Numerical test of Kramers reaction rate theory in two dimensions

Abstract

The Fokker–Planck–Kramers equation for a system composed by a reactive coordinate x coupled to a solvent coordinate y is employed to study the effect of additional degrees of freedom on the dynamics of reactive events. The system is studied numerically in the diffusional regimes of both coordinates, for different topologies of the bistable potential function and anisotropies of friction. The eigenvalue spectrum is evaluated by representing the time evolution operator over a basis set of orthonormal functions. A detailed analysis of the effect of the explicit consideration of the slow nonreactive mode is carried on to show that a variation of qualitative picture (scenario) of the reaction dynamics occurs when friction along different directions is strongly anisotropic, depending also on the structure of the two-dimensional potential surface. The numerical study supports both the qualitative picture of the reaction dynamics and the rate constant expressions obtained analytically. For those cases where the Langer theory has a restricted range of applicability because of the change in the reaction dynamics scenario, this fact has been numerically demonstrated. Here the Langer expression for the rate constant is replaced by the one obtained as a result of the consideration of the effective one-dimensional problem along the solvent coordinate, characterized by a smaller activation energy than that in the initial problem. All of these facts were confirmed by the numerical test, which shows a satisfactory agreement with the analytical results.