Born-Oppenheimer Dynamics, Electronic Friction, and the Inclusion of Electron-Electron Interactions

Abstract

We present a universal expression for the electronic friction as felt by a set of classical nuclear degrees of freedom (DoF's) coupled to a manifold of quantum electronic DoF's; no assumptions are made regarding the nature of the electronic Hamiltonian and electron-electron repulsions are allowed. Our derivation is based on a quantum-classical Liouville equation (QCLE) for the coupled electronic-nuclear motion, followed by an adiabatic approximation whereby electronic transitions are assumed to equilibrate faster than nuclear movement. The resulting form of friction is completely general, but does reduce to previously published expressions for the quadratic Hamiltonian (i.e. Hamiltonians without electronic correlation). At equilibrium, the second fluctuation-dissipation theorem is satisfied and the frictional matrix is symmetric. To demonstrate the importance of electron-electron correlation, we study electronic friction within the Anderson-Holstein model, where a proper treatment of electron-electron interactions shows signatures of a Kondo resonance and a mean-field treatment is completely inadequate.Introduction.-The Born-Oppenheimer (BO) approximation is probably the most important framework underlying modern physics and chemistry. According to the BO approximation, for a system of nuclei and electrons, we split up the total Hamiltonian into the nuclear kinetic energy (T nuc ) and the electronic HamiltonianĤ:

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

“…55 and confirmed in Ref. 53 , the fluctuating force and other finite temperature effects can also be included 54 . Apart from the HGT expression, another important (and very different) type of electronic friction has been given by von Oppen et al, using a NEGF and scattering matrix formalism 47 .…”

“…Furthermore, there has been no conclusion in the literature as to the differences between these friction tensors. For instance, on the one hand, are there perhaps different forms for electronic friction, just as there are many slightly different master equations 53,[63][64][65][66][67] ? When studying rate crossings near a metal surface, Hynes el al derived a non-Markovian friction tensor and argued that this friction tensor was "quite distinct [from the HGT] Markovian electronic friction, acting on a particle impinging from the gas phase on a metallic surface."…”

“…In a recent letter, we have argued that there is a unique, universal electronic friction tensor 53 that is valid in and out of equilibrium, with or without el-el interactions:…”

“…However, the derivation is rather complicated for those without expertise in nonequilibrium field theory, especially when non-Condon effects are incorporated 56 . Furthermore, both the HGT and von Oppen NEGF models effectively assume non-interacting electrons, where electron-electron (el-el) correlation is missing; el-el interactions can give very new and interesting physics at low temperature 26,53,62 . Finally, a third form for electronic friction can also be found in the literature: the friction can be expressed in terms of response/correlation functions 45,53,59 .…”

“…In the literature, quite a variety of friction models have been derived using different methods, some systematic and some ad hoc. Techniques for deriving friction include path integrals 45 , nonequilibrium Green's functions (NEGF) and scattering approaches [46][47][48] , influence functionals 49,50 , Meyer-Miller (MM) mappings 51,52 , quantum-classical Liouville equation projections 53 , as well as simple Fermi-Golden rule arguments 54,55 . The final expression for electronic friction can also be expressed in a variety of different forms, i.e.…”