Energy transfer as a random walk. II. Two-dimensional regular lattices

In this paper, we study the incoherent energy transfer in two-dimensional regular systems where the intermolecular interactions are due to Coulombic and exchange forces. This work follows our treatment for three-dimensional lattices [Blumen and Zumofen, J. Chem. Phys. 75, 892 (1981)], where long-range transfer steps were also taken into account. We analyze the random walk of the excitation for the triangular, square, and hexagonal lattices both analytically and numerically, using series and matrix inversions as well as simulation techniques. We show that in a first passage model the survival probability of a random walker on a lattice with randomly distributed traps is known if the distribution of the number Rn of distinct visited sites is given. The decay law is expressed as a cumulant expansion which involves moments of Rn. The first few moments of Rn are obtained from the numerical procedures; here the simulation is particularly helpful. The determined mean Sn and variance σn2 agree well with analytical asymptotic expressions; we tabulate the results for all lattices and interaction types using convenient analytical forms. Also, the mean number Mn of visits to the origin is given. All results behave smoothly with respect to changes in the interaction range and in coordination number, a fact apparent from a diffusion model which we analyze. From the distribution of Rn, we calculate the survival probability and compare it to approximate forms of current use, thus checking their regions of validity.