Theory of liquid-state activated barrier crossing: The instantaneous potential and the parabolic model.

This paper gives a theoretical treatment of liquid-phase activated barrier crossing that is valid for chemical reactions which occur on typical (e.g., high activation barrier) potential-energy surfaces. This treatment is based on our general approach [S. A. Adelman, Adv. Chem. Phys. 53, 61 (1983)] to problems in liquid-phase chemical dynamics. We focus on the early-time regime [times short compared to the relaxation time of <F(t)F>0, the fluctuating force autocorrelation function of the reaction coordinate] in which the solvent is nearly ‘‘frozen.’’ This regime has been shown to be important for the determination of the rate constant in the molecular-dynamics simulations of model aqueous SN2 reactions due to Wilson and co-workers. Our treatment is based on a generalized Langevin equation of motion which naturally represents the physics of the early-time regime.In this regime the main features of the reaction dynamics are governed by the instantaneous potential WIP[y,F], which accounts for the cage confinement forces which dominate the liquid-phase effects at early times, rather than by the familiar potential of mean force. The instantaneous potential is derived from the t→0 limit of the equation of motion and its properties are developed for both symmetric and nonsymmetric reactions. The potential is then shown to account for both the early-time barrier recrossing processes found to determine the transmission coefficient κ in the SN2 simulations and the dependence of these processes on environmental fluctuations modeled by F. Making the parabolic approximation for the gas-phase part of WIP[y,F] yields the following result for the transmission coefficient: κ=ω-1PMFx+=ω-1PMFωMIP[1+ω-2 MIPΘ(x+)]1/2≠ ω-1PMFω MIP[1+ (1)/(2) ω-2MIPΘ(ωMIP)], where ωMIP and ωPMF are, respectively, the barrier frequencies of WIP[y,F=0] and of the potential of mean force, and... [ABSTRACT FROM AUTHOR]