A general formula for the rate of resonant transfer of energy between two electric multipole moments of arbitrary order using molecular quantum electrodynamics

A general expression is derived for the matrix element for the resonant transfer of energy between an initially excited donor species and an acceptor moiety in the ground state, with each entity possessing an electric multipole moment of arbitrary order. In the quantum electrodynamical framework employed, the coupling between the pair is mediated by the exchange of a single virtual photon. The probability amplitude found from second-order perturbation theory is a product of the electric moments located at each center and the resonant multipole-multipole interaction tensor. Using the Fermi golden rule, a general formula for the rate of energy transfer is obtained. As an illustration of the efficacy of the theory developed, rates of excitation energy exchange are calculated for systems interacting through dipole-quadrupole, dipole-octupole, quadrupole-quadrupole, and the familiar dipole-dipole coupling. For each of the cases examined, the near- and far-zone limits of the migration rate are calculated from the result valid for all donor-acceptor separations beyond wave function overlap. Expression of the octupole contribution to the transfer rate in terms of its irreducible components of weights 1 and 3 leads to new features. The octupole weight-1 term is found to contribute only when the interaction is retarded, while the dipole-octupole weight-1 contribution appears as a higher-order correction term to the dipole-dipole rate. Order of magnitude estimates are given for the contributions of dipole-quadrupole and dipole-octupole terms relative to the leading dipole-dipole rate for near-, intermediate-, and far-zone separations to further understand the role played by higher multipole moments in the transfer of excitation and the mechanism dominating the process.