Frictional effects near a metal surface

Abstract

When a classical master equation (CME) is used to describe the nonadiabatic dynamics of a molecule at metal surfaces, we show that in the regime of reasonably strong molecule-metal couplings, the CME can be reduced to a Fokker-Planck equation with an explicit form of electronic friction. For a single metal substrate at thermal equilibrium, the electronic friction and random force satisfy the fluctuation-dissipation theorem. When we investigate the time scale for an electron transfer (ET) event between the molecule and metal surface, we find that the ET rates show a turnover effect (just as in Kramer's theory) as a function of frictional damping.

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

“…Several other research groups have also identified the same effective friction tensor [28] using different methodologies, some using influence functionals [29] and some using non-equilibrium Green's functions (NEGF) [30,31]. The effects of non-Condon terms have also been considered [32][33][34] at finite temperature.…”

Lindblad equation and its semiclassical limit of the Anderson-Holstein model

“…where we follow the notation of [5]:d andd † are the annihilation and creation operators for the two-level electron state of the molecule,ĉ k andĉ † k are the annihilation and creation operators for electron states in the bath, E k is the energy level of those states, µ is the Fermi level, and V k is the coupling strength between the molecule and the k-th mode in bath, assumed to be real. The Hilbert space corresponds to the molecule is thus L 2 (R) ⊗ C 2 = L 2 (R) ⊗ span |0〉, |1〉 , so that we haved |1〉 = |0〉 andd † |0〉 = |1〉.…”

“…In the first part, we attempt to justify the formal derivation of [5] in a more mathematical way. In the second part, more importantly, we attempt to study the phase space counterparts of Lindblad equation by applying Wigner transformation.…”

“…In the perspective of quantum master equation, which is mostly related to the current work, the quantum master equation in Redfield flavor for Anderson-Holstein model has been derived in [14][15][16]. The semiclassical limit of the Redfield equation, known as the classical master equation (CME), has been considered in [5], which also proposed a numerical method based on surface hopping. The CME perspective has then been used to study various physical aspects of Anderson-Holstein model, e.g., broadening, Marcus rate [17][18][19].…”

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

“…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."…”