Quantum Effects on the Rate Constant

Highly accurate prediction of transmission coefficients including many degrees of freedom is a very difficult quantum mechanical problem. A simplifying approximation is to consider tunneling only in the degree of freedom corresponding to the reaction coordinate. Within this one-dimensional fonnalism, various levels of approximation are available. 

The simplest approximation is that of Wigner (1932), which takes 

A more robust approximation to k has been provided by Skodje and Truhlar (1981), generalizing earlier work by Bell (1959) for parabolic barriers. For notational convenience we take 

An inspection of the power series expansion for the exponential in Eqs. (15.40) and (15.41) indicates that neither expression diverges as a and become arbitrarily close to equal (an analogous consideration of the power series expansion for the sine function in Eq. (15.40) indicates the first term on the r.h.s. to be similarly free from singularities). 

It should be noted, however, tliat even tlie best one-dimensional tunneling estimate is still likely to underestimate the full tunneling contribution, since tunneling may occur through dimensions of the PES other than die reaction coordinate. Multi-dimensional tunneling approximations are sufficiently complex, however, tliat tliey will not be further discussed here. 

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