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

Abstract

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.

Hybrid exciton state in a quantum dot–dendrite system: Green functions

“…To solve the Schwinger-Dyson equations for the dendrimer system with many generations, it is convenient to use the "electron transfer graph" method developed in [13,14], where the method is used to study electron transfer between localized sites. Then for the SchwingerDyson equation (7), every site corresponds to a graph vertex.…”

“…The details of the method and the proof of some formulae can be found in [13]. In this section we just write briefly about definitions and terminology, the main method and results of the authors [13] which we will use in the following sections.…”

“…For illustration, if the chain of sites is three sites as in Fig.3b, the cyclic term and the cyclic value of the graph will be as in the Fig.3c, or [13]:…”

“…In this case it was proved in [13] that the diagonal element of the Green's function of side-graph can be expressed through the Green's function of subgraph in the continued fraction representation. The value of the loop with the sidegroup could be expanded into continuous fraction of loops values of extended side graphγ-the subgraph of the graph.…”

“…We apply here the " real space" Green's function method developed in [13,14]. The double-time Green's functions are writtten as…”