscopy. At the room temperature, most bonds occupy the lowest-energy vibrational levels. However, during an exothermic reaction, higher-level vibrational modes can become occupied as the potential energy converts into the vibrational kinetic energy. In an IR spectroscopy, the higher-level vibrational modes appear as an overtone signal as 1->2 and 2->3 etc. transitions become detected. As the vibrational modes relax back to the ground state, the overtone signal disappears.

=== Polanyi's empirical rules ===
The Polanyi rules state for reactants given momentum along the direction of the reaction path, translational energy, as opposed to vibrational energy, is preferred to overcome the early transition state in an exothermic reaction.

From the previous section, H2 + F -> H + HF is an exothermic reaction, with the activation energy of +1.119 kJ/mol. The series of trajectories at ~1.5kJ/mol but with different angle of projection was explored.
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|<nowiki>-1.69</nowiki>
|0.00
|0.0
|1.503
|Yes
|-
|<nowiki>-1.49</nowiki>
|0.20
|7.6
|1.506
|Yes
|-
|<nowiki>-1.26</nowiki>
|0.40
|17.6
|1.500
|Yes
|-
|<nowiki>-1.01</nowiki>
|0.60
|30.7
|1.503
|Yes
|-
|<nowiki>-0.73</nowiki>
|0.80
|47.6
|1.504
|Yes
|-
|<nowiki>-0.41</nowiki>
|1.00
|67.7
|1.498
|No
|-
|<nowiki>-0.05</nowiki>
|1.20
|87.6
|1.501
|No
|}
At a low angle of projection, the trajectory is directed straight at the TS along the MEP. The higher angle indicates the trajectory with lower velocity toward the TS, but with greater oscillations. The reaction favours the low angle of projection (i.e. a system with greater distribution of translational energy compared to vibrational energy) is preferred to overcome the TS.

MRD:aj3318

2020-05-08T21:52:28Z

Aj3318: /* Exercise 2: F + H2 System */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 is an exothermic and HF + H is an endothermic reaction. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

The approximate position of TS is at '''r1'''=181.1 and '''r2'''=74.5 pm, which represent the intermolecular distance between F-H and H-H respectively. The eigenvalues of the Hessian matrix were -0.002 and +0.332 for the eigenvectors (+1.000, -0.023), (-0.023, +1.000) respectively. The potential energy at the TS is -433.980 kJ mol-1.

The activation energy of the system was estimated by comparing the potential energy of the TS to the potential energy of the products when the system is slightly perturbed from TS. The following demonstrates how the calculation is performed.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Figure
|-
|MEP
|1000
|0.5
|181.1
|74.5
|0
|0
|15
|-
|MEP
|1000
|0.5
|181.2
|74.5
|0
|0
|16
|}
[[File:Figure 15 Ea of H HF to HH F.PNG|thumb|450px|none|Figure 15. (a) Potential energy surface of H+HF to H2+F. (b) Energy/Time graph. ]]

[[File:Figure 16 Ea of HH F to H HF.PNG|thumb|450px|none|Figure 16. (a) Potential energy surface of H2+F to H+HF. (b) Energy/Time graph. ]]

By iterating the calculations, the potential energy was estimated to the 6 significant figures. This involved finding the trajectory down the MEP from an initial condition, inserting the last geometry as the initial condition, and repeating until the potential energy stabilises.
{| class="wikitable"
!System
!Ep/kJmol-1
|-
|HF + H
|<nowiki>-560.699</nowiki>
|-
|F + H2
|<nowiki>-435.099</nowiki>
|-
|F-H-H
|<nowiki>-433.980</nowiki>
|}
The activation energy of HF + H -> H + H2 is +126.719 kJ mol-1, and F + H2 -> HF + H is +1.119 kJ mol-1. 

=== Reaction Dynamics ===
The following reaction condition resulted in the reaction from H2 + F -> HF + H.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
|-
|Dynamics
|1000
|0.5
|200
|75
|<nowiki>-1.0</nowiki>
|0.3
|}
[[File:Figure 17a EXO example.PNG|thumb|450px|none|Figure 17. (a) Potential energy surface of an exothermic reaction. (b) Distance/Time graph]]

Following an exothermic reaction, the loss in potential energy becomes replaced by the increase in kinetic energy. The kinetic energy can be in translational, rotational or vibrational form. However, since we are working on a 1D system, the rotational motion can be neglected.

The overall gain in the kinetic energy, including both translational and vibrational energy, can be measured using a bomb calorimeter. The collision of products with high translational and vibrational energy collides to the wall of the calorimeter, transferring the energy as heat. The heat produced from the reaction can subsequently be measured using a thermal sensor.

The vibrational energy can specifically be measured using an IR spectroscopy. At the room temperature, most bonds occupy the lowest-energy vibrational levels. However, during an exothermic reaction, higher-level vibrational modes can become occupied as the potential energy converts into the vibrational kinetic energy. In an IR spectroscopy, the higher-level vibrational modes appear as an overtone signal as 1->2 and 2->3 etc. transitions become detected. As the vibrational modes relax back to the ground state, the overtone signal disappears.

=== Polanyi's empirical rules ===
The Polanyi rules state for reactants given momentum along the direction of the reaction path, translational energy, as opposed to vibrational energy, is preferred to overcome the early transition state in an exothermic reaction.

From the previous section, H2 + F -> H + HF is an exothermic reaction, with the activation energy of +1.119 kJ/mol. The series of trajectories at ~1.5kJ/mol but with different angle of projection was explored.
{| class="wikitable"
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Angle of Projection/deg
!Kinetic Energy/kJ/mol
!Reaction
|-
|<nowiki>-1.69</nowiki>
|0.00
|0.0
|1.503
|Yes
|-
|<nowiki>-1.49</nowiki>
|0.20
|7.6
|1.506
|Yes
|-
|<nowiki>-1.26</nowiki>
|0.40
|17.6
|1.500
|Yes
|-
|<nowiki>-1.01</nowiki>
|0.60
|30.7
|1.503
|Yes
|-
|<nowiki>-0.73</nowiki>
|0.80
|47.6
|1.504
|Yes
|-
|<nowiki>-0.41</nowiki>
|1.00
|67.7
|1.498
|No
|-
|<nowiki>-0.05</nowiki>
|1.20
|87.6
|1.501
|No
|}
At a low angle of projection, the trajectory is directed straight at the TS along the MEP. The higher angle indicates the trajectory with lower velocity toward the TS, but with greater oscillations. The reaction favours the low angle of projection (i.e. a system with greater distribution of translational energy compared to vibrational energy) is preferred to overcome the TS.

2020-05-08T20:25:27Z

Aj3318:

File:Figure 15 Ea of H HF to HH F.PNG

2020-05-08T20:24:46Z

Aj3318:

MRD:aj3318

2020-05-08T20:21:05Z

Aj3318: /* Exercise 2: F + H2 System */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 is an exothermic and HF + H is an endothermic reaction. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

<nowiki>The approximate position of TS is at </nowiki>'''r1'''=181.1 and '''r2'''=74.5 pm, which represent the intermolecular distance between F-H and H-H respectively. <nowiki></nowiki>The eigenvalues of the Hessian matrix were -0.002 and +0.332 for the eigenvectors (+1.000, -0.023), (-0.023, +1.000) respectively. The potential energy at the TS is -433.980 kJ mol-1.

The activation energy of the system was estimated by comparing the potential energy of the TS to the potential energy of the products when the system is slightly perturbed from TS. The following demonstrates how the calculation is performed.
{| class="wikitable"
!Calculation Type
!Steps
!Time/fs
!rFH/pm
!rHH/pm
!pFH/gmol-1pmfs-1
!pHH/gmol-1pmfs-1
!Figure
|-
|MEP
|1000
|0.5
|181.1
|74.5
|0
|0
|15
|-
|MEP
|1000
|0.5
|181.2
|74.5
|0
|0
|16
|}
[fig 15]

[fig 16]

By iterating the calculations, the potential energy was estimated to the 6 significant figures. This involved finding the trajectory down the MEP from an initial condition, inserting the last geometry as the initial condition, and repeating until the potential energy stabilises.
{| class="wikitable"
!System
!Ep/kJmol-1
|-
|HF + H
|<nowiki>-560.699</nowiki>
|-
|F + H2
|<nowiki>-435.099</nowiki>
|-
|F-H-H
|<nowiki>-433.980</nowiki>
|}
<nowiki>The activation energy of HF + H -> H + H2 is +126.719 kJ mol</nowiki>-1, and F + H2 -> HF + H is +1.119 kJ mol-1. <nowiki></nowiki>

=== Reaction Dynamics ===

MRD:aj3318

2020-05-08T19:06:44Z

Aj3318: /* Potential Energy Surface */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 and HF + H are exothermic and endothermic respectively. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H.

The energetics of the reactions F + H2 and the reverse H + HF were investigated, with the following properties observed:
{| class="wikitable"

! Reaction
! Transition State Energy / kcal mol-1 
! Reactants Energy / kcal mol-1 
! Activation Energy / kcal mol-1 
! Classification

|-
| F + H2
| - 103.753
| - 133.945
| + 30.192
| Exothermic

|-
| H + HF
| - 103.744
| - 103.987
| + 0.243
| Endothermic
|}
F + H2 is exothermic as the H-F bond is stronger than the H-H bond by virtue of the higher electronegativity of fluorine relative to hydrogen (Pauling Electronegativities of: F = 3.98 and H = 2.20).[5] The fluorine atom in H-F pulls electrons closer to itself, leading to a highly polarised H-F bond and a strong ionic contribution to the bonding strength as well as increased orbital overlap. The H-F bond is therefore stronger than the purely covalent H-H bond, (bond dissociation energies of: H-F = + 567 kJ mol1 and H-H = + 436 kJ mol1.[6] The formation of the H-F bond in the forward reaction F + H2 is therefore energetically favourable and overcompensates for the loss of the H-H bond, with an exothermic release of energy.[5] The reverse reaction, H + HF, is endothermic and so requires the input of energy as the energy released by the formation of the H-H bond does not outweigh the energy used to break the H-F bond.[5]

According to Hammond's postulate, the transition state between two interconverting structures will most closely resemble the structure it is closer in energy to.[7] For an exothermic reaction, the transition state would be early and closely resemble the reactants, therefore having only a marginally higher energy than the reactants, i.e. a small activation energy.[7]
{| class="wikitable"

! Calculation Type
! Steps
! rH-F / Å
! rH-H / Å
! pHF
! pHH

|-
| Dynamics
| 500
| 1.811
| 0.745
| 0
| 0
|}
The approximate position of the transition state for both F + H2 and H + HF, was found to be '''rH-F = 1.811 Å and rH-H = 0.745 Å'''.

The potential surface plot in Figure 20 and energy vs time plot in Figure 21 shows that the transition state has been attained, with no change in potential energy as expected at the saddle point and no movement towards either products or reactants. The potential surface plot also shows the minima in the reactants channel to be higher in energy than the products channel, indicating an exothermic reaction in the event of a reactive trajectory.

By inspecting the potential energy surfaces, classify the F + H2 and H + HF reactions according to their energetics (endothermic or exothermic). How does this relate to the bond strength of the chemical species involved?

F + H2 is exothermic, and H + HF is endothermic. This means that the H-F bond is stronger than the H-H bond.

Locate the approximate position of the transition state.

Because the activation energy for one of the reactions is so small, it is not easy to locate the transition state immediately. Use the Hammond postulate to guide your search.

Report the activation energy for both reactions.

You will be able to report a reasonable estimate by performing a mep (with a sufficient number of steps) from a structure neighbouring the transition state, and choosing to plot the appropriate quantity as a function of "time".

MRD:aj3318

2020-05-08T19:05:58Z

Aj3318: /* Potential Energy Surface */

= Introduction =
In this Wiki page, the molecular reaction dynamics of two simple triatomic systems are investigated: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2<0). This differs from the local minimum where the Hessian matrix is a definite negative (i.e. fx=0, fy=0, fxx>0, fxxfyy-fxy2>0). Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate. 

[[File:Figure 4 distance vs time plot for TS.png|thumb|450px|none|Figure 4. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At a perfectly symmetrical system of H-H-H, the system cannot "roll down" to H2 + H or H + H2, but will equilibrate at the most stable state of H-H-H (i.e. transition state). This is illustrated below. 

[[File:Figure 5 potential energy surface for TS.png|thumb|450px|none|Figure 5. Intermolecular distance vs Time plot for finding TS state. Initial condition: r1=200, r2=200, p1=p2=0. ]]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77 pm. Hence, the transition state of H + H2 system is the state where both atoms are 90.77 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using MEP and dynamic calculations.
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|0.1
|91.8
|90.8
|0
|0
|}

The minimum energy path (MEP) is a trajectory where the change in potential is assumed to be lost as "heat", resulting in the overall loss of energy from the system (dV=dq). Hence, the kinetic energy is always at zero and therefore the momentum is also always zero. The dynamic simulation is the trajectory where the change in potential is converted to kinetic energy (dV=dK). In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space. The MEP is useful to find the reaction coordinate (i.e. lowest-energy path).

The trajectories from MEP and dynamic calculations are shown below.

[[File:Figure 6 PES of MEP and dynamic.PNG|thumb|700px|none|Figure 6. (a) Potential energy surface from MEP. (b) Potential energy surface from dynamics. ]]

The potential energy surface of dynamic simulation have the "oscillating" movements caused by the interplay of kinetic and potential energy of the system, whereas the MEP simulation only moves downhill due to the absence of excess kinetic energy (i.e. zero momentum). Mathematically, the motion at MEP is directed toward '''p''' = -'''grad'''(V)dt, whereas the dynamic motion must consider '''p''' = -ʃ'''grad'''(V)dt. The motion in distance vs time plot of both motions is also shown below.

[[File:Figure 7 distance time MEP and dynamic.PNG|thumb|700px|none|Figure 7. (a)Distance/Time graph from MEP. (b) Distance/Time graph from dynamics. ]]

Note that the gradient of Fig. 7b, representing velocity, generally increases over time as the kinetic energy is added to the system, whereas the gradient of Fig. 7a only depends on the slope of the potential surface. 

The energy of the trajectories is shown below. As stated by the assumption, we find that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[[File:Figure 8 energy time MEP dynamic.PNG|thumb|700px|none|Figure 8. (a)Energy/Time graph from MEP. (b) Energy/Time graph from dynamics. ]]

=== Reactive and Unreactive Trajectories ===
One primary aim of the molecular reaction dynamics is to predict the reactivity of a system of atoms. Using the potential surface plot, we can predict whether a reaction will occur at a specific condition by simulating whether the collision would result in the trajectory from one local minimum to another via the transition state. The illustration of this is demonstrated below. Series of simulation was carried out in the following conditions: 
{| class="wikitable"
!Steps number
!Size/fs
!r1/pm
!r2/pm
!p1
!p2
!Etot
|-
|1000
|0.1
|74
|200
|shown below in gmol-1pmfs-1
|shown below in gmol-1pmfs-1
|shown below in kJ mol-1
|}

{| class="wikitable" border="1"
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 ||-414.280||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. The amplitude of vibration at the reactant is greater compared to the product, which indicates that some translational energy became converted to vibrational energy. ||[[File:Figure 9 table 3 row 1.png|thumb|300px|none]]

|-
| -3.1 || -4.1 ||-420.077||No||The trajectory is not reactive as the system does not cross the transition state. The atom C collides into AB and bounces off. The little/no change in the amplitude of the oscillation shows that the collision resulted in no energy exchange between AB and C. ||[[File:Figure 10 table 3 row 2.png|thumb|300px|none]]
|-
| -3.1 || -5.1 ||-413.977||Yes||The trajectory is reactive as the system has a high enough energy to cross the transition state. Compared to the first case, the initial momentum is greater, so the vibrational energy of both reactant and products are higher. ||[[File:Figure 11 table 3 row 3.png|thumb|300px|none]]
|-
| -5.1 || -10.1 ||-357.277||No||The trajectory is not reactive, as the system crosses the energy barrier twice. The atom C collides into AB with a high momentum, forming an A-B-C structure. This quickly dissociates back to AB + C due to the high vibrational energy at BC. The translational energy of C is partially transferred into the vibrational energy of AB, resulting in a greater amplitude of oscillation. ||[[File:Figure 12 table 3 row 4.png|thumb|300px|none]]
|-
| -5.1 || -10.6 ||-349.477||Yes||The trajectory is reactive, as the system crosses the energy barrier three times. The atom C collides into AB with a high momentum, forming an A-B-C structure. The bonds A-B and B-C stretches and contracts 3 times before the atom A dissociates. The translational energy of C is partially transferred into the vibrational energy of BC, and into the translational energy of A. ||[[File:Figure 13 table 3 row 5.png|thumb|300px|none]]
|}

From the table above, one can conclude that the reactivity of a collision may not be possible to predict just by looking at the total energy of the system. The reactivity of a collision depends on both energy and the angle of trajectories, and there are possibilities that a system with enough energy would cross the energy barrier twice or more and bounce back to the original structure.

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

The transition state theory (TST) predicts the rate of reaction in the macroscopic scale based on the statistical thermodynamics. The TST states that the rate of reaction, kTST, of a reaction A+B->P can be expressed as the following:

<math>k_{TST}=R_{cl}\frac{Q_TS/V}{(Q_{\mathcal{A}}/V)(Q_{\mathcal{B}}/V)}e^{-\frac{E_a}{k_B T}}</math>

Where QA, QB and QTS are the partition functions of A, B and the transition state respectively. V is the volume of the mixtures, Rcl is the rate of crossing the energy barrier, Ea is the activation energy, kB is the Botzmann constant and T is the temperature.

There are 5 key assumptions in the TST:
# The nuclear and electronic motions are decoupled (BO approximation)
# The energy of the reactants follow the Boltzmann distribution
# Molecular systems that have crossed the transition state and formed products cannot return to the reactants
# In the transition state, the motion along the reaction coordinate can be treated classically
# The energy of the transition states that are becoming the product follow the Boltzmann distribution.
In experiments, all molecular systems in macroscopic scale are in equilibrium, meaning that both forward and backward reactions may occur. This violates the third assumption of TST, causing the TST to overestimate the rate of reaction. Another source of error could arise from the third assumption. Although minor, the tunnelling effect could make an observable difference to the experimental rate of reaction in a small molecular system.

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
The potential energy surface of the F-H-H system is as follows.

[[File:Figure 14 potential energy surface of FHH.png|thumb|450px|none|Figure 14. Potential energy surface of F-H-H.]]

On the potential energy surface of F-H-H, it is clear that the F + H2 system has higher potential energy compared to the HF + H system. Consequently, F + H2 and HF + H are exothermic and endothermic respectively. This indicates that the bond strength of HF is stronger than that of H2. Intuitively, the fluorine atom is significantly more electronegative compared to the hydrogen atom, causing the H-F bond to be highly polarised. This leads to the increased ionic contribution in the H-F bond, resulting in a stronger H-F bond compared to H-H. 

The energetics of the reactions F + H2 and the reverse H + HF were investigated, with the following properties observed:
{| class="wikitable"

! Reaction
! Transition State Energy / kcal mol-1 
! Reactants Energy / kcal mol-1 
! Activation Energy / kcal mol-1 
! Classification

|-
| F + H2
| - 103.753
| - 133.945
| + 30.192
| Exothermic

|-
| H + HF
| - 103.744
| - 103.987
| + 0.243
| Endothermic
|}
F + H2 is exothermic as the H-F bond is stronger than the H-H bond by virtue of the higher electronegativity of fluorine relative to hydrogen (Pauling Electronegativities of: F = 3.98 and H = 2.20).[5] The fluorine atom in H-F pulls electrons closer to itself, leading to a highly polarised H-F bond and a strong ionic contribution to the bonding strength as well as increased orbital overlap. The H-F bond is therefore stronger than the purely covalent H-H bond, (bond dissociation energies of: H-F = + 567 kJ mol1 and H-H = + 436 kJ mol1.[6] The formation of the H-F bond in the forward reaction F + H2 is therefore energetically favourable and overcompensates for the loss of the H-H bond, with an exothermic release of energy.[5] The reverse reaction, H + HF, is endothermic and so requires the input of energy as the energy released by the formation of the H-H bond does not outweigh the energy used to break the H-F bond.[5]

According to Hammond's postulate, the transition state between two interconverting structures will most closely resemble the structure it is closer in energy to.[7] For an exothermic reaction, the transition state would be early and closely resemble the reactants, therefore having only a marginally higher energy than the reactants, i.e. a small activation energy.[7]
{| class="wikitable"

! Calculation Type
! Steps
! rH-F / Å
! rH-H / Å
! pHF
! pHH

|-
| Dynamics
| 500
| 1.811
| 0.745
| 0
| 0
|}
The approximate position of the transition state for both F + H2 and H + HF, was found to be '''rH-F = 1.811 Å and rH-H = 0.745 Å'''.

The potential surface plot in Figure 20 and energy vs time plot in Figure 21 shows that the transition state has been attained, with no change in potential energy as expected at the saddle point and no movement towards either products or reactants. The potential surface plot also shows the minima in the reactants channel to be higher in energy than the products channel, indicating an exothermic reaction in the event of a reactive trajectory.

By inspecting the potential energy surfaces, classify the F + H2 and H + HF reactions according to their energetics (endothermic or exothermic). How does this relate to the bond strength of the chemical species involved?

F + H2 is exothermic, and H + HF is endothermic. This means that the H-F bond is stronger than the H-H bond.

Locate the approximate position of the transition state.

Because the activation energy for one of the reactions is so small, it is not easy to locate the transition state immediately. Use the Hammond postulate to guide your search.

Report the activation energy for both reactions.

You will be able to report a reasonable estimate by performing a mep (with a sufficient number of steps) from a structure neighbouring the transition state, and choosing to plot the appropriate quantity as a function of "time".

File:Figure 14 potential energy surface of FHH.png

2020-05-08T19:02:59Z

Aj3318:

MRD:aj3318

File:Figure 4 distance vs time plot for TS.png

2020-05-07T18:22:23Z

Aj3318:

MRD:aj3318

2020-05-07T18:22:02Z

Aj3318:

= Introduction =
This Wiki page investigates the molecular reaction dynamics of two simple triatomic systems: H + H2 and F + H2. For simplicity, atoms are assumed to be at 180° with each other, reducing the system into 1D space. The key variables to be explored include intermolecular distances, momentum, and the potential energy. The three atoms are labelled A, B and C. The vectors '''r1''' and '''r2''' are the vectors '''BA''' and '''BC''' in pm respectively. The momenta are noted '''p1''' and '''p2 '''in g mol-1 pm fs-1. The potential energy is the energy of the system relative to the state where the three atoms are infinitely separated from each other, estimated based on the London-Eyring-Polanyi-Sato (LEPS) function.

[[File:wl4015aDiagram1.png|thumb|450px|none|Figure 1. Triatomic system, where atom A reacts with the diatomic molecule BC. Diagram from https://wiki.ch.ic.ac.uk/wiki/index.php?title=CP3MD (accessed May 2020).]]

This Wiki explores the series of reaction dynamics in the triatomic systems. The potential surface, intermolecular distance/time, momentum/time, energy/time are simulated using two methods of calculation: minimum energy path (MEP) and dynamics. The details of these calculations are discussed.

The answers to questions are highlighted in blue. 

= Exercise 1: H + H2 System =

=== Potential Energy Surfaces ===
Potential energy surface is the plot of potential energy in respect to the geometric parameters. In a complicated system with a large number of degree of freedom, multi-dimensional analysis may be necessary. In this example, the three atoms are assumed to be restricted to 1D motion in free space, allowing the full system to be analysed in 3-dimensional coordinates. The potential energy surface of the H + H2 system is shown below.

[[File:Figure_2_potential_energy_surface.png|thumb|450px|none|Figure 2. Potential energy surface diagram of H-H-H triatomic system. ]]

The set of valleys on the potential energy diagram are often used to represent the "reaction coordinate" in chemical energy diagrams. The maximum of the lowest-energy path is the "transition state", which, on a 2D surface in 3-dimensional space, is represented as the saddle point. The saddle point is the point where the Hessian matrix has both negative and positive eigenvalues (i.e. fx=0, fy=0, fxxfyy-fxy2>0). This differs from the local minimum where the Hessian matrix is a definite negative (. Intuitively, saddle point is a turning point that is a relative minimum in one direction but maximum in another direction, whereas the local minimum is a minimum in all directions. 

[[File:Figure 3 saddle point.png|thumb|450px|none|Figure 3. Illustration of the local maximum, minimum and saddle point. Adapted from https://en.wikipedia.org/wiki/Saddle_point#/media/File:Saddle_Point_between_maxima.svg (Accessed May 2020).]]

=== Locating Transition State ===
The most general method to locate the transition state is to solve the differential equations numerically. In this particular system, the transition state can easily be located by setting the atoms A and C equidistance from each other, and running the simulation under minimum energy path (MEP - further discussed later). The system will approach the transition state. This is possible because the transition state is perfectly symmetrical, as predicted by the Hammond's postulate.

[Potential surface at transition state]

At a perfectly symmetrical system, the potential energy cannot tilt in any direction, and become stuck at the unstable equilibrium. The position where this approaches to at MEP is the transition state. 

[Internuclear Distances vs Time]

At the end of the simulation, the coordinate (r1, r2) approaches r1 = r2 = 90.77424978754546 pm.

Hence, the transition state of H + H2 system is the state where both atoms are 90.8 pm away from the central atom.

=== Calculating Reaction Path ===
The reaction path at the following parameters were analysed using both mep and dynamic calculations.
{| class="wikitable"
!Steps
!r1/pm
!r2/pm
!p1/gmol-1pmfs-1
!p2/gmol-1pmfs-1
|-
|500
|91.8
|90.8
|0
|0
|-
|
|
|
|
|
|-
|
|
|
|
|
|}

The minimum energy path (MEP) is the movement of atoms where all change in potential is assumed to be lost as a "work done", where the energy is dissipated to the surroundings. Hence, the kinetic energy is kept at zero and therefore the momentum is zero. The dynamic simulation is the movement of atoms where the change in potential is converted to kinetic energy. In this simulation, none of the energy is lost to the surroundings and the total energy is conserved. The dynamic simulation is closer to the real intermolecular interactions in the free space, where there is no other particle that can exchange the kinetic energy. The MEP is useful to find the reaction coordinate and the transition state.

The trajectories from mep and dynamics are shown below.

[mep trajectory potential surface] [dynamics trajectories potential surface]

The potential surface plot of dynamics have the "oscillating" movements caused by the interplay of kinetic and potential energy, whereas the MEP only moves downhill due to the zero kinetic energy. Mathematically, the direction of the movement of two methods differ such that in MEP, the movement is directed to Fdt = -grad(V)dt. On the other hand, the dynamic trajectory is directed Fdt = -grad(V)dt + p where p is the momentum.

[mep distance vs time plot] [dynamics distance vs time plot]

The momentum and velocity of dynamics show an oscillating movement, whereas in MEP, both momentum and velocity are zero at all times. 

[dynamics momentum vs time] [dynamics velocity vs plot]

We can also see that the total energy is constant at dynamics, and the kinetic energy is zero at MEP.

[dynamics energy vs time] [mep energy vs time]

=== Reactive and Unreactive Trajectories ===
One primary use of the analysis of molecular reaction dynamics is to predict whether a system of atoms is reactive or not. In the potential surface plot, we can predict whether a reaction will occur at specific conditions by predicting whether the system has enough energy and the correct angle of collision to result in the movement from one local minimum to another via the transition state. The following table illustrates the example of simulating the reaction dynamics.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || || || ||
|-
| -3.1 || -4.1 || || || ||
|-
| -3.1 || -5.1 || || || ||
|-
| -5.1 || -10.1 || || || ||
|-
| -5.1 || -10.6 || || || ||
|}

=== Transition State Theory ===
Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?

Transition State Theory connects the collision energy, angle of collision,

= Exercise 2: F + H2 System =

=== Potential Energy Surface ===
By inspecting the potential energy surfaces, classify the F + H2 and H + HF reactions according to their energetics (endothermic or exothermic). How does this relate to the bond strength of the chemical species involved?

F + H2 is exothermic, and H + HF is endothermic. This means that the H-F bond is stronger than the H-H bond.

Locate the approximate position of the transition state.

Because the activation energy for one of the reactions is so small, it is not easy to locate the transition state immediately. Use the Hammond postulate to guide your search.

Report the activation energy for both reactions.

You will be able to report a reasonable estimate by performing a mep (with a sufficient number of steps) from a structure neighbouring the transition state, and choosing to plot the appropriate quantity as a function of "time".