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:

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.

We investigate a simple surface hopping (SH) approach for modeling a single impurity level coupled to a single phonon and an electronic (metal) bath (i.e., the Anderson-Holstein model). The phonon degree of freedom is treated classically with motion along--and hops between--diabatic potential energy surfaces. The hopping rate is determined by the dynamics of the electronic bath (which are treated implicitly). For the case of one electronic bath, in the limit of small coupling to the bath, SH recovers phonon relaxation to thermal equilibrium and yields the correct impurity electron population (as compared with numerical renormalization group). For the case of out of equilibrium dynamics, SH current-voltage (I-V) curve is compared with the quantum master equation (QME) over a range of parameters, spanning the quantum region to the classical region. In the limit of large temperature, SH and QME agree. Furthermore, we can show that, in the limit of low temperature, the QME agrees with real-time path integral calculations. As such, the simple procedure described here should be useful in many other contexts.

Electronic friction is a correction to the Born-Oppenheimer approximation, whereby nuclei in motion experience a drag in the presence of a manifold of electronic states. The notion of electronic friction has a long history and has been (re-)discovered in the context of a wide variety of different chemical and physical systems including, but not limited to, surface scattering events, surface reactions or chemisorption, electrochemistry, and conduction through molecular-(or nano-) junctions. Over the years, quite a few different forms of electronic friction have been offered in the literature. In this perspective, we briefly review these developments of electronic friction, highlighting the fact that we can now isolate a single, unifying form for (Markovian) electronic friction. We also focus on the role of electron-electron interactions for understanding frictional effects and offer our thoughts on the strengths and weaknesses of using electronic friction to model dynamics in general.

We present a protocol for the study of the dynamics and thermodynamics of quantum systems strongly coupled to a bath and subject to an external modulation. Our protocol quantifies the evolution of the system-bath composite by expanding the full density matrix as a series in the powers of the modulation rate, from which the functional form of work, heat and entropy rates can be obtained. Under slow driving, thermodynamic laws are established. The entropy production rate is positive and is found to be related to the excess work dissipated by friction, at least up to second order in the driving speed. As an example of the present methodology, we reproduce the results for the quantum thermodynamics of the driven resonance level model. We also emphasize that our formalism is quite general and allows for electron-electron interactions, which can give rise to exotic Kondo resonances appearing in thermodynamic quantities. arXiv:1808.08176v1 [cond-mat.stat-mech]

A broadened classical master equation (BCME) is proposed for modeling nonadiabatic dynamics for molecules near metal surfaces over a wide range of parameter values and with arbitrary initial conditions. Compared with a standard classical master equation-which is valid in the limit of weak molecule-metal couplings-this BCME should be valid for both weak and strong molecule-metal couplings. (The BCME can be mapped to a Fokker-Planck equation that captures level broadening correctly.) Finally, our BCME can be solved with a simple surface hopping algorithm; numerical tests of equilibrium and dynamical observables look very promising.

In a previous paper [Dou et al., J. Chem. Phys. 142, 084110 (2015)], we have introduced a surface hopping (SH) approach to deal with the Anderson-Holstein model. Here, we address some interesting aspects that have not been discussed previously, including transient phenomena and extensions to arbitrary impurity-bath couplings. In particular, in this paper we show that the SH approach captures phonon coherence beyond the secular approximation, and that SH rates agree with Marcus theory at steady state. Finally, we show that, in cases where the electronic tunneling rate depends on nuclear position, a straightforward use of Marcus theory rates yields a useful starting point for capturing level broadening. For a simple such model, we find I-V curves that exhibit negative differential resistance.

Dynamics at molecule−metal interfaces are a subject of intense current interest and come in many different flavors of experiments: gas-phase scattering, chemisorption, electrochemistry, nanojunction transport, and heterogeneous catalysis, to name a few. These dynamics involve nuclear degrees of freedom entangled with many electronic degrees of freedom (in the metal), and as such there is always the possibility for nonadiabatic phenomena to appear: the nuclei do not necessarily need to move slower than the electrons to break the Born−Oppenheimer (BO) approximation. In this Feature Article, we review a set of dynamical methods developed recently to deal with such nonadiabatic phenomena at a metal surface, methods that serve as alternatives to Tully's independent electron surface hopping (IESH) model. In the weak molecule−metal coupling regime, a classical master equation (CME) can be derived and a simple surface hopping approach is proposed to propagate nuclear and electronic dynamics stochastically. In the strong molecule−metal interaction regime, a Fokker−Planck equation can be derived for the nuclear dynamics, with electronic DoFs incorporated into the overall friction and random force. Lastly, a broadened classical master equation (BCME) can interpolate between the weak and strong molecule−metal interactions. Here, we briefly review these methods and the relevant benchmarking data, showing in particular how the methods can be used to calculate nonequilibrium transport properties. We highlight several open questions and pose several avenues for future study.