Quantum theory of electronic friction

Electronic friction is an important energy loss channel for atoms and molecules scattering off, reacting, or simply vibrating at metallic surfaces. It is usually well described by mixed classical-quantum approaches where the nuclei evolve classically according to Langevin-type equations of motion, and Born-Oppenheimer forces and friction kernels are obtained from first-principles electronic structure calculations. However, classical dynamics falls short when light atoms are involved, which is also the situation where electronic friction becomes the dominant dissipation channel and its role in the dynamics can be unambiguously assessed. Furthermore, the interplay between electronic friction and nuclear quantum effects in molecular processes at surfaces is largely unknown; in fact, it is not even clear how to include electronic friction in a quantum setting. Here we fill this gap by developing a fully quantum theory of electronic friction at $T=0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The electronic bath is considered to be entirely general and can be made of interacting electrons, potentially in a strongly correlated state. The derived friction kernel agrees with a recently obtained mixed quantum-classical result [Dou, Miao, and Subotnik, Phys. Rev. Lett. 119, 046001 (2017)], except for a pseudomagnetic contribution in the latter that is removed here. The ensuing equation of motion for the nuclear wave function is a nonlinear Schr\"odinger equation with a frictional vector potential that depends on the past wave function behavior. The equation becomes local-in-time in the typical situation where the electrons respond rapidly on the slow timescale of the nuclear dynamics (Markov limit) and generalizes previously known Schr\"odinger-Langevin equations to coordinate-dependent, tensorial friction kernels.