Quasiclassical approaches to the generalized quantum master equation.

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin-boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.

An Accurate Linearized Semiclassical Approach for Calculating Cavity-Modified Charge Transfer Rate Constants.

We show that combining the linearized semiclasscial approximation with Fermi's golden rule (FGR) rate theory gives rise to a general-purpose cost-effective and scalable computational framework that can accurately capture the cavity-induced rate enhancement of charge transfer reactions that occurs when the molecular system is placed inside a microcavity. Both partial linearization with respect to the nuclear and photonic degrees of freedom and full linerization with respect to nuclear, photonic, and electronic degrees of freedom (the latter within the mapping Hamiltonian approach) are shown to be highly accurate, provided that the Wigner transforms of the product (WoP) of operators at the initial time is not replaced by the product of their Wigner transforms. We also show that the partial linearization method yields the quantum-mechanically exact cavity-modified FGR rate constant for a model system in which the donor and acceptor potential energy surfaces are harmonic and identical except for a shift in the equilibrium energy and geometry, if WoP is applied.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Cavity-Modified Molecular Dynamics.

Recent experimental realizations of strong coupling between optical cavity modes and molecular matter placed inside the cavity have opened exciting new routes for controlling chemical processes. Simulating the cavity-modified dynamics of complex chemical systems calls for the development of accurate, flexible, and cost-effective approximate numerical methods that scale favorably with system size and complexity. In this Letter, we test the ability of quasiclassical mapping Hamiltonian methods to serve this purpose. We simulated the spontaneous emission dynamics of an atom confined to a microcavity via five different variations of the linearized semiclassical (LSC) method. Our main finding is that recently proposed LSC-based methods which use a modified form of the identity operator are reasonably accurate and perform significantly better than the Ehrenfest and standard LSC methods, without significantly increasing computational costs. These methods are therefore highly promising as a general purpose tool for simulating cavity-modified dynamics of complex chemical systems.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics.

Quasi-classical mapping Hamiltonian methods have recently emerged as a promising approach for simulating electronically nonadiabatic molecular dynamics. The classical-like dynamics of the overall system within these methods makes them computationally feasible, and they can be derived based on well-defined semiclassical approximations. However, the existence of a variety of different quasi-classical mapping Hamiltonian methods necessitates a systematic comparison of their respective advantages and limitations. Such a benchmark comparison is presented in this paper. The approaches compared include the Ehrenfest method, the symmetrical quasi-classical (SQC) method, and five variations of the linearized semiclassical (LSC) method, three of which employ a modified identity operator. The comparison is based on a number of popular nonadiabatic model systems; the spin-boson model, a Frenkel biexciton model, and Tully's scattering models 1 and 2. The relative accuracy of the different methods is tested by comparing with quantum-mechanically exact results for the dynamics of the electronic populations and coherences. We find that LSC with the modified identity operator typically performs better than the Ehrenfest and standard LSC approaches. In comparison to SQC, these modified methods appear to be slightly more accurate for condensed phase problems, but for scattering models there is little distinction between them.

Improved population operators for multi-state nonadiabatic dynamics with the mixed quantum-classical mapping approach.

The mapping approach addresses the mismatch between the continuous nuclear phase space and discrete electronic states by creating an extended, fully continuous phase space using a set of harmonic oscillators to encode the populations and coherences of the electronic states. Existing quasiclassical dynamics methods based on mapping, such as the linearised semiclassical initial value representation (LSC-IVR) and Poisson bracket mapping equation (PBME) approaches, have been shown to fail in predicting the correct relaxation of electronic-state populations following an initial excitation. Here we generalise our recently published modification to the standard quasiclassical approximation for simulating quantum correlation functions. We show that the electronic-state population operator in any system can be exactly rewritten as a sum of a traceless operator and the identity operator. We show that by treating the latter at a quantum level instead of using the mapping approach, the accuracy of traditional quasiclassical dynamics methods can be drastically improved, without changes to their underlying equations of motion. We demonstrate this approach for the seven-state Frenkel-exciton model of the Fenna-Matthews-Olson light harvesting complex, showing that our modification significantly improves the accuracy of traditional mapping approaches when compared to numerically exact quantum results.

On the identity of the identity operator in nonadiabatic linearized semiclassical dynamics.

Simulating the nonadiabatic dynamics of condensed-phase systems continues to pose a significant challenge for quantum dynamics methods. Approaches based on sampling classical trajectories within the mapping formalism, such as the linearized semiclassical initial value representation (LSC-IVR), can be used to approximate quantum correlation functions in dissipative environments. Such semiclassical methods however commonly fail in quantitatively predicting the electronic-state populations in the long-time limit. Here we present a suggestion to minimize this difficulty by splitting the problem into two parts, one of which involves the identity and treating this operator by quantum-mechanical principles rather than with classical approximations. This strategy is applied to numerical simulations of spin-boson model systems, showing its potential to drastically improve the performance of LSC-IVR and related methods with no change in the equations of motion or the algorithm in general, but rather by simply using different functional forms of the observables.

Quantum Dynamics with Short-Time Trajectories and Minimal Adaptive Basis Sets.

Methods for solving the time-dependent Schrödinger equation via basis set expansion of the wave function can generally be categorized as having either static (time-independent) or dynamic (time-dependent) basis functions. We have recently introduced an alternative simulation approach which represents a middle road between these two extremes, employing dynamic (classical-like) trajectories to create a static basis set of Gaussian wavepackets in regions of phase-space relevant to future propagation of the wave function [J. Chem. Theory Comput., 11, 8 (2015)]. Here, we propose and test a modification of our methodology which aims to reduce the size of basis sets generated in our original scheme. In particular, we employ short-time classical trajectories to continuously generate new basis functions for short-time quantum propagation of the wave function; to avoid the continued growth of the basis set describing the time-dependent wave function, we employ Matching Pursuit to periodically minimize the number of basis functions required to accurately describe the wave function. Overall, this approach generates a basis set which is adapted to evolution of the wave function while also being as small as possible. In applications to challenging benchmark problems, namely a 4-dimensional model of photoexcited pyrazine and three different double-well tunnelling problems, we find that our new scheme enables accurate wave function propagation with basis sets which are around an order-of-magnitude smaller than our original trajectory-guided basis set methodology, highlighting the benefits of adaptive strategies for wave function propagation.

Basis set generation for quantum dynamics simulations using simple trajectory-based methods.

Methods for solving the time-dependent Schrödinger equation generally employ either a global static basis set, which is fixed at the outset, or a dynamic basis set, which evolves according to classical-like or variational equations of motion; the former approach results in the well-known exponential scaling with system size, while the latter can suffer from challenging numerical problems, such as singular matrices, as well as violation of energy conservation. Here, we suggest a middle road: building a basis set using trajectories to place time-independent basis functions in the regions of phase space relevant to wave function propagation. This simple approach, which potentially circumvents many of the problems traditionally associated with global or dynamic basis sets, is successfully demonstrated for two challenging benchmark problems in quantum dynamics, namely, relaxation dynamics following photoexcitation in pyrazine, and the spin Boson model.