Friction and velocity in Kramers’ theory of chemical kinetics

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Abstract
The Fokker–Planck equation for the phase space distribution function is studied as a means of calculating rate constants for chemical reactions. For the case of a symmetric double minimum potential and moderate friction, the eigenfunction of interest is found via a similarity transformation coupled with singular perturbation methods, and the corresponding eigenvalue is obtained by using a variational formula. The resulting expression for the rate constant is seen to be a generalization of the well known Kramers formula. Its range of validity is determined by solving the Fokker–Planck equation numerically, and it is found that the new formula is quite accurate if the barrier height and the friction constant are at least moderately large. For the case of low friction, exact numerical values of the rate constant are obtained and these are used to develop a semitheoretical formula which is valid over the entire range of friction constants. This then permits an evaluation of the range of applicability of the rate constant predicted by the transition state method.