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In a previous study (Martinez, F.; Hanna, G. Chem. Phys. Lett. 2013, 573, 77-83), we demonstrated the ability of two approximate solutions of the quantum-classical Liouville equation (QCLE) for qualitatively capturing the electronic dynamics in the pump-probe transient absorption (TA) signal of a model of a condensed phase photoinduced electron transfer reaction whose ground and excited donor states have the same equilibrium geometry. However, the question remained as to the ability of these solutions to treat the more complex situation in which the electronic states are coupled to a low-frequency inner-sphere harmonic vibrational mode (representing an intramolecular mode of the donor-acceptor complex) that shifts their equilibrium geometries with respect to each other and thereby gives rise to signatures of vibrational dynamics in the TA signal. Thus, in this study, we investigated this situation by treating the vibrational mode both quantum mechanically and classically within the context of the approximate Poisson bracket mapping equation (PBME) and forward-backward trajectory solutions (FBTS) of the QCLE. Depending on the definition of the quantum subsystem, both PBME and FBTS are capable of qualitatively capturing several of the main features in the exact TA signal and quantitatively capturing the characteristic time scale of the vibrational dynamics, despite the moderately strong subsystem-bath coupling in this model. Particularly, we found that treating the vibrational mode quantum mechanically using either PBME or FBTS better captures the signatures of the vibrational dynamics, while treating it classically using FBTS better captures the decay in the signal. These findings underscore the utility of the PBME and FBTS approaches for efficiently modeling and interpreting TA signals.

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.

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