Kramers’s theory of chemical kinetics: Eigenvalue and eigenfunction analysis

Abstract

The Smoluchowski equation for a double‐minimum potential is studied as a means of calculating rate constants for chemical reactions. In the one‐dimensional case, a symmetric fourth‐degree potential is used, and the solution is obtained in terms of eigenvalues and eigenfunctions. Asymptotic expressions for the lower eigenfunctions are found by means of singular perturbation theory, and the corresponding eigenvalues are obtained via a variational principle. A quantum‐mechanical analogy is used to generate the higher eigenvalues and eigenfunctions. The expression for the rate constant is seen to be a generalization of earlier results, and its range of validity is determined by solving the appropriate equations numerically. It is found that the new formula is accurate over a rather wide range of barrier heights. In the three‐dimensional case, the rate constant is again found by means of a variational calculation, using as a trial function the eigenfunction for a separable potential. The result is seen to reduce to the more familiar expression in the appropriate limit, and its range of validity is investigated through direct numerical computations of the eigenvalues.