Green’s functions in quantum transition state theory

Tunneling corrections to the classical transition state theory rate are considered for multidimensional anharmonic potentials. A theory is proposed for systematic accounting for the reaction barrier anharmonicity and associated curvature of the reaction path in terms of thermodynamic Green’s functions. A corresponding diagrammatic technique is developed for calculation of the rate constant. The theory allows summation of an infinite number of perturbation terms in a self-consistent manner. It is shown that as a result of self-consistent treatment, a renormalized potential function of the barrier can be introduced in place of the original potential which is the average of the latter over the thermal de Broglie wavelength of the tunneling particle. Even simple approximations for the rate calculated for such averaged potential barrier automatically takes into account a large body of anharmonic and quantum effects. The harmonic approximation of the renormalized potential coincides with the ‘‘best fit’’ quadratic potential surface introduced in the paper of G. Voth, D. Chandler, and W. H. Miller [J. Chem. Phys. 91, 7749 (1989)]. The theory also provides a method for calculation of further anharmonic corrections to this renormalized harmonic approximation.