Wenjie Dou

Surface hopping with a manifold of electronic states. II. Application to the many-body Anderson-Holstein model

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.

Frictional effects near a metal surface

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 Kramers theory) as a function of frictional damping.

Surface hopping with a manifold of electronic states. III. Transients, broadening, and the Marcus picture

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.

A broadened classical master equation approach for nonadiabatic dynamics at metal surfaces: Beyond the weak molecule-metal coupling limit

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.

Surface hopping with a manifold of electronic states. I. Incorporating surface-leaking to capture lifetimes

We investigate the incorporation of the surface-leaking (SL) algorithm into Tullys fewest-switches surface hopping (FSSH) algorithm to simulate some electronic relaxation induced by an electronic bath in conjunction with some electronic transitions between discrete states. The resulting SL-FSSH algorithm is benchmarked against exact quantum scattering calculations for three one-dimensional model problems. The results show excellent agreement between SL-FSSH and exact quantum dynamics in the wide band limit, suggesting the potential for a SL-FSSH algorithm. Discrepancies and failures are investigated in detail to understand the factors that will limit the reliability of SL-FSSH, especially the wide band approximation. Considering the easiness of implementation and the low computational cost, we expect this method to be useful in studying processes involving both a continuum of electronic states (where electronic dynamics are probabilistic) and processes involving only a few electronic states (where non-adiabatic processes cannot ignore short-time coherence).

Electronic friction near metal surfaces: A case where molecule-metal couplings depend on nuclear coordinates

We derive an explicit form for the electronic friction as felt by a molecule near a metal surface for the general case that molecule-metal couplings depend on nuclear coordinates. Our work generalizes a previous study by von Oppen et al [Beilstein Journal of Nanotechnology, 3, 144, 2012], where we now go beyond the Condon approximation (i.e. molecule-metal couplings are not held constant). Using a non-equilibrium Greens function formalism in the adiabatic limit, we show that fluctuating metal-molecule couplings lead to new frictional damping terms and random forces, plus a correction to the potential of mean force. Numerical tests are performed and compared with a modified classical master equation; our results indicate that violating the Condon approximation can have a large effect on dynamics.

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

We present a universal expression for the electronic friction as felt by a set of classical nuclear degrees of freedom (DOFs) coupled to a manifold of quantum electronic DOFs; 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 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.

A many-body states picture of electronic friction: The case of multiple orbitals and multiple electronic states

We present a very general form of electronic friction as present when a molecule with multiple orbitals hybridizes with a metal electrode. To develop this picture of friction, we embed the quantum-classical Liouville equation (QCLE) within a classical master equation (CME). Thus, this article extends our previous work analyzing the case of one electronic level, as we may now treat the case of multiple levels and many electronic molecular states. We show that, in the adiabatic limit, where electron transitions are much faster than nuclear motion, the QCLE-CME reduces to a Fokker-Planck equation, such that nuclei feel an average force as well as friction and a random force-as caused by their interaction with the metallic electrons. Finally, we show numerically and analytically that our frictional results agree with other published results calculated using non-equilibrium Greens functions. Numerical recipes for solving this QCLE-CME will be provided in a subsequent paper.

Dynamics of Barrier Crossings for the Generalized Anderson–Holstein Model: Beyond Electronic Friction and Conventional Surface Hopping

We investigate barrier crossings within the context of the Anderson-Holstein model, as relevant to coupled nuclear-electronic dynamics near a metal surface. Beyond standard electronic friction or conventional surface-hopping dynamics, we show that a broadened classical master equation can recover both the correct nonadiabatic and the correct adiabatic dynamics for a general escape problem (even with possibly multiple escape channels). In the case of a large barrier with only a single escape channel, we also find a surprising conclusion: electronic friction can recover Marcuss nonadiabatic theory of electron transfer in the limit of small molecule-metal couplings. The latter conclusion establishes a hidden connection between Marcuss nonadiabatic theory and Kramers adiabatic theory of rate constants.

Molecular electronic states near metal surfaces at equilibrium using potential of mean force and numerical renormalization group methods: Hysteresis revisited

We investigate equilibrium observables for molecules near metals by employing a potential of mean force (PMF) that takes level broadening into account. Through comparison with exact data, we demonstrate that this PMF approach performs quite well, even for cases where molecule-electrode couplings depend on nuclear position. As an application, we reexamine the possibility of hysteresis effects within the Anderson-Holstein model (i.e., an impurity coupled both to a metal surface and a nuclear oscillator). As compared against the standard mean field approach by Galperin et al. [Nano Lett. 5, 125 (2005)], our PMF approach agrees much better with exact results for average electronic populations both at zero and finite temperature; we find, however, that mean field theory can be very useful for predicting the onset of dynamical instabilities, metastable states, and hysteresis.