  The effect of a thermal environment on electron (or hole) transfer through
molecular bridges and on the electron conduction properties of such bridges is
studied. Our steady state formalism based on an extension of the Redfield
theory (D. Segal et al, J. Phys. Chem. B 104 (2000) 3817; Chem. Phys. 268
(2001) 315) is extended in two ways: First, a better description of the weak
coupling limit, which accounts for the asymmetry of the energy dependence of
the quasi-elastic component of the transmission is employed. Secondly, for
strong coupling to the thermal bath the small polaron transformation is
employed prior to the Redfield expansion. It is shown that the thermal coupling
is mainly characterized by two physical parameters: The reorganization energy
that measures the coupling strength and the correlation time (or its inverse -
the spectral width) of the thermal bath. Implications for the observed
dependence of the bridge-length dependence of the transmissions are discussed.
It is argued that in the intermediate regime between tunneling behavior and
site-to-site thermally induced hopping, the transport properties may depend on
the interplay between the local relaxation rate and the transmission dynamics.
