Reaction probabilities harmonic oscillators

The assumption about a uniform probability for any distribution of the energy between the harmonic oscillators may now be used to determine the probability Pet >e (E). It can be expressed as the ratio between the density of states corresponding to the situation where the energy exceeds the threshold energy in the reaction coordinate and the total density of states at energy E, that is, N(E) of Eq. (7.36). 

Here we look at some details of the cumulative reaction probability of the reaction of H + H2 and discuss what picture can be obtained from the analytical expressions of the current theory. Figure 7.7 shows the cumulative reaction probability N E) as a function of the energy E both in linear and log scales. The result of the harmonic approximation (dotted curve) and that of the NF theory (solid curve) are compared. The calculation uses the potential energy surface of Mielke et al. The total angular momentum is fixed to zero for simplicity. This allows us to take the zeroth order approximation as in eqn (7.11), i.e. a collection of harmonic oscillators and a parabolic barrier. Readers interested in the extension of the theory to include the rotational motions should refer to the literature. ... 

Finally, we also emphasize that it is probably not a good idea to take all the vibrational modes as the bath , for a few of them will probably be strongly coupled to the reaction coordinate and this would introduce complicated dynamical structure into the "friction" and "random" forces. If one wishes to apply this procedure quantitatively, then it is probably much more realistic to keep at least the one or two modes most strongly coupled to the reaction coordinate — as measured by the size of the coupling elements l] (s) or (s) — as part of the "system", so that the "bath" then truly is a set of free harmonic oscillators only weakly coupled to it. 

In general, in the above considerations the coordinate x is presumed to describe nuclear motion normal to the intersection line L of the diabatic.potential energy surfaces of reactants and products. In particular cases, however, the coordinate x can coincide with a dynamically separable reaction coordinate. Then, the whole manydimensional problem of calculating the transition probability for any energy value is simply reduced to a one-dimensional one. Such is, for instance, the situation in a system of oscillators making harmonic vibrations with the same frequency in both the initial and final state /67/ which we considered in Sec.3.1.1. The diabatic surfaces (50.1) then represent two similar (N+1>dimensional rotational paraboloids which intersect in a N-dimensional plane S, and the intersection... 

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