We employ the functional renormalization group to study the effects of phonon-assisted tunneling on the nonequilibrium steady-state transport through a single level molecular quantum dot coupled to electronic leads. Within the framework of the spinless Anderson-Holstein model, we focus on small to intermediate electron-phonon couplings, and we explore the evolution from the adiabatic to the antiadiabatic limit and also from the low-temperature non-perturbative regime to the high temperature perturbative one. We identify the phononic signatures in the bias-voltage dependence of the electrical current and the differential conductance. Considering a temperature gradient between the electronic leads, we further investigate the interplay between the transport of charge and heat. Within the linear response regime, we compare the temperature dependence of various thermoelectric coefficients to our earlier results obtained within the numerical renormalization group [Phys.~Rev.~B {bf 96}, 195156 (2017)]. Beyond the linear response regime, in the context of thermoelectric generators, we discuss the influence of molecular vibrations on the output power and the efficiency. We find that the molecular energy dissipation, which is inevitable in the presence of phonons, is significantly suppressed in the antiadiabatic limit resulting in the enhancement of the thermoelectric efficiency.
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