Electronic Dynamics through Conical Intersections via Quasiclassical Mapping Hamiltonian Methods

Nonadiabatic Field with Triangle Window Functions on Quantum Phase Space

Recent progress on the constraint coordinate-momentum phase space (CPS) formulation of finite-state quantum systems has revealed that the triangle window function approach is an isomorphic representation of the exact population-population correlation function of the two-state system. We use the triangle window (TW) function and the CPS mapping kernel element to formulate a novel useful representation of discrete electronic degrees of freedom (DOFs). When it is employed with nonadiabatic field (NaF) dynamics, a new variant of the NaF approach (i.e., NaF-TW) is proposed. The NaF-TW expression of the population of any adiabatic state is always positive semidefinite. Extensive benchmark tests of model systems in both the condensed phase and gas phase demonstrate that the NaF-TW approach is able to faithfully capture the dynamical interplay between electronic and nuclear DOFs in a broad region, including where the states remain coupled all the time, as well as where the bifurcation characteristic of nuclear motion is important.

Detailed balance in mixed quantum-classical mapping approaches

The violation of detailed balance poses a serious problem for the majority of current quasiclassical methods for simulating nonadiabatic dynamics. In order to analyze the severity of the problem, we predict the long-time limits of the electronic populations according to various quasiclassical mapping approaches by applying arguments from classical ergodic theory. Our analysis confirms that regions of the mapping space that correspond to negative populations, which most mapping approaches introduce in order to go beyond the Ehrenfest approximation, pose the most serious issue for reproducing the correct thermalization behavior. This is because inverted potentials, which arise from negative electronic populations entering the nuclear force, can result in trajectories unphysically accelerating off to infinity. The recently developed mapping approach to surface hopping (MASH) provides a simple way of avoiding inverted potentials while retaining an accurate description of the dynamics. We prove that MASH, unlike any other quasiclassical approach, is guaranteed to describe the exact thermalization behavior of all quantum-classical systems, confirming it as one of the most promising methods for simulating nonadiabatic dynamics in real condensed-phase systems.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

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.

All-Atom Nonadiabatic Semiclassical Mapping Dynamics for Photoinduced Charge Transfer of Organic Photovoltaic Molecules in Explicit Solvents

Direct all-atom simulation of nonadiabatic dynamics in disordered condensed phases like liquid solutions and amorphous solids has been challenging. The first all-atom simulation of the photoinduced charge-transfer dynamics of a prototypical organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad in explicit tetrahydrofuran is presented. Based on the Meyer-Miller mapping Hamiltonian, various semiclassical and mixed quantum-classical dynamics are employed, including the linearized semiclassical, symmetrical quasiclassical, mean-field Ehrenfest, classical mapping model, and spin-mapping model approaches. The all-atom nonadiabatic dynamics were compared to multi-state harmonic models with a globally shared bath, and the models built using the ensemble averages on the initial electronic state could reproduce the all-atom results. The solvent effect was found to be critical for the photoinduced charge transfer, and the time-dependent solute-solvent radial distribution functions revealed that only the nonadiabatic dynamics started with the effective forces on the initial electronic state could capture the correct nuclear dynamics. The proposed strategy for modeling condensed-phase nonadiabatic dynamics with atomistic details is readily applied to complex condensed-phase systems.

Electronic Absorption Spectra from Off-Diagonal Quantum Master Equations

Quantum master equations (QMEs) provide a general framework for describing electronic dynamics within a complex molecular system. Off-diagonal QMEs (OD-QMEs) correspond to a family of QMEs that describe the electronic dynamics in the interaction picture based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. The fact that OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrodinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrodinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrodinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained in this way is studied in the context of a biexciton benchmark model, by comparing spectra calculated based on time-local and time-nonlocal OD-QMEs to spectra calculated based on a Redfield-type QME and the non-perturbative and quantum-mechanically exact hierarchical equations of motion (HEOM) method.

2D electronic-vibrational spectroscopy with classical trajectories

Two-dimensional electronic-vibrational (2DEV) spectra have the capacity to probe electron–nuclear interactions in molecules by measuring correlations between initial electronic excitations and vibrational transitions at a later time. The trajectory-based semiclassical optimized mean trajectory approach is applied to compute 2DEV spectra for a system with excitonically coupled electronic excited states vibronically coupled to a chromophore vibration. The chromophore mode is in turn coupled to a bath, inducing redistribution of vibrational populations. The lineshapes and delay-time dynamics of the resulting spectra compare well with benchmark calculations, both at the level of the observable and with respect to contributions from distinct spectroscopic processes.

Calculating Multidimensional Optical Spectra from Classical Trajectories

Multidimensional optical spectra are measured from the response of a material system to a sequence of laser pulses and have the capacity to elucidate specific molecular interactions and dynamics whose influences are absent or obscured in a conventional linear absorption spectrum. Interpretation of complex spectra is supported by theoretical modeling of the spectroscopic observable, requiring implementation of quantum dynamics for coupled electrons and nuclei. Performing numerically correct quantum dynamics in this context may pose computational challenges, particularly in the condensed phase. Semiclassical methods based on calculating classical trajectories offer a practical alternative. Here I review the recent application of some semiclassical, trajectory-based methods to nonlinear molecular vibrational and electronic spectra.

Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Forecasting nonadiabatic dynamics using hybrid convolutional neural network/long short-term memory network

Modeling nonadiabatic dynamics in complex molecular or condensed-phase systems has been challenging, especially for the long-time dynamics. In this work, we propose a time series machine learning scheme based on the hybrid convolutional neural network/long short-term memory (CNN-LSTM) framework for predicting the long-time quantum behavior, given only the short-time dynamics. This scheme takes advantage of both the powerful local feature extraction ability of CNN and the long-term global sequential pattern recognition ability of LSTM. With feature fusion of individually trained CNN-LSTM models for the quantum population and coherence dynamics, the proposed scheme is shown to have high accuracy and robustness in predicting the linearized semiclassical and symmetrical quasiclassical mapping dynamics as well as the mixed quantum-classical Liouville dynamics of various spin-boson models with learning time up to 0.3 ps. Furthermore, if the hybrid network has learned the dynamics of a system, this knowledge is transferable that could significantly enhance the accuracy in predicting the dynamics of a similar system. The hybrid CNN-LSTM network is thus believed to have high predictive power in forecasting the nonadiabatic dynamics in realistic charge and energy transfer processes in photoinduced energy conversion.

Unified Formulation of Phase Space Mapping Approaches for Nonadiabatic Quantum Dynamics

Nonadiabatic dynamical processes are one of the most important quantum mechanical phenomena in chemical, materials, biological, and environmental molecular systems, where the coupling between different electronic states is either inherent in the molecular structure or induced by the (intense) external field. The curse of dimensionality indicates the intractable exponential scaling of calculation effort with system size and restricts the implementation of "numerically exact" approaches for realistic large systems. The phase space formulation of quantum mechanics offers an important theoretical framework for constructing practical approximate trajectory-based methods for quantum dynamics. This Account reviews our recent progress in phase space mapping theory: a unified framework for constructing the mapping Hamiltonian on phase space for coupled F-state systems where the renowned Meyer-Miller Hamiltonian model is a special case, a general phase space formulation of quantum mechanics for nonadiabatic systems where the electronic degrees of freedom are mapped onto constraint space and the nuclear degrees of freedom are mapped onto infinite space, and an isomorphism between the mapping phase space approach for nonadiabatic systems and that for nonequilibrium electron transport processes. While the zero-point-energy parameter is conventionally assumed to be positive, we show that the constraint implied in the conventional Meyer-Miller mapping Hamiltonian requires that such a parameter can be negative as well and lies in (-1/F, +∞) for each electronic degree of freedom. More importantly, the zero-point-energy parameter should be interpreted as a special case of a commutator matrix in the comprehensive phase space mapping Hamiltonian for nonadiabatic systems. From the rigorous formulation of mapping phase space, we propose approximate but practical trajectory-based nonadiabatic dynamics methods. The applications to both gas phase and condensed phase problems include the spin-boson model for condensed phase dissipative two-state systems, the three-state photodissociation models, the seven-site model of the Fenna-Matthews-Olson monomer in photosynthesis of green sulfur bacteria, the strongly coupled molecular/atomic matter-optical cavity systems designed for controlling and manipulating chemical dynamical processes, and the Landauer model for a quantum dot state coupled with two electrodes. In these applications the overall performance of our phase space mapping dynamics approach is superior to two prevailing trajectory-based methods, Ehrenfest dynamics and fewest switches surface hopping.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

Multi-state harmonic models with globally shared bath for nonadiabatic dynamics in the condensed phase

Model Hamiltonians constructed from quantum chemistry calculations and molecular dynamics simulations are widely used for simulating nonadiabatic dynamics in the condensed phase. The most popular two-state spin-boson model could be built by mapping the all-atom anharmonic Hamiltonian onto a two-level system bilinearly coupled to a harmonic bath using the energy gap time correlation function. However, for more than two states, there lacks a general strategy to construct multi-state harmonic (MSH) models since the energy gaps between different pairs of electronic states are not entirely independent and need to be considered consistently. In this paper, we extend the previously proposed approach for building three-state harmonic models for photoinduced charge transfer to the arbitrary number of electronic states with a globally shared bath and the system-bath couplings are scaled differently according to the reorganization energies between each pair of states. We demonstrate the MSH model construction for an organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad dissolved in explicit tetrahydrofuran solvent. Nonadiabatic dynamics was simulated using mixed quantum-classical techniques, including the linearized semiclassical and symmetrical quasiclassical dynamics with the mapping Hamiltonians, mean-field Ehrenfest, and mixed quantum-classical Liouville dynamics in two-state, three-state, and four-state harmonic models of the triad system. The MSH models are shown to provide a general and flexible framework for simulating nonadiabatic dynamics in complex systems.

Commutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics

We show that a novel, general phase space mapping Hamiltonian for nonadiabatic systems, which is reminiscent of the renowned Meyer-Miller mapping Hamiltonian, involves a commutator variable matrix rather than the conventional zero-point-energy parameter. In the exact mapping formulation on constraint space for phase space approaches for nonadiabatic dynamics, the general mapping Hamiltonian with commutator variables can be employed to generate approximate trajectory-based dynamics. Various benchmark model tests, which range from gas phase to condensed phase systems, suggest that the overall performance of the general mapping Hamiltonian is better than that of the conventional Meyer-Miller Hamiltonian.

Simulating energy transfer dynamics in the Fenna-Matthews-Olson complex via the modified generalized quantum master equation

The generalized quantum master equation (GQME) provides a general and formally exact framework for simulating the reduced dynamics of open quantum systems. The recently introduced modified approach to the GQME (M-GQME) corresponds to a specific implementation of the GQME that is geared toward simulating the dynamics of the electronic reduced density matrix in systems governed by an excitonic Hamiltonian. Such a Hamiltonian, which is often used for describing energy and charge transfer dynamics in complex molecular systems, is given in terms of diabatic electronic states that are coupled to each other and correspond to different nuclear Hamiltonians. Within the M-GQME approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic density matrix is fully captured by a memory kernel superoperator, which can be obtained from short-lived (compared to the time scale of energy/charge transfer) projection-free inputs. In this paper, we test the ability of the M-GQME to predict the energy transfer dynamics within a seven-state benchmark model of the Fenna-Matthews-Olson (FMO) complex, with the short-lived projection-free inputs obtained via the Ehrenfest method. The M-GQME with Ehrenfest-based inputs is shown to yield accurate results across a wide parameter range. It is also found to dramatically outperform the direct application of the Ehrenfest method and to provide better-behaved convergence with respect to memory time in comparison to an alternative implementation of the GQME approach previously applied to the same FMO model.

On-the-Fly Symmetrical Quasi-Classical Dynamics with Meyer-Miller Mapping Hamiltonian for the Treatment of Nonadiabatic Dynamics at Conical Intersections

The on-the-fly version of the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian (SQC/MM) is implemented to study the nonadiabatic dynamics at conical intersections of polyatomic systems. The current on-the-fly implementation of the SQC/MM method is based on the adiabatic representation and the dressed momentum. To include the zero-point energy (ZPE) correction of the electronic mapping variables, we employ both the γ-adjusted and γ-fixed approaches. Nonadiabatic dynamics of the methaniminium cation (CH₂NH₂⁺) and azomethane are simulated using the on-the-fly SQC/MM method. For CH₂NH₂⁺, both ZPE correction approaches give reasonable and consistent results. However, for azomethane, the γ-adjusted version of the SQC/MM dynamics behaves much better than the γ-fixed version. Further analysis indicates that it is always recommended to use the γ-adjusted SQC/MM dynamics in the on-the-fly simulation of photoinduced dynamics of polyatomic systems, particularly when the excited state is well separated from the ground state in the Franck-Condon region. This work indicates that the on-the-fly SQC/MM method is a powerful simulation protocol to deal with the nonadiabatic dynamics of realistic polyatomic systems.

Two-dimensional vibrational-electronic spectra with semiclassical mechanics

Two-dimensional vibrational-electronic (2DVE) spectra probe the effects on vibronic spectra of initial vibrational excitation in an electronic ground state. The optimized mean trajectory (OMT) approximation is a semiclassical method for computing nonlinear spectra from response functions. Ensembles of classical trajectories are subject to semiclassical quantization conditions, with the radiation-matter interaction inducing discontinuous transitions. This approach has been previously applied to two-dimensional infrared and electronic spectra and is extended here to 2DVE spectra. For a system including excitonic coupling, vibronic coupling, and interaction of a chromophore vibration with a resonant environment, the OMT method is shown to well approximate exact quantum dynamics.

Three-state harmonic models for photoinduced charge transfer

A widely used strategy for simulating the charge transfer between donor and acceptor electronic states in an all-atom anharmonic condensed-phase system is based on invoking linear response theory to describe the system in terms of an effective spin-boson model Hamiltonian. Extending this strategy to photoinduced charge transfer processes requires also taking into consideration the ground electronic state in addition to the excited donor and acceptor electronic states. In this paper, we revisit the problem of describing such nonequilibrium processes in terms of an effective three-state harmonic model. We do so within the framework of nonequilibrium Fermi's golden rule (NE-FGR) in the context of photoinduced charge transfer in the carotenoid-porphyrin-C₆₀ (CPC₆₀) molecular triad dissolved in explicit tetrahydrofuran (THF). To this end, we consider different ways for obtaining a three-state harmonic model from the equilibrium autocorrelation functions of the donor-acceptor, donor-ground, and acceptor-ground energy gaps, as obtained from all-atom molecular dynamics simulations of the CPC₆₀/THF system. The quantum-mechanically exact time-dependent NE-FGR rate coefficients for two different charge transfer processes in two different triad conformations are then calculated using the effective three-state model Hamiltonians as well as a hierarchy of more approximate expressions that lead to the instantaneous Marcus theory limit. Our results show that the photoinduced charge transfer in CPC₆₀/THF can be described accurately by the effective harmonic three-state models and that nuclear quantum effects are small in this system.

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.

Spectroscopic response theory with classical mapping Hamiltonians

Exact quantum dynamics with a time-independent Hamiltonian in a discrete state space can be computed using classical mechanics through the classical Meyer-Miller-Stock-Thoss mapping Hamiltonian. In order to compute quantum response functions from classical dynamics, we extend this mapping to a quantum Hamiltonian with time-dependence arising from a classical field. This generalization requires attention to time-ordering in quantum and classical propagators. Quantum response theory with the original quantum Hamiltonian is equivalent to classical response theory with the classical mapping Hamiltonian. We elucidate the structure of classical response theory with the mapping Hamiltonian, thereby generating classical versions of the two-sided quantum density operator diagrams conventionally used to describe spectroscopic processes. This formal development can provide a foundation for new semiclassical approximations to spectroscopic observables for models in which classical nuclear degrees of freedom are introduced into a mapping Hamiltonian describing electronic states. Calculations of the temperature-dependence of two-dimensional electronic spectra for an exciton dimer using two semiclassical approaches are compared with benchmark calculations using the hierarchical equations of motion method.