Orbital Surface Hopping with an Electron Thermostat Yields Accurate Dynamics and Detailed Balance

In mixed quantum-classical simulations of molecule-metal surface interactions, the discretization of the metallic electronic continuum typically results in a closed-system representation that fails to capture the open-system nature of the true physical process. This approximation can introduce significant artifacts, including deviations in the dynamical evolution and a violation of the principle of detailed balance. To address this fundamental challenge, we introduce an electronic thermostat into our previously developed orbital surface hopping (OSH) framework, generalizing the method to efficiently handle many discrete electronic states. We first outline the derivation of electronic thermostat orbital surface hopping, where the amplitude of the electronic thermostat is well justified. We then demonstrate that this method can reproduce accurate dynamics and detailed balance in long time, whereas without electronic thermostat the detailed balance is violated. Thus, this method offers a reliable tool for studying nonadiabatic dynamics near metal surfaces.