Vibration-induced correction to the current through a single molecule

We provide analytical results for the perturbative correction to the current-voltage relation through a vibrating molecule for weak electron-phonon coupling. The nonlinear conductance exhibits a step-like feature at $eV=\hbar \omega_0$, where $\omega_0$ is the vibration frequency. We establish criteria for the sign change of the step in the conductance (up or down). This transition turns out to be nonuniversal and is governed by essentially all system parameters.