Assessment of approximate solutions of the quantum–classical Liouville equation for dynamics simulations of quantum subsystems embedded in classical environments

Abstract

The quantum-classical Liouville equation (QCLE) provides a rigorous approach for modelling the dynamics of systems that can be effectively partitioned into a quantum subsystem and a classical environment. Several surface-hopping algorithms have been developed for solving the QCLE and successfully applied to simple model systems, but simulating the long-time dynamics of complex, realistic systems using these schemes has proven to be computationally demanding. Motivated by the need for computationally efficient algorithms, two approximate solutions of the QCLE, the Poisson bracket mapping equation (PBME) solution and the forward-backward trajectory solution (FBTS), were developed. These solutions involve simple algorithms in which both the quantum and classical degrees of freedom are described in terms of continuous variables and evolve according to classical-like equations of motion. However, since these schemes are approximate, they must be benchmarked against the exact quantum and QCLE surface-hopping solutions for a variety of simple and complex systems to determine the conditions under which they are valid. To illustrate the validity of the PBME and FBTS approaches, we review the results of a simple model for a condensed-phase photo-induced electron transfer and present new results for a realistic model for a proton transfer in a hydrogen-bonded complex dissolved in a polar nanocluster. Overall, the results demonstrate that caution must be taken when applying these approximate methods, since they can manifest nonphysical behaviour for systems where a mean-field-like description is not valid.

Self-Consistent Filtering Scheme for Efficient Calculations of Observables via the Mixed Quantum-Classical Liouville Approach

“…On the other hand, the Poisson Bracket Mapping Equation (PBME) − and the Forward–Backward Trajectory Solution (FBTS) algorithms involve generating an ensemble of trajectories in which both the quantum and classical DOF conveniently evolve according to classical-like equations of motion. Despite their simplicity and efficiency, these methods solve the QCLE more approximately and can yield inaccurate results in situations where a mean-field picture of the dynamics is not valid. − Thus, efforts toward making the SSTP and TBQC algorithms more efficient are worthwhile.…”

Coupled forward-backward trajectory approach for nonequilibrium electron-ion dynamics

“…However, these trajectories are different from Ehrenfest trajectories, but their dynamics is based on an approximation to the formal solution to the quantum classical Liouville equation. Theoretical and numerical details of the FBTS method are described elsewhere [23,24,47]. In both the MTEF and the FBTS methods, the ensemble average of trajectories can be performed by Monte Carlo sampling for the initial condition of each trajectory.…”

Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

“…The quantum-classical equations of motion herein discussed can be implemented in silico using a variety of simulation algorithms [79,[114][115][116][117][118][119][120][121][122][123][124]. We will sketch out one such integration algorithm, which unfolds the quantum-classical dynamics of the operator-valued quasi-probability function in terms of piecewise-deterministic trajectories evolving on the adiabatic energy surfaces of the system under study [79,114].…”