Energy transfer as a random walk on regular lattices

In this paper we study the incoherent energy transfer in molecular crystals, where the intermolecular interactions are due to electrostatic forces. We analyze the random walk of the excitation both analytically and numerically and we take into account the occurence of long-range transfer steps. For five lattices (simple-cubic, body-centered-cubic, face-centered-cubic, diamond, and a sparse-cubic lattice) we present the average number Mn of returns to the origin and the mean number Sn of distinct sites visited on an n-step walk, where the steps are due to different multipolar interactions. The computations are carried out using both simulation and matrix inversion methods and they are checked against analytical asymptotic forms. We give the coefficients of these analytical expressions obtained from the first four terms in the expansion of the generating functions for nearest-neighbor walks. The results obtained behave smoothly with respect to changes in the range of the interaction and in the coordination number of the lattice (number of nearest neighbors). This allows us to use the continuous description of the random walk in terms of a diffusion model to obtain convenient semiquantitative expressions for Mn and Sn for different lattices. Since the methods presented vary greatly in their degree of sophistication, we are also able to demonstrate the advantages and drawbacks of the different approaches under realistic conditions.