Accelerated calculation of integrals in the theory of non-thermal electron transfer

In this article we’ve carried out a modification of the expression for calculating the probability of non-thermal electron transfer in a two-level system obtained in the framework of the nonstationary second-order perturbation theory for electron interaction. This expression takes into account the reorganization of the environment (Debye model) and high-frequency intramolecular vibrational modes (quantum mechanical description). Since the “Debye” part is characterized by slow relaxation times, which significantly exceed the period of intramolecular vibrations, the “quantum” part will be a rapidly oscillating function included in the integral. As a result, computing two-dimensional integral using simple grid methods may require significant computational resources. To speed up the calculation of this integral, we propose a Gauss quadrature formula. It is shown that, in the high-temperature limit, the quadrature formula can be successfully applied in the theory of non-thermal electron transfer and increases the calculation speed of the integral by two orders of magnitude.