Variational transition state theory for reactions in condensed phases

Abstract

A generalization of the Kramers–Grote–Hynes theory for reaction rates in the spatial diffusion limit is derived for a general class of Hamiltonians. Previous restrictions to harmonic baths and bilinear system bath couplings are removed. The key ingredient is the systematic use of variational transition state theory (VTST) to identify the optimal dividing surface. A pair of collective modes are defined as a linear combination of all system and bath modes. A free energy surface is defined in the two degree of freedom collective mode phase space. The VTST estimate for the rate of reaction on this surface is shown to be an upper bound to the exact rate. The optimal definition of the collective modes is obtained by minimizing the rate. The resulting rate expression is formally identical to the Kramers–Grote–Hynes theory. However, the minimization procedure leads to a new definition of the time dependent friction. In consistence with transition state theory, this time dependent friction is constructed from equilibrium properties of the composite system and does not call for any dynamical computations. The friction parameters are determined from equilibrium centroid averages of partial derivatives of the full potential at the barrier of the potential of mean force. This removes previous ambiguities as to the definition of time dependent friction in condensed matter systems. A procedure is presented for finding collective modes along which the friction exerted by the bath is minimized. This result may be of substantial interest in the study of complex dynamical systems in biology, chemistry, and physics.