Activated rate processes: Generalization of the Kramers–Grote–Hynes and Langer theories

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Abstract
The variational transition state theory approach for dissipative systems is extended in a new direction. An explicit solution is provided for the optimal planar dividing surface for multidimensional dissipative systems whose equations of motion are given in terms of coupled generalized Langevin equations. In addition to the usual dependence on friction, the optimal planar dividing surface is temperature dependent. This temperature dependence leads to a temperature dependent barrier frequency whose zero temperature limit in the one dimensional case is just the usual Kramers–Grote–Hynes reactive frequency. In this way, the Kramers–Grote–Hynes equation for the barrier frequency is generalized to include the effect of nonlinearities in the system potential. Consideration of the optimal planar dividing surface leads to a unified treatment of a variety of problems. These are (a) extension of the Kramers–Grote–Hynes theory for the transmission coefficient to include finite barrier heights, (b) generalization of Langer’s theory for multidimensional systems to include both memory friction and finite barrier height corrections, (c) Langer’s equation for the reactive frequency in the multidimensional case is generalized to include the dependence on friction and the nonlinearity of the multidimensional potential, (d) derivation of the non‐Kramers limit for the transmission coefficient in the case of anisotropic friction, (e) the generalized theory allows for the possibility of a shift of the optimal planar dividing surface away from the saddle point, this shift is friction and temperature dependent, (f) a perturbative solution of the generalized equations is presented for the one and two dimensional cases and applied to cubic and quartic potentials.