  The Wiedemann-Franz (WF) law is a fundamental result in solid-state physics
that relates the thermal and electrical conductivity of a metal. It is derived
from the predominant origin of energy conversion in metals: the motion of
quasi-free charge-carrying particles. Here, an equivalent WF relationship is
developed for molecular systems in which charge carriers are moving not as free
particles but instead hop between redox sites. We derive a concise analytical
relationship between the electrical and thermal conductivity generated by
electron hopping in molecular systems and find that the linear temperature
dependence of their ratio as expressed in the standard WF law is replaced by a
linear dependence on the nuclear reorganization energy associated with the
electron hopping process. The robustness of the molecular WF relation is
confirmed by examining the conductance properties of a paradigmatic molecular
junction. This result opens a new way to analyze conductivity in molecular
systems, with possible applications advancing the design of molecular
technologies that derive their function from electrical and/or thermal
conductance.
