Unified model of energy transfer for arbitrary Franck-Condon offset and temperature

The Huang-Rhys-Pekar W_p(T, S) function is known to give the thermal-Franck-Condon (FC) weight of single-center radiative and non-radiative transitions for arbitrary energy mismatch p?ω₀, temperature T, and FC offset S. The thermal-FC weight for energy transfer is shown also to be W_p(T, S), with S enlarged to be the sum of the S values of the energy donor and energy acceptor. This result generalizes an earlier result of Miyakawa and Dexter for low-temperature Poisson functions. The W_p function with enlarged S is equivalent to the overlap of the donor emission and acceptor absorption bands in the Förster-Dexter theory. Experimental verification is good except for the p dependence of non-radiative and energy-transfer rates for S small (e.g. 4f → 4f transitions). Here, if the smooth, exponential p dependences (exponential energy-gap laws) that have been drawn through many 4f → 4f transition rates in certain hosts (Y₂O₃, LaF₃) are fitted with a W_p(T, S) function with a single S value, this S value is too large to describe 4f → 4f transitions.