Diagram techniques for solving Schwinger-Dyson equations: Electron transfer pathways in biological molecules

A graph method is developed to solve Schwinger-Dyson equations for electron transfer reactions in biological molecules. Feynman diagrams provide a convenient technique for the calculation of self-energy. Multiple pathway mechanisms of electron transfer can be examined by splitting of the graphical representation into clusters in case of rate-limiting steps. The approximation of weak hopping greatly simplifies the problem of calculating Green functions, which powerfully express a number of characteristics of the process of electron transfer such as the spectral density of states and the correlational function. Rules of graph transformations are derived, and applied to calculate Green matrix elements corresponding to a single hydrogen bond-coupled path in polypeptides, and to the case of the through-backbone pathway. The relation between cluster graphs and Feynman diagrams in locator representation is discussed. Formulas up to the second-order perturbations for linear structure of the cluster graph are given. Calculations of the electron transfer rate dependence on donor-acceptor distance are presented. It is shown that taking into account the second-order perturbation makes the dependence of the logarithm of the electron transfer rate on donor-acceptor distance nonlinear. This effect is especially significant for large distances.