Derivation of Kramers’ formula for condensed phase reaction rates using the method of reactive flux

Abstract

Kramers’ formula for the rate of barrier crossing as a function of solvent friction is here rederived using the method of reactive flux. In the reactive flux formalism trajectories are started at the top of the barrier and propagated forward for a short time, to determine whether they are reactive or not. In isolated molecules it is customary to associate with each set of initial conditions a reactivity index (traditionally known as the characteristic function), which is 1 for a reactive trajectory and 0 for a nonreactive trajectory. In this paper we suggest that if the solvent interaction with the system is treated stochastically, it is appropriate to generalize the reactivity index to fractional values between 0 and 1, to take into account an ensemble average over different stochastic histories. We show how this fractional reactivity index can be calculated analytically, by using an analytic solution of the phase space Fokker–Planck equation. Starting with the distribution δ(x)δ(u−u0) that originates at the top of a parabolic barrier (x=0) at t=0, the fraction of the distribution function that is to the right of x=0, in the limit that t→∞, is the fractional reactivity index. The analytical expression for the fractional reactivity index leads immediately to Kramers’ expression for the rate constant. The derivation shows explicitly that the dynamical origin of Kramers’ prefactor is trajectories that recross the barrier. The evolution of the phase space distribution that originates at the top of the barrier highlights an interesting underlying phase space structure of this system, which may be considered as a paradigm for dissipative systems whose underlying dynamics is unstable.