Theory of activated rate processes: A new derivation of Kramers’ expression

Abstract

The generalized Langevin equation of motion for a particle trapped in a one-dimensional well with a barrier height V0 and coupled to a dissipative medium is modeled by a harmonic bath. Using the properties of the bath and a normal mode analysis we prove that the reactive frequency defined by Grote and Hynes for averaged motion across the barrier is actually a renormalized effective barrier frequency. We then show that the Kramers–Grote–Hynes expression for the rate of escape over the barrier is just the continuum limit of the usual gas phase harmonic transition state theory expression.