Semiclassical small-polaron hopping in a generalized molecular-crystal model

The thermally activated rates for high-temperature adiabatic and nonadiabatic small-(bi)polaron hopping are calculated for a generalization of Holstein's molecular-crystal model. In the expanded model a carrier occupying a molecule is coupled to many molecular vibrational modes rather than to just the single vibrational mode envisioned in the original model. This generalization of the molecular-crystal model does not significantly affect the semiclassical small-(bi)polaron jump rates. In particular, for the generalized model the hopping activation energy becomes a sum of contributions associated with each of the vibrational modes to which the carrier is coupled. The vibrational frequency that is the preexponential factor for adiabatic small-(bi)polaron hopping is the square root of the sum of the squares of the vibrational frequencies weighted by their relative contributions to the net hopping activation energy.