  We investigate the spinless Anderson-Holstein model routinely employed to
describe the basic physics of phonon-assisted tunneling in molecular devices.
Our focus is on small to intermediate electron-phonon coupling; we complement a
recent strong coupling study [Phys.~Rev.~B {87}, 075319 (2013)]. The entire
crossover from the antiadiabatic regime to the adiabatic one is considered. Our
analysis using the essentially analytical functional renormalization group
approach backed-up by numerical renormalization group calculations goes beyond
lowest order perturbation theory in the electron-phonon coupling. In
particular, we provide an analytic expression for the effective tunneling
coupling at particle-hole symmetry valid for all ratios of the bare tunnel
coupling and the phonon frequency. It contains the exponential polaronic as
well as the power-law renormalization; the latter can be traced back to x-ray
edge-like physics. In the antiadiabatic and the adiabatic limit this expression
agrees with the known ones obtained by mapping to an effective interacting
resonant level model and lowest order perturbation theory, respectively. Away
from particle-hole symmetry, we discuss and compare results from several
approaches for the zero temperature electrical conductance of the model.
