Transition state theory, transmission probabilities

For another perspective we mention a second approach of which the reader should be aware. In this approach the dividing surface of transition state theory is defined not in terms of a classical mechanical reaction coordinate but rather in terms of the centroid coordinate of a path integral (path integral quantum TST, or PI-QTST) [96-99] or the average coordinate of a quanta wave packet. In model studies of a symmetric reaction, it was shown that the PI-QTST approach agrees well with the multidimensional transmission coefScient approach used here when the frequency of the bath is high, but both approaches are less accurate when the frequency is low, probably due to anharmonicity [98] and the path centroid constraint [97[. However, further analysis is needed to develop practical PI-QTST-type methods for asymmetric reactions [99]. 

A more sophisticated reaction path approach is to replace in eq. (40) by 8jjq(sJj This is the essence of the approach taken by Garrett and Truhlar and generalized by Miller et al. and Skodje and Truhlar for polyatomic reactions. Truhlar and coworkers have proposed one-dimensional paths which deviate from the reaction path in order to compute accurate tunneling probabilities from which transmission coefficients (see below) are then used to correct their version of variational transition state theory, the so-called improved canonical variational [transition state] theory (ICVT) (also see below). 

Given the n-th adiabatic barrier height En one can easily formulate an adiabatic transition state theory for the reaction probability, from the n-th reagents vibrational state pA (Ex)- The simplest estimate is unit transmission probability for translational energies (E-p) greater than the barrier height and zero otherwise ... 

We saw in Chapter 7 that the transmission coefficient k takes into account the fact that the activated complex does not always pass through to the transition state and the term kT/h arises from consideration of motions that lead to the decay of the activated complex into products. It follows that, in the case of an electron transfer process, K kT/h) can be thought of as a measure of the probability that an electron will move from D to A in the transition state. The theory due to R.A. Marcus supposes that this probability decreases with increasing distance between D and A in the DA complex. More specifically, for given values of the temperature and A G, the rate constant varies with the edge-to-edge distance... 

Equation (156.111) is valid, however, also in the case of a dynamically non-separable reaction coordinate if the classical motion along it is very fast, so that the condition (72.Ill) is fulfilled from reactants to transition region (if there is one), which assures the possibility of a non-adiabatic separation of the reaction coordinate, as dicussed in Sec.4.2.II. In this situation the semiclassical collision theory expression (78.Ill) is justified in which the transmission coefficient X is defined by (78 lII). Thetransition probability is then independent of the quantum state of reactants and may be computed, according to (55 III), by setting 12 x where... 

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