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 approximately(t)F approximately>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 S(N)2 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 W(IP)[y,F approximately], 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 kappa in the S(N)2 simulations and the dependence of these processes on environmental fluctuations modeled by F approximately. Making the parabolic approximation for the gas-phase part of W(IP)[y,F approximately] yields the following result for the transmission coefficient: kappa = omega-PMF-1x+ = omega-PMF-1-omega-MIP [1 + omega-MIP-2-THETA(x+)]1/2 equal sign with dot omega-PMF-1-omega-MIP [1 + 1/2-omega-MIP-2-THETA(omega-MIP)], where omega-MIP and omega-PMF are, respectively, the barrier frequencies of W(IP)[y,F approximately = 0] and of the potential of mean force, and where THETA(x+) = integral-0/infinity exp(- x+t)-THETA-(t)dt with THETA(t) = (k(B)T)-1<F approximately(t)F approximately>0. This result for kappa, 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 S(N)2 systems.
