An imaginary energy method-based formulation of a quantum rate theory

Abstract

An imaginary energy method is used to derive a rate constant expression that leads to the WKB tunneling rate at low temperature and to the quantum transition state rate at high temperature. For weak coupling, the imaginary energy method also leads to Fermi’s Golden Rule formula. The rate expression is analyzed for the model of one-dimensional motion in a metastable potential profile with linear coupling to a bath of harmonic oscillators (Kramers’ problem). We recover the results of the Kramers, Grote-Hynes, and Wolynes (KGHW) theory from the rate expression. Our rate expression is given, in part, in terms of partition functions and is suited to a path integral treatment. The path integral evaluation of the rate constant also yields the KGHW formula. We use the same expression to analyze the low-temperature behavior of tunneling in a double-well potential for a system linearly coupled to a bath, to provide a new expression for electron transfer in the case of strong coupling of the electron to its two sites.