Jegajeg01108428

Molecular Reaction Dynamics

H + H₂ system

Dynamics from the transition state region

Figure 1:The reaction trajectory before transition state is reached

Figure 2: The reaction trajectory after transition state is reached

Figure 3: Transition state is indicated by the arrow. The minimum points are the troughs at the end of the surface plot

Figure 4: The whole reaction trajectory

Question 1: Total gradient has a value of zero at minimum and transition structure. This is because both points are stationary points with a first derivative of the potential energy of zero. The curvature of the plot can be used to differentiate these points where the minima is a trough and the transition structure is a peak. The second derivative of potential energy is thus positive for the minima and negative for the transition structure.

Trajectories from r₁ = r₂: locating the transition state

Question 2: Transition state is found at a position where the A-B and B-C distances are 0.908 Å as shown in Figure 5. This is found by trial-and-error method where the r₁ and r₂ distances are changed until a single point near the maximum of the minimum energy path is achieved. As C is leaving and A is forming a bond with B, a transition state is achieved when B forms bonds of equal lengths to A and C. This agrees with the values obtained.This is also confirmed by analysing the internuclear distance versus time curve where the straightest line is obtained by setting the distance to be 0.908 Å. A straight line indicates that there is no vibration and that indicates a transition state.

Figure 5:Surface plot showing the transition state

Figure 6: Internuclear distance versus Time plot

Calculating reaction path

The Internuclear Distances vs Time graph and Internuclear Momenta vs Time graph for trajectories from r₁ = r_ts+0.01, r₂ = r_ts are shown below:

Figure 7:Internuclear distance vs Time plot

Figure 8 :Internuclear momenta vs Time plot

The final values of the positions r1(t)  r2(t)  are 6.00 Å and 5.28 Å. 
The average momenta   p1(t)  p2(t) at large t positions are:
          B-C: 2.48
          A-B: 1.22
          A-C: 0.00

Trajectories from r₁ = r_ts+δ, r₂ = r_ts

Figure 10:Surface plot with MEP calculation

Figure 11:Surface plot with Dynamics calculation

Question 3: In MEP calculation, the trajectory towards the product channel is seen only near the transition state. No vibration is observed. This is because in MEP calculations, the velocity is set to zero in each time step, the energy is mainly translational. Inertia of the atoms are not taken into account. The velocity starts from zero, increases and returns to zero again which results in the small trajectory observed.In dynamic view, the trajectory extends all the way from the transition state to the product channel.Also, the product is observed to have more vibrational energy. The inertia of the products are taken into account, thus the velocity is sustained for a longer path.

Reactive and unreactive trajectories

Question 4: Trajectory of the reaction with different value of momentum were investigated and the results are tabulated.

Reactivity at different momentum

p1	p2	Reactivity
-1.25	-2.5	reactive
-1.5	-2.0	unreactive
-1.5	-2.5	reactive
-2.5	-5.0	unreactive
-2.5	-5.2	reactive

Figure 11: It could be seen that the reaction proceeds passing through the transition state. The reactants have enough kinetic energy to overcome the activation barrier, thus transition state is reached and product is formed.There is almost no vibration when A approaches the diatomic molecule B-C indicating that the energy is mainly translational. Some of the translational energy is then converted to the vibrational energy of the product A-B.

Figure 12: The reaction does not pass through the transition state and product is not formed. The total momentum ( p₁+ p₂) of the reactants is not high enough, thus the reactants do not have enough kinetic energy to overcome the activation barrier. Reactants are formed again. The vibration of B-C is however larger than the vibration in Figure 11 because p₂ is higher in this case.The extra energy is in the form of vibrational energy.

Figure 13: It could be seen that the reaction proceeds passing through the transition state. The reactants have enough kinetic energy to overcome the activation barrier, thus transition state is reached and product is formed.Compared to Figure 11, the vibration of B-C is larger because p₁ is higher. The total momentum of the reaction is high enough to drive the reaction to completion by overcoming the activation barrier.

Figure 14: The total momentum in this example is high enough to overcome the activation barrier thus reach the transition state. However, the kinetic energy is very high that the trajectory passes through the transition state again and returns to the reactant channel.Products are not formed.

Figure 15: The activation barrier is overcome and the transition state can be reached because the reactants have enough kinetic energy. The transition state is crossed again because the reactants have very high kinetic energy (the highest among the examples).However, the trajectory eventually reaches the product channel after crossing the transition state again.

It could be seen from Figure 14 and Figure 15 that eventhough the total momentum in the last example is higher than the penultimate example, the reaction still goes to completion. This indicates that the total energy of a reaction only does not determine the reaction trajectory but the form of energy as well. In Figure 11, Figure 13 and Figure 15, the energy is largely translational and the reaction goes to completion. However, in Figure 12 and Figure 14, the energy is mainly vibrational and the reaction does not go through completion although supplied with very high kinetic energy. It can thus be deduced that H + H₂ reaction is an exothermic reaction, as the translational energy promotes the formation of an early transition state. This conclusion however needs to be verified by observing reaction trajectories at different values of momentum and amount of vibrational energy

Question 5: The main assumptions of the transition state theory are as follow:

i- When the molecules occupy different energy levels, vibrational energy states for example,they obey the Boltzmann distribution. ii- The motion of the reactants or products obey classical mechanics, quantum tunneling does not occur. iii-Transition state theory assumes that once the transition state is reached, it must form products. iv- Rate of the reaction is dependent only on the transition state. v- The structure at the transition state is in quasi-equilibrium with the reactants

The third assumption of the Transition State theory (once the transition state is reached, it must form products) can be used to compare the predictions with experimental values.

The reactions with the first three set of momenta (Figure 11,Figure 12 and Figure 13) follow this assumption, the last two do not. It could be seen from Figure 14 and Figure 15 that after the transition state is reached, it is crossed again. This is not in agreement with the Transition State theory prediction which states that there is no recrossing of the transition state

F - H - H system

PES inspection

Question 6:

The enthalpy of the reaction can be determined from bond strengths using the formula, ΔH = (energy needed to break bonds)- (energy needed to form bonds).

The strength of the bonds are as shown:

H-F : 570 kJ/mol
H-H : 436 kJ/mol

The enthalpy of reaction for these molecules are then:

a) F + H₂ ,

ΔH= 436-570

 = -134 kJ/mol (exothermic)

b) H + HF

ΔH= 570-436

 = +134 kJ/mol (endothermic)

Question 7: Approximate position of the transition structure of the reactions are as follow:

F + H₂:

  H-F : 1.810 Å
  H-H : 0.745 Å

H + HF

  H-H : 1.810 Å
  H-F : 0.745 Å

This is again determined by looking at the Internuclear distance versus Time plot which gives the straightest possible line.

Figure 16: Internuclear distance vs Time plot for F + H₂ reaction

Figure 17: Internuclear distance vs Time plot for H+ HF reaction

Question 8: The activation energy,E_a,of the reaction is the difference in energy between the reactants and the transition state. The respective values of these energies are determined from the surface plot as shown:

Figure 18: Surface plot showing the energy of the reactants

Figure 19: Surface plot showing the energy of the transition state

Figure 20: Surface plot showing the energy of the products

1. E_a of F + H₂ reaction = -102.7-9-(-103.8)= 1.1 kcal/mol

E_a of H + HF reaction can be calculated from the surface plot of the F + H₂ reaction as it is just a reverse of the reaction.

2. E_a of H + HF reaction = -102.7-9-(-133.9)= 31.2 kcal/mol

Reaction dynamics

Question 9: The principle of conservation of energy states that energy cannot be created nor destroyed.Thus, the energy is converted from one form to another. Potential energy is converted into kinetic energy and vice versa. In an exothermic reaction where some of the potential and kinetic energy is converted into heat energy which is released to the surroundings. When the reaction is carried out in a closed system, the temperature change could be measured which could be related to heat energy absorbed or released.The temperature change could be measured using a thermometer and the heat released or absorbed can be calculated using appropriate equations such as H= mcϴ. The heat released or absorbed can be quantified using calorimetry methods as well. Vibrational energy gained or lost can be quantified using spectroscopy methods such as IR spectroscopy. The wavenumber,v,of the peaks on the IR spectrum can be related to the energy using the equation E=hv where h is the Planck's constant.Thus, from the energy values of the reactants and products, amount of vibrational energy converted into other forms of energy or amount of energy converted into vibrational energy can be determined.

Question 10: The distribution of energy in the form of translation and vibration affects the efficiency of the reaction based on the position of the transition state. Polanyi's empirical rules can be invoked in explaining this where the rule states that vibrational energy promotes the attainment of a late transition state while translational energy promotes the attainment of early transition state. Thus, if a reaction has an early transition state, providing energy mainly in the form of translation will result in an efficient reaction because transition state is reached easily and products is formed. However, if the translation state is late and it is supplied with translational energy, the transition state attainment will be slow or not complete, resulting in a less efficient reaction.^[2]

In this particular example, it could be seen from the surface plot that for the F + H₂ reaction which is exothermic, when the energy is supplied has a large proportion of vibrational energy, the transition state state is reached,and then crossed again multiple times, showing that the reaction is inefficient. Energy might be lost to reorganise the structure of the molecule again and again.

Figure 21: Surface plot of the F + H₂ reaction

For the endothermic H + HF reaction, when the energy supplied is mainly translational, it could be seen again that the transition state is crossed again a few times before the product is formed. The reaction is thus less efficient.

Figure 22: Surface plot of the H + HF reaction