The total gradient of a potential energy/internuclear distance plot at minima and at transition states will be zero. Minima and transition states can be identified from these plots by inspection, as the minima will occur at the lowest potential energy values where δV/δ/r=0. Transition states will be seen at the points where δV/δr=0 and the potential energy values are at their highest along the trajectory. The identity of potential transition states can be confirmed by starting the trajectory from this point and checking to see whether the system falls towards the entrance or exit channel (Figs. 2, 3). These images were obtained by determining distances AB and BC at the proposed transition state, and inputting the initial momentum and distance conditions as stated. Fig. 2 clearly shows the formation of the HAHB bond and subsequent vibration where initial BC momentum is -2.7 kgms-1, and vice versa for fig. 3.
Fig. 1: Potential energy vs internuclear distance plot of a triatomic hydrogen system. Transition state is marked with an X.
Fig. 2: Potential energy vs internuclear distance plot of a triatomic hydrogen system, where initial AB distance and BC distance are 0.9Å, AB momentum is 0.0 kgms-1 and BC momentum is -2.7 kgms-1.
Fig. 3: Potential energy vs internuclear distance plot of a triatomic hydrogen system, where initial AB distance and BC distance are 0.9Å, AB momentum is -2.7 kgms-1 and BC momentum is 0.0 kgms-1.
(Is it possible to distinguish transition states and minima mathematically? Lt912 (talk) 07:59, 9 June 2017 (BST))
Predicting Geometry at Transition State
Transition state geometry was determined by testing different values of r1=r2 at p1-p2=0. The best estimate for AB/BC distance was 0.90775Å. This is shown in fig. 4 by an internuclear distance/time plot, where it is seen that there is no variation in AB/BC distances over time at 0.90775Å. Fig. 4 shows no vibration, however this is because the programme cannot display such small variations in internuclear distance. If the precision of the estimate is lowered to 0.9Å, periodic vibrations can be observed, showing the triatomic system vibrating at the transition state.
Fig. 4: internuclear distance against time, where r1=r2=0.90775Å.
Fig. 5: internuclear distance against time, where r1=r2=0.9Å.
Calculating the Reaction Path
The MEP and dynamic trajectory were calculated. On the surface plot, the MEP can simply be shown as a smooth line following the minimum exit channel from the transition state (Fig. 6). This is because the MEP simply shows the potential energy of the most stable structure, which remains at a minimum as the velocity term is consistently set to 0 at every step. The dynamic trajectory shows some small vibrations (Fig. 7) in the BC bond, which are confirmed when viewing the animated simulation.
Fig 6: MEP surface plot of triatomic hydrogen system.Fig 7: dynamic surface plot of triatomic hydrogen system.
Reactive and Unreactive Trajectories
Testing Reactivity of Trajectory at Different Momenta
p1
p2
Reactive?
-1.25
-2.5
yes
-1.5
-2.0
no
-1.5
-2.5
yes
-2.5
-5.0
no
-2.5
-5.2
yes
Fig 8: dynamic surface plot of triatomic hydrogen system with p1 = -1.25 kgms-1 and p2 = -2.5 kgms-1.Fig 9: dynamic surface plot of triatomic hydrogen system with p1 = -1.5 kgms-1 and p2 = -2.0 kgms-1.Fig 10: dynamic surface plot of triatomic hydrogen system with p1 = -1.5 kgms-1 and p2 = -2.5 kgms-1.Fig 11: dynamic surface plot of triatomic hydrogen system with p1 = -2.5 kgms-1 and p2 = -5.0 kgms-1.Fig 12: dynamic surface plot of triatomic hydrogen system with p1 = -2.5 kgms-1 and p2 = -5.2 kgms-1.
Fig. 8 shows a reaction in which p1=-1.25 kgms-1 and p2= -2.5 kgms-1. The collision here between HA and HB is reactive. This can be seen as the trajectory passes through the transition state. The trajectory approaches the transition state as HA approaches HB and then oscillations can be seen as HC moves increasingly further away from HB, which show the HA/HB bond vibrating.
Fig. 9 shows an unreactive collision. Here, HA is shown to be approaching HB, and the trajectory oscillates as there are small changes in r2 as the HB/HC bond vibrates. The trajectory does not reach the transition state, but instead doubles back on itself and HA is repelled. This occurs when r2 is approximately 1 Å. This is because the repulsive Coulombic force between nuclei A and B is stronger than the force due to the collision, and hence the energy in the collision is not sufficient to overcome the transition state.
Fig. 10 shows a reactive collision. This is very similar to the collision shown in Fig. 8, except p1 = -1.5 kgms-1 instead of -1.25 kgms-1. Hence the collision is slightly higher energy, but not such a high energy as to prevent a reactive collision.
Fig. 11 shows an unreactive collision. Here, the force of HA on HB is so great that the internuclear distance is reduced to ~0.6 Å. At internuclear distances this small, the electrostatic force is great enough such that HA and HB repel each other as per Newton's third law. Therefore, HB accelerates back towards HC and reforms the BC bond. Hence, the overall collision is unreactive.
Fig. 12 shows a reactive collision. This collision begins in a similar fashion to that described by fig. 11, however, in this case the repulsive force between HA and HB is so great that as HB accelerates back towards HC the BC internuclear distance is reduced to ~0.5 Å. Hence, HC and HB also experience a strong Coulombic repulsion that causes HB to accelerate back towards HA. However, because the momentum of HB colliding with HA this second time round is lower, the nuclei do not get close enough together to repel again, and the bond is formed.
Transition State Theory
Transition state theory assumes that:
Transition states more similar in energy to reactants have a more similar structure to the reactants than the products, and vice-versa.
Atoms behave according to classical mechanics.
Reaction from reactants to products is irreversible.
A general equation for reaction rate is given by d[P]/dt = k[A][B]
The Arrhenius equation allows k (rate constant) to be calculated by k = Aexp(-Ea/RT)
There is a small error between the transition state theory and the calculated trajectories shown in figures 8-11, which is that TST treats the motion of the atoms classically. However, this does not have such a large impact on these reactions, as there is no electron tunneling occurring. The main error here is that TST assumes that the products will form and the reactions are irreversible. This is not the case, as fig. 10 and fig. 11 show that there is some barrier recrossing - the system passes through the transition state more than once before finally not having enough energy to bypass it. This will have an effect on how reliable the TST is when comparing to experimental results. In a reaction, the reactants will have a range of energy values at any one time as dictated by the Boltzmann distribution. This means that some reactants will have momenta high enough to allow for collisions such as those seen in figs. 10 and 11, where the transition state is passed through multiple times. As the TST does not take this into account, it means that the experimental rate will be lower than that predicted by the TST, because some reactants will require multiple collisions before they reach the product state, particularly if they pass through the transition state and then revert to reactants, as in fig. 10. TST does not take into account the fact that both the reactants and products are in equilibrium with the transition state.
(is hammonds postulate an assumption of TST? Lt912 (talk) 08:04, 9 June 2017 (BST))
F-H-H System
Fig 13: Surface plot of an FHH system, with initial estimate of Ea based on inspection.Fig 14: MEP analysis of the FHH system using parameters defined by coordinates of Ea obtained by inspection.Fig 15: Coordinates showing energy value of reactant.Fig 16: Surface plot of HHF system.Fig 17: First estimate of transition state of HHF system.Fig 18: Value to the right of proposed TS, showing TS is a small maximum.Fig 19: Value to the left of proposed TS, showing TS is a small maximum.Fig 20: Side view of MEP analysis of the HHF system using parameters defined by coordinates of Ea obtained by inspection.Fig 21: Top view of MEP analysis of the HHF system using parameters defined by coordinates of Ea obtained by inspection.Fig 21: Energy coorindates of reactants, enabling calculation of Ea.
The surface plots show that the FHH system (H-H+F) is exothermic, and the HHF system (HF+H) is endothermic. This indicates that the H-F bond is stronger than the H-H bond, because lower energy structures are more stable. The HFH system is different; as the products are the same as the reactants, as the two hydrogen atoms are indistinguishable. The transition state for the FHH system is closer to the reactants than products because the reaction is exothermic, as per the Hammond postulate. The transition state for the HHF system is closer to the products because the reaction is endothermic. Hence, in both of these cases the transition state leans towards the higher energy structure, the H-H molecule. Fig. 13 shows an initial prediction for the position of the transition state, gained by inspection using the coordinate marker in MATLAB to search for small increases in potential energy that may indicate the presence of the transition state. After locating this, the AB/BC distances were input into an mep function to see whether or not the trajectory would flow smoothly from reactants to products, as this would indicate the approximate transition state had been found. Fig. 15 shows coordinates before the transition state, hence enabling calculation of Ea. The same procedure was implemented to work out the activation energy of the HHF system. Activation energy of the FHH system was 0.1 kcal mol-1. Activation energy of the HHF system was 30.1 kcal mol-1.
Reaction Dynamics
Fig 22: Surface plot of reactive FHH system.Fig 23: Momentum vs time plot of reactive FHH system.
Fig. 23 illustrates the changes in internuclear momenta in the system. The initial state of the system, where the species present are H-H and F, there are small fluctuations in internuclear momentum between the two hydrogen atoms as they have vibration energy. As the fluoride approaches, there is a big change in the potential energy, as the F-H bond is much stronger than the H-H. The large decrease in potential energy is therefore transferred to vibration energy in the F-H bond. This is seen in Fig. 23 as the changes in internuclear momenta are much larger when the F-H bond is formed than when the H-H bond is present. This could be confirmed experimentally by using IR spectroscopy.
(What anslysis of the IR spectrum would you carry out to confirm this? Lt912 (talk) 08:07, 9 June 2017 (BST))
Polanyi Rules
The Polanyi rules state that vibrational energy is better at promoting a late barrier reaction than translational energy.[1]
Fig 24: Surface plot of reactive FHH system, at low vibrational energy and high translational.Fig 25: Surface plot of reactive FHH system, at low translational energy and high vibrational.Fig 26: Surface plot of reactive HHF system, at low translational energy and high vibrational.Fig 27: Surface plot of reactive HHF system, at low vibrational energy and high translational.
The distribution of energy between the modes affects the efficiency of the reaction because they favour different types of reaction. Translational energy being dominant favours the early transition state, where the structure of the TS is more similar to the reactants than the products. This is because the main barrier to reaction here is the movement of the reagent reaching the reactant. For example, in the FHH/HHF cases, the F-H bond is stronger than the H-H bond and therefore the main barrier to the transition state for HHF is the breaking of the H-F bond, which is aided by strong vibrations in the bond. Conversely, the H-H bond is much weaker, and the transition state is much more similar in structure and energy to the H-H so it takes less vibrational motion for the bond to break easily. Hence in this case, the main barrier to the transition state is the translational motion of the fluoride towards the hydrogen.[2] This is illustrated in figures 24 and 25, using the FHH system. The reaction is successful at low vibrational and high translational energy, but unsuccessful where translational energy is low and vibrational energy is high. The position of the TS therefore affects the reaction because late and early stage transition states affect the type of energy which is favourable.
Bibliography
[1] Zhaojun Zhang, Yong Zhou, and Dong H. Zhang, Journal of Physical Chemistry, 2012, , 3, 3416−3419
[2] Polanyi J.C., Science, 1987, 236, 680-690
