  Computationally inexpensive approximations describing electron-phonon
scattering in molecular-scale conductors are derived from the non-equilibrium
Green's function method. The accuracy is demonstrated with a first principles
calculation on an atomic gold wire. Quantitative agreement between the full
non-equilibrium Green's function calculation and the newly derived expressions
is obtained while simplifying the computational burden by several orders of
magnitude. In addition, analytical models provide intuitive understanding of
the conductance including non-equilibrium heating and provide a convenient way
of parameterizing the physics. This is exemplified by fitting the expressions
to the experimentally observed conductances through both an atomic gold wire
and a hydrogen molecule.
