A unified framework for semiclassical reaction rate theory

Abstract

A general semiclassical theory for the calculation of reaction rate constants is developed. The theory can be understood as a formal framework that encompasses existing semiclassical methods: instanton theory and semiclassical transition state theory (SCTST). Unlike SCTST, the present formalism does not start from the concept of "good" action-angle variables. Instead, it is based on a conjectured connection between the cumulative reaction probability and the instanton contribution to the formally exact generalisation of Gutzwiller's formula for the trace of the Green's function. The formalism effectively generalises the "imaginary free-energy" formulation of instanton theory to microcanonical scattering rates and all orders in $\hbar$. In one dimension, explicit expressions are derived for the generalised reduced action up to $O(\hbar^4)$ using exact WKB/quantum Hamilton-Jacobi theory. The connection between the present formalism and the standard second order vibrational perturbation theory (VPT2) version of SCTST is explored. It is also shown that the standard thermal instanton rate theory, as well as higher order (dividing surface independent) "perturbative" corrections can be straightforwardly derived from the framework. Above the crossover temperature, first-order corrections in $\hbar$ to the parabolic barrier ("sphaleron") rate are also derived. A simple anharmonic transition state theory and anharmonic version of the Wigner tunneling correction are presented. Finally, the potential for the development of new and improved semiclassical methods for modelling reaction kinetics is discussed.