Activated rate processes: Finite-barrier expansion for the rate in the spatial-diffusion limit

A dynamically corrected variational transition-state theory is formulated for the thermally activated escape of a particle trapped in a potential well separated from a different well or continuum by a barrier and coupled to a heat bath. The theory is based on the Hamiltonian-equivalent formulation of the generalized Langevin equation. The dynamical corrections are obtained by utilizing the reactive-flux method in which the choice of dividing surface is guided by minimization of the transition-state flux. Analytic correction formulas, valid for memory friction, are obtained for the Kramers-Grote-Hynes estimate of the rate in the range from moderate friction to the large-friction limit. The analytic expansion is in terms of the inverse barrier height (1/β${\mathit{V}}^{\mathrm{‡}}$). For the special case of an extended Smoluchowski equation containing finite damping corrections, the exact expansion is also obtained using the mean-first-passage-time formulation. The dynamically corrected variational transition-state-theory expansion is shown to be identical to the mean-first-passage-time result.