Testing the Limits of Approximate Solutions of the Quantum-Classical Liouville Equation for Modelling Quantum Processes Occurring in Condensed Phase Environments

Abstract

Many methods exist for simulating the nonadiabatic dynamics of mixed quantum-classical systems, some of which are more accurate than others. Some of the most accurate methods are based on solving the quantum-classical Liouville equation (QCLE) represented in the adiabatic basis, in terms of an ensemble of surface-hopping trajectories. However, long-time dynamics simulations of observables using these methods are computationally demanding as very large numbers of trajectories are required for convergence. Motivated by the need for more efficient approaches, more approximate methods were previously developed, but starting from a representation of the QCLE in the so-called mapping basis. These methods, known as the Poisson Bracket Mapping Equation (PBME) and Forward-Backward Trajectory Solution (FBTS), treat both the quantum and classical degrees of freedom in terms of continuous variables that evolve according to classical-like equations of motion. Owing to the approximate nature of these methods, it is necessary to understand the conditions under which they are valid. In this thesis, three studies were conducted to shed light on this matter. The first study was concerned with the laying down and testing of a formalism for calculating nonlinear spectroscopic signals efficiently using PBME and FBTS dynamics. In particular, expressions for simulating a time-integrated pump-probe transient absorption (TA) signal were first derived based on the so-called equation-of-motion phase-matching approach and then used to calculate the TA signal in a reduced model of a condensed phase photo-induced electron transfer (PIET) reaction. In the second study, calculations of TA signals were carried out for a more complex PIET model, in which the PIET takes place in a complex with an inner sphere vibrational mode. The details of how PBME and FBTS can be implemented for vibronic systems were worked out for two cases: one in which the vibrational mode is quantized and the other in which it is treated classically. In the third study, PBME simulations of a realistic model of a proton transfer (PT) reaction in a phenol-trimethylamine complex dissolved in a polar nanocluster are performed and analyzed. Expressions for calculating free energies as a function of both a classical and quantum reaction coordinate are derived and then evaluated for the PT reaction. The results of these studies collectively demonstrate that it is possible to extract useful information from PBME/FBTS simulations of nonlinear spectroscopic signals and about the dynamical behaviour of more realistic systems. However, great care must be taken in choosing which systems/conditions to apply these methods to. Beyond the limits of their underlying approximations, PBME/FBTS can yield highly inaccurate results and even portray a very different qualitative picture of a system’s dynamical behaviour.