Nonadiabatic dynamics at metal surfaces: Surface hopping and electronic friction

The coupled electron-nuclear dynamics at molecule-metal interfaces that involve electrons transfer as well as nuclear motions are intrinsically nonadiabatic and difficult to model. Here we present two approaches to treat such nonadiabatic dynamics. In the weak molecule-metal interaction regime, we propose a surface-hopping scheme, where nuclei evolve on diabatic potential energy surfaces with stochastic hopping between them. We demonstrate that such a surface-hopping scheme gives correct detailed balance and recovers Marcus electron transfer rate at molecule-metal interfaces. In the strong molecule-metal interaction regime, we derive a Langevin equation, where classical nuclear degrees of freedom evolve on an adiabatic potential energy surface, while experiencing electronic friction and random force from electronic response. Our form of electronic friction is completely general, but does reduce to previously published expressions without electron-electron interaction. When electron-electron correlations are included, we show electronic friction exhibits Kondo resonance, whereas a mean-field treatment is completely inadequate.