Reactant well

In the limit that the barrier height is large compared with the thermal energy, it is standard practice to expand the potential n ar the reactant well... 

We can approximate this firaction of states in the reactant well, by expanding the potential in a harmonic approximation and assuming that the tempera ture is low compared with the barrier height. This leads to an estimate for the rate constant... 

In my experience, this is a "true statement. Please remember to keep the reactants well iced, though. Now, to begin ... 

For example, when the energy barrier is high compared to the thermal energy, we can assume that when a reactant state is prepared there will be many oscillations in the reactant well before the system concentrates enough energy in the reaction coordinate ... 

We counted the contribution of only those trajectories that have a positive momentum at the transition state. Trajectories with negative momentum at the transition state are moving from product to reactant. If any of those trajectories were deactivated as products, their contribution would need to be subtracted from the total. Why Because those trajectories are ones that originated from the product state, crossed the transition state twice, and were deactivated in the product state. In the TST approximation, only those trajectories that originate in the reactant well are deactivated as product and contribute to the reactive flux. We return to this point later in discussing dynamic corrections to TST. 

Suppose that the reactant well can be approximated as harmonic and the activation energy is much larger than the thermal energy. In that case we can approximate the rate constant... 

If the assumptions underlying the TST are satisfied, the trajectories with initially positive momenta will be trapped in the product well and those with initially negative momentum will be trapped in the reactant well. That will result in a value of k(t) = 1 and the rate constant k =... 

We assume that when the activated reactants cross the transition state a fraction P are deactivated as product and the remaining fraction 1 — f recross the transition state surface [8,24]. If each fraction has roughly the same distribution of momenta as the original fraction, we can say that of the fraction 1 — f that recross, P( — P) will be deactivated in the reactant well and the remaining (1 — P)- will recross the transition state into the... 

But this is not the whole story We not only need to know that a trajectory that crosses the transition state surface is eventually deactivated as product, we also need to know whether it originated from the reactant well A trajectory that originates from the product well and ends up as product won t contribute to the forward rate of reaction. Some of the trajectories did originate as product. We need to find that fraction and subtract it. 

If the wavelengths of the reacting nuclei become comparable to barrier widths, that is, the distance nuclei must move to go from reactant well to product well, then there is some probability that the nuclear wave functions extend to the other side of the barrier. Thus, the quantum nature of the nuclei allows the possibility that molecules tunnel through, rather than pass over, a barrier. 

As in the molecular beam experiment, the reactor volume, pumping speed, and rate of introduction of reactants have values which lead to a flux of reactants well defined in time. Strozier, however, simply doses gas into the vacuum system (reactor) rather than using a molecular beam. He studied CO oxidation, which has nonlinearities in the surface rate equation, so that computer rather than analytic solutions are necessary. The results are represented at constant frequency and varying temperature as shown in Fig. 8, which is a computer simulation (37). 

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

Figure 1. An example of ground state nuclear tunneling along the reaction coordinate (R.C.). The reactant well (R) is on the left side and the product well (P) is on the right. The blue and red lines describe a light and a heavy isotope probability function, respectively.

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

A common approach for the study of activated barrier crossing reactions is the transition state theory (TST), in which the transfer rate over the activation barrier V is given by (0)R/2jt)e where 0)r (the oscillation frequency of the reaction coordinate at the reactant well) is an attempt frequency to overcome the activation barrier. For reactions in solution a multi-dimensional version of TST is used, in which the transfer rate is given by... 

The criterion for spontaneity of a reaction is the value of AG, not AG °. A reaction with a positive AG ° can go in the forward direction if AG is negative. This is possible if the term RT In ([products]/[reactants]) in Equation 13-3 is negative and has a larger absolute value than AG °. For example, the immediate removal of the products of a reaction can keep the ratio [prod-ucts]/[reactants] well below 1, such that the term RT In ([products]/[reactants]) has a large, negative value. 

Here, coR is the frequency of motion in the reactant well, and Eb is the height of the transition-state barrier. Xr is the effective barrier frequency with which the reactant molecule passes, by diffusive Brownian motions through the barrier region and is given by the following self-consistent relation... 

The distance xB - xR and the abscissa x, of the inflexion point are uniquely determined by the barrier height QB and the angular frequencies wR and (oB of the reactant well and the barrier potential. 

The spatial equilibrium is assumed to be near the bottom of the reactant well ... 

Table II Comparison of the ratio k/k E °f quantum rate k over k, which is the TST result corrected for zero-point energy in the reactant well. Also shown are the Landail-Zener and centroid calculations67 and the molecular dynamics with quantum transition result.68...

Bottom this overlap can equivalently be viewed as an interaction energy, H g, between reactant and product surfaces, leading to an avoided crossing, (a) When H g is large (>100 cm ) the reaction remains on the lower surface, and the reaction is "adiabatic", (b) When H g is small, some trajectories may cross to the upper "R" surface and return to the reactant well without making products. 

The energy difference X between the bottom of the reactant well and the product well at the same reaction coordinate is known as the reorganization energy within the assumption of parabolic wells of equal force constants this is related to the activation barrier AG simply by AG = X/4 [8]. 

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