Kinetic Theory Expression for the Rate Kernel

Although the analysis in terms of the propagators for independent motion gL is convenient for displaying the content of the kinetic theory expression for the rate kernel, calculations based on (10.4), which contains the propagator for the correlated motion of the AB pair, are probably more convenient to carry out. In kinetic theory, such rate kernel expressions are usually evaluated by projections onto basis functions in velocity space. (We carry out such a calculation in Section X.B). Hence the problem reduces to calculation of matrix elements of (coupled AB motion in a nonreactive system) and subsequent summation of the series. This emphasizes the point that a knowledge of the correlated motion of a pair of molecules for short distance and time scales is crucial for an understanding of the dynamic processes that contribute to the rate kernel. 

This discussion was intended to provide some insight into the dynamic events that are incorporated into the kinetic theory expression for the rate kernel. These include all the events that are normally associated with qualitative ideas concerning caging effects on reaction dynamics. We next indicate how such kinetic theory results might be analyzed further, and how they are related to the diffusion equation results discussed earlier. 

The calculations presented here have simply served to show that a limiting form of the kinetic theory expression for the rate kernel can yield the results of configuration space approaches. However, the real promise of the kinetic theory method lies in the fact that it is not restricted to a description in terms of diffusive propagators, and the consequent motion on these space and time scales. 

The previous sections have attempted to provide some insight into the form of the microscopic expressions for the rate kernels and rate coefficients that characterize condensed-phase reactions. Although the equilibrium one-way flux rate coefficient ky is relatively easy to calculate and under certain circumstances may yield an adequate description of the rate, a variety of important dynamic effects are contained in the relaxing part of the rate kernel, In this section, we describe a kinetic theory that... 

The dynamic process that enter into the rate kemal expression [(9.46) or (8.9)] are, of course, those that have been included in the kinetic equation, as discussed briefly in Section VII. We discuss now the specific processes, which are relevant for the rate kernel, in more detail. The kinetic theory expression contains all the collision events that one might anticipate would be important for liquid state reactions. The analysis of the rate kernel in the limit where velocity relaxation effects are neglected bears a strong similarity to the derivation of Stokes law from kinetic theory, and we also explore this relationship. 

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