  In molecular devices electronic degrees of freedom are coupled to vibrational
modes of the molecule, offering an opportunity to study fundamental aspects of
this coupling between at the nanoscale. To this end we consider the
nonequilibrium heat exchange between a conduction band and a bosonic bath
mediated by a single molecule. For molecules large enough so that on-site
interactions can be dropped we carry out an asymptotically exact calculation of
the heat current, governed by the smallness of the electron-phonon coupling,
and obtain the steady state heat current driven by a finite temperature drop.
At low temperatures the heat current is found to have a power-law behavior with
respect to the temperature difference with the power depending on the nature of
the bosonic bath. At high temperatures, on the other hand, the current is
linear in the temperature difference for all types of bosonic baths. The
crossover between these behaviors is described. Some of the results are given a
physical explanation by comparing to a perturbative Master equation calculation
(whose limitation we examine).
