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.
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