  A framework for estimating heating and expected temperature rise in current
carrying molecular junctions is described. Our approach is based on applying
the Redfield approximation to a tight binding model for the molecular bridge
supplemented by coupling to a phonon bath. This model, used previously to study
thermal relaxation effects on electron transfer and conduction in molecular
junctions, is extended and used to evaluate the fraction of available energy,
i.e. of the potential drop, that is released as heat on the molecular bridge.
Classical heat conduction theory is then applied to estimate the expected
temperature rise. For a reasonable choice of molecular parameters and for
junctions carrying currents in the nA range, we find the temperature rise to be
a modest few degrees. It is argued, however, that using classical theory to
describe heat transport away from the junction may underestimate the heating
effect.
