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

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Abstract
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 〈F̃(t)F̃〉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,F̃], 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 F̃. Making the parabolic approximation for the gas-phase part of WIP[y,F̃] yields the following result for the transmission coefficient: κ=ω−1PMFx+=ω−1PMFωMIP[1+ω−2 MIPΘ̂(x+)]1/2≠ ω−1PMFω MIP[1+ (1)/(2) ω−2MIPΘ̂(ωMIP)], where ωMIP and ωPMF are, respectively, the barrier frequencies of WIP[y,F̃=0] and of the potential of mean force, and where Θ̂(x+)=∫∞0 exp(−x+t)Θ(t)dt with Θ(t)≡(kBT)−1〈F̃(t)2F〉0. This result for κ, which is equivalent to a result of Grote and Hynes, but which more naturally represents the physics of the early-time regime, permits a straightforward interpretation of the variation of the transmission coefficients for the model SN2 systems.