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In a recent publication [W. Dou, G. Miao, and J. E. Subotnik, Phys. Rev.Lett. 119, 046001 (2017)], using the quantum-classical Liouville equation(QCLE), we derived a very general form for the electronic friction felt by amolecule moving near one or many metal surfaces. Moreover, we have alreadyproved the equivalence of the QCLE electronic friction with theHead-Gordon--Tully model as well as a generalized version of von Oppen'snonequilibrium Green's function (NEGF) method at equilibrium [W. Dou and J. E.Subotnik, Phys. Rev. B 96, 104305 (2017)]. In the present paper, we now furtherprove the equivalence between the QCLE friction and the NEGF friction for thecase of multiple metal surfaces and an out-of-equilibrium electronic current.The present results conclude our recent claim that there is only one universalelectronic friction tensor arising from the Born-Oppenheimer approximation.

physical review. b./physical review. b

The universality of electronic friction II: Equivalence of the  quantum-classical Liouville equation approach with von Oppen's nonequilibrium  Green's function methods out of equilibrium