Electronic friction near metal surfaces: A case where molecule-metal couplings depend on nuclear coordinates

We derive an explicit form for the electronic friction as felt by a molecule near a metal surface for the general case that molecule-metal couplings depend on nuclear coordinates. Our work generalizes a previous study by von Oppen et al. [Beilstein J. Nanotechnol. 3, 144 (2012)], where we now go beyond the Condon approximation (i.e., molecule-metal couplings are not held constant). Using a non-equilibrium Green's function formalism in the adiabatic limit, we show that fluctuating metal-molecule couplings lead to new frictional damping terms and random forces, plus a correction to the potential of mean force. Numerical tests are performed and compared with a modified classical master equation; our results indicate that violating the Condon approximation can have a large effect on dynamics.