A General Non-Markovian Electronic Friction and Langevin Dynamics Framework for Nonequilibrium Systems

Technical Summary: A General Non-Markovian Electronic Friction and Langevin Dynamics Framework for Nonequilibrium Systems

Problem Statement
Simulating the vibronic dynamics of molecules interacting with metal surfaces is essential for understanding phenomena such as molecular scattering, surface reactions, and nonequilibrium charge transport. However, these systems often operate in strongly coupled regimes where dynamics are highly non-Markovian and non-perturbative. While fully quantum methods like Hierarchical Equations of Motion (HEOM) and Multilayer Multiconfigurational Time-Dependent Hartree (ML-MCTDH) provide accurate benchmarks, they are computationally prohibitive for systems with many vibrational modes or large excitations.

Consequently, mixed quantum-classical approaches, specifically Electronic Friction and Langevin Dynamics (EFLD), are widely used. These methods typically rely on a Markovian approximation, where electronic noise is treated as white noise and electronic friction decays instantaneously. While successful in specific limits (e.g., heavy nuclei or strong coupling), growing evidence suggests that Markovian approximations fail in systems with many-body interactions or bias-driven inelastic transitions. Specifically, the Markovian limit can lead to unphysical instabilities, such as negative electronic friction, and fails to capture the full dynamical structure of electronic back-action in nonequilibrium regimes. The primary challenge in extending EFLD to the non-Markovian regime is the difficulty of sampling coordinate-dependent, quantum colored noise, particularly when the fluctuation-dissipation theorem (FDT) is violated out of equilibrium.

Methodology
The authors present a general framework for simulating nonequilibrium vibronic dynamics using Non-Markovian Electronic Friction and Langevin Dynamics (NM-EFLD). The core of the method is a Markovian embedding scheme that transforms the generalized Langevin equation (GLE) with memory kernels into a set of coupled Markovian stochastic differential equations.

Key methodological components include:

1. Generalized Langevin Equation: The vibrational dynamics are governed by a GLE where the electronic influence is represented by a mean adiabatic force, a non-Markovian friction term, and a Gaussian stochastic force with colored noise.
2. Independent Pole Decompositions: Unlike previous embedding schemes that often assume equilibrium or factorized coordinate-time dependence, this framework performs independent exponential decompositions for both the electronic friction kernel, $\gamma(x, t)$, and the stochastic force correlation function, $D(x, t)$.
   • $\gamma(x, t) = \text{Re} \left\{ \sum a_j(x) e^{-b_j(x)t} \right\}$
   • $D(x, t) = \text{Re} \left\{ \sum c_j(x) e^{-d_j(x)t} \right\}$
     This allows the method to handle nonequilibrium scenarios where the quantum FDT is violated and $\gamma$ and $D$ are not simply related.
3. Auxiliary Variables: The memory kernels are mapped to deterministic auxiliary variables ($u_j$) representing the friction history and stochastic auxiliary variables ($w_j$) representing the colored noise. These variables evolve according to coupled Ornstein-Uhlenbeck processes, retaining the full coordinate dependence of the electronic forces.
4. Propagation Algorithm: The authors employ an updated ABOBA algorithm (a second-order symmetric decomposition of the stochastic propagator) to integrate the coupled equations of motion. The algorithm includes specific operator splitting to handle the time evolution of the auxiliary variables alongside the physical vibrational degrees of freedom.

Key Contributions

• General Nonequilibrium Framework: The paper introduces a method that retains the full coordinate dependence of electronic forces and treats quantum colored noise without requiring the classical FDT, making it applicable to biased, nonequilibrium transport.
• Independent Decomposition: The approach decouples the decomposition of the friction kernel and the noise correlation function, acknowledging that in nonequilibrium, these quantities require different numbers of exponential terms and cannot be linked by simple equilibrium relations.
• Stability and Accuracy: The method is demonstrated to be numerically stable in parameter regimes where the conventional Markovian EFLD (M-EFLD) fails, specifically where M-EFLD predicts unphysical negative friction leading to instability.

Results
The method was validated using a vibrationally coupled donor-acceptor molecular junction model. The results were compared against numerically exact HEOM simulations:

• Equilibrium Validation: At zero bias, the NM-EFLD approach correctly reproduces the quantum thermal expectation values for vibrational energy and the exact correlation functions of the stochastic force, confirming the validity of the auxiliary variable representation.
• Nonequilibrium Performance:
  • For a model with positive energy splitting ($\Delta > 0$), the NM-EFLD approach accurately predicts steady-state vibrational excitation at high bias voltages, closely matching the HEOM benchmark. In contrast, M-EFLD underpredicts the excitation.
  • For a model with negative energy splitting ($\Delta < 0$), the M-EFLD approach becomes unstable at moderate bias voltages ($\Phi \approx 0.6$ V), predicting vibrational excitation orders of magnitude larger than the exact quantum result. The NM-EFLD approach remains stable across the entire bias range and quantitatively reproduces the HEOM results.
• Mechanism of Instability: Analysis of the electronic force spectra revealed that while the zero-frequency (Markovian) friction coefficient could be negative (suggesting driving), the full frequency-dependent friction was positive and dominant. The Markovian approximation, by neglecting finite-frequency components, qualitatively misrepresents the energy exchange, leading to instability. The NM-EFLD framework captures these finite-frequency memory effects, correctly stabilizing the dynamics.

Significance and Claims
The paper claims that non-Markovian electronic forces play a decisive role in nonequilibrium vibrational dynamics, particularly in stabilizing regimes where the Markovian approximation fails. The authors argue that electronic relaxation in these systems is governed not only by the molecule-metal coupling but also by the electronic energy splitting, creating memory effects that span multiple timescales.

The significance of the work lies in providing a practical, general framework to incorporate these memory effects into large-scale simulations of current-driven molecular transport, vibrational energy transfer, and surface scattering. By utilizing Markovian embedding with independent pole decompositions, the method bridges the gap between computationally expensive fully quantum methods and the widely used but often inaccurate Markovian EFLD, offering a robust tool for studying complex molecule-metal interfaces out of equilibrium.