Dynamical effects in electron transfer reactions . II . Numerical solution

In part I a reaction-diffusion equation was introduced for the description of electron transfer reactions which are induced by fluctuations in both the solvent polarization and in the intramolecular vibrational coordinates. We analyze the model employing a generalized moment expansion for the time behavior of the survival probability Q(t), i.e., for the fraction of molecules that have not transferred their electron at time t. Numerical and, in the narrow reaction window limit, analytical solutions are given for the average survival times r. When the contribution of the intramolecular coordinates is appreciable an approximate power-law behavior r a: 7j_, with 0 <a< 1, is found for the dependence of r on the solvent dielectric relaxation time rL, in the large rL regime. Within the framework of the generalized moment description Q(t) is approximated as a superposition of several optimized exponential functions. In the small and intermediate rL regimes it is found that a singleor hi-exponential description, respectively, is sufficient. Simple formulas for such approximations in terms of the average survival times are given. Furthermore it is demonstrated that in the large r L regime a truly multiexponential time behavior for the survival probability is encountered which, over a certain range of time, can appear to be algebraic, i.e., Q(t) a: tr. The relation of these results to experimental data is discussed.