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 non-physical behaviour for systems where a mean-field-like description is not valid.