Molecular electronic states near metal surfaces at equilibrium using potential of mean force and numerical renormalization group methods: Hysteresis revisited.

We investigate equilibrium observables for molecules near metals by employing a potential of mean force (PMF) that takes level broadening into account. Through comparison with exact data, we demonstrate that this PMF approach performs quite well, even for cases where molecule-electrode couplings depend on nuclear position. As an application, we reexamine the possibility of hysteresis effects within the Anderson-Holstein model (i.e., an impurity coupled both to a metal surface and a nuclear oscillator). As compared against the standard mean field approach by Galperin et al. [Nano Lett. 5, 125 (2005)], our PMF approach agrees much better with exact results for average electronic populations both at zero and finite temperature; we find, however, that mean field theory can be very useful for predicting the onset of dynamical instabilities, metastable states, and hysteresis.