A generalization of Kramers’ rate formula to include some anharmonic effects

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Abstract
Consideration from the Langevin approach of Brownian‐motion effects on a particle in a parabolic barrier potential leads to a transmission function which gives the probability that the particle will surmount the barrier. When used in conjunction with an approximate low‐temperature normalization condition, the Kramers rate formula, originally derived using the Fokker–Planck approach, is reproduced. The rate formula is then generalized by including anharmonic effects due to the presence of the barrier as they enter in an exact normalization condition. The generalized Kramers formula has a temperature dependence of the frequency factor which is verified by computer simulation for a periodic and double‐well potential. Data from computer experiments are fitted using both the original and generalized formulas. The generalized formula is found to be useful in extracting information on the barrier height and friction coefficient from the experimental data.