DERIVATION OF KRAMERS FORMULA FOR CONDENSED-PHASE REACTION-RATES USING THE METHOD OF REACTIVE FLUX

Kramers' formula for the rate of barrier crossing as a function of sol vent 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 as sociate with each set of initial conditions a reactivity index (tradit ionally known as the characteristic function), which is 1 for a reacti ve trajectory and 0 for a nonreactive trajectory. In this paper we sug gest that if the solvent interaction with the system is treated stocha stically, it is appropriate to generalize the reactivity index to frac tional values between 0 and 1, to take into account an ensemble averag e over different stochastic histories. We show how this fractional rea ctivity index can be calculated analytically, by using an analytic sol ution of the phase space Fokker-Planck equation. Starting with the dis tribution delta(x) delta(u - u0) that originates at the top of a parab olic barrier (x=0) at t=0, the fraction of the distribution function t hat is to the right of x=0, in the limit that t-->infinity, is the fra ctional reactivity index. The analytical expression for the fractional reactivity index leads immediately to Kramers' expression for the rat e constant. The derivation shows explicitly that the dynamical origin of Kramers' prefactor is trajectories that recross the barrier. The ev olution 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.