Multi-dimensional tunneling approximation

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. 

In eqn (12.19), the effects of nonadiabatic transition including the nuclear tunneling are properly taken into account by rj and naturally the main task is to evaluate the thermally averaged transition probability P fl, < ), which has to be evaluated using the Monte Carlo technique for multi-dimensional systems. It is easily shown that the Marcus-Hush formula in adiabatic and nonadiabatic limits can be recovered from eqn (12.19) and eqn (12.20) within the high-temperature approximation. 

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