Universality of electronic friction: Equivalence of von Oppen's nonequilibrium Green's function approach and the Head-Gordon-Tully model at equilibrium

For a molecule moving near a single metal surface at equilibrium, following von Oppen and coworkers [N. Bode, S. V. Kusminskiy, R. Egger, and F. von Oppen, Beilstein J. Nanotechnol. 3, 144 (2012)] and using a nonequilibrium Green's-function (NEGF) approach, we derive a very general form of electronic friction that includes non-Condon effects. We then demonstrate that the resulting NEGF friction tensor agrees exactly with the Head-Gordon-Tully model, provided that finite temperature effects are incorporated correctly. The present results are in agreement with our recent claim that there is only one universal electronic friction tensor arising from the Born-Oppenheimer approximation [W. Dou, G. Miao, and J. E. Subotnik, Phys. Rev. Lett. 119, 046001 (2017)].