Accuracies of the empirical theories of the escape probability based on Eigen model and Braun model compared with the exact extension of Onsager theory

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Abstract
This paper deals with the exact extension of the original Onsager theory of the escape probability to the case of finite recombination rate at nonzero reaction radius. The empirical theories based on the Eigen model and the Braun model, which are applicable in the absence and presence of an external electric field, respectively, are based on a wrong assumption that both recombination and separation processes in geminate recombination follow exponential kinetics. The accuracies of the empirical theories are examined against the exact extension of the Onsager theory. The Eigen model gives the escape probability in the absence of an electric field, which is different by a factor of 3 from the exact one. We have shown that this difference can be removed by operationally redefining the volume occupied by the dissociating partner before dissociation, which appears in the Eigen model as a parameter. The Braun model gives the escape probability in the presence of an electric field, which is significantly different from the exact one over the whole range of electric fields. Appropriate modification of the original Braun model removes the discrepancy at zero or low electric fields, but it does not affect the discrepancy at high electric fields. In all the above theories it is assumed that recombination takes place only at the reaction radius. The escape probability in the case when recombination takes place over a range of distances is also calculated and compared with that in the case of recombination only at the reaction radius.