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The generalized quantum master equation (GQME) provides a powerful framework for simulating electronically nonadiabatic molecular dynamics. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. In this paper, we consider two different procedures for calculating the memory kernel of the GQME from projection-free inputs obtained via the combination of the mapping Hamiltonian (MH) approach and the linearized semiclassical (LSC) approximation. The accuracy and feasibility of the two procedures are demonstrated on the spin-boson model. We find that although simulating the electronic dynamics by direct application of the two LSC-based procedures leads to qualitatively different results that become increasingly less accurate with increasing time, restricting their use to calculating the memory kernel leads to an accurate description of the electronic dynamics. Comparison with a previously proposed procedure for calculating the memory kernel via the Ehrenfest method reveals that MH/LSC methods produce memory kernels that are better behaved at long times and lead to more accurate electronic dynamics.

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... [1][2][3][4][5][6][7][8][9][10][11] The ability to accurately simulate photoinduced CT dynamics could therefore offer invaluable insights toward the discovery of more efficient artificial energy-conversion materials. Many quantum dynamical methods have been proposed to study electronic transitions in the condensed phase, such as the semiclassical initial value representation, [12][13][14] mean-field Ehrenfest, [15][16][17][18] fewest switches surface hopping, [19][20][21] mixed quantum-classical Liouville, [22][23][24][25] generalized quantum master equation, [26][27][28][29][30][31][32][33][34][35] and hierarchical equations of motion, [36][37][38][39][40][41] to name a few. Despite these advances, simulating CT dynamics in realistic complex systems rigorously and accurately remains highly challenging. ...

... In summary, we have demonstrated the feasibility of three strategies to construct three-state harmonic models that could be used to investigate the significance of the nonequilibrium effects due to initial nuclear preparation and the nuclear quantum dynamical effects in the photoinduced charge transfer dynamics in the condensed phase. As a natural extension of this work, it would be desirable to have a comprehensive study for applying the models proposed here to more complicated multi-level molecular systems and combining with other condensed-phase quantum dynamical methods such as tensor-train split-operator Fourier transform, 94 mixed quantum-classical Liouville, 95 quasiclassical mapping Hamiltonian, 96,97 and generalized quantum master equation, 34,35 as well as hierarchical equations of motion [39][40][41] and stochastic equations of motion. 98,99 It is also desirable to extend the current treatment of NE-FGR to more than three states, such as incorporating CT1 → CT2 transition. ...

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.

... The numerical comparisons between these two approaches have been extensively discussed in the recent work, [109][110][111] and recent development on choosing the identity operator 109,111,112 has also shown to significantly improve the population dynamics, even just using a less accurate Liouvillian L [1] LSC . Along the same direction, one can use the mapping action variable's Wigner transform 113 to construct [Â] W and [B] W and engineer various shapes of Window functions for these estimators. ...

We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future.

... In the future we expect that the accuracy can be extended to longer times by combining the dynamics with a general-ized quantum master equation, as has been successfully done for other mapping approaches. [78][79][80][81] Another natural extension would be to develop an FBTS or PLDM method based on spin mapping. Finally, we also believe that the spin mapping will be relevant in the search for a nonadiabatic extension to ring-polymer molecular dynamics. ...

We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system and a spin-1/2 particle. In the present paper, we generalize this method to describe the dynamics of $N$-level systems. This is done via a mapping to a classical phase space that preserves the $SU(N)$-symmetry of the original quantum problem. The theory reproduces the standard Meyer--Miller--Stock--Thoss Hamiltonian without invoking an extended phase space, and we thus avoid leakage from the physical subspace. In contrast with the standard derivation of this Hamiltonian, the generalized spin mapping leads to an $N$-dependent value of the zero-point energy parameter that is uniquely determined by the Casimir invariant of the $N$-level system. Based on this mapping, we derive a simple way to approximate correlation functions in complex nonadiabatic molecular systems via classical trajectories, and present benchmark calculations on the seven-state Fenna--Matthews--Olson complex. The results are significantly more accurate than conventional Ehrenfest dynamics, at a comparable computational cost, and can compete in accuracy with other state-of-the-art mapping approaches.

... As a natural extension of this work, it would be desirable to have a comprehensive comparison of the approximate semiclassical and quasiclassical mapping dynamics based on the MSH model proposed here and the widely used isolated bath model, such as the seven-state model for the photosynthetic Fenna-Matthews-Olson (FMO) complex. 4,5 Moreover, more accurate or numerically exact methods such as the tensor-train split-operator Fourier transform, 92 generalized quantum master equation, 40,93 or hierarchical equations of motion 94-96 could be used for benchmarking the MSH models. ...

Model Hamiltonians constructed from quantum chemistry calculations and molecular dynamics simulations are widely used for sim- ulating 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 correla- tion 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 Hamiltoni- ans, 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.

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. E. Runeson and J. O. Richardson, J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system and a spin-1/2 particle. In the present paper, we generalize this method to describe the dynamics of N-level systems. This is done via a mapping to a classical phase space that preserves the SU(N)-symmetry of the original quantum problem. The theory reproduces the standard Meyer–Miller–Stock–Thoss Hamiltonian without invoking an extended phase space, and we thus avoid leakage from the physical subspace. In contrast to the standard derivation of this Hamiltonian, the generalized spin mapping leads to an N-dependent value of the zero-point energy parameter that is uniquely determined by the Casimir invariant of the N-level system. Based on this mapping, we derive a simple way to approximate correlation functions in complex nonadiabatic molecular systems via classical trajectories and present benchmark calculations on the seven-state Fenna–Matthews–Olson light-harvesting complex. The results are significantly more accurate than conventional Ehrenfest dynamics, at a comparable computational cost, and can compete in accuracy with other state-of-the-art mapping approaches.

In this paper, we investigate the ability of different quasi-classical mapping Hamiltonian methods to simulate the dynamics of electronic transitions through conical intersections. The analysis is carried out within the framework of the linear vibronic coupling (LVC) model. The methods compared are the Ehrenfest method, the symmetrical quasi-classical method and several variations of the linearized semiclassical (LSC) method, including ones that are based on the recently introduced modified representation of the identity operator. The accuracy of the various methods is tested by comparing their predictions to quantum-mechanically exact results obtained via the MCTDH method. The LVC model is found to be a nontrivial benchmark model that can differentiate between different approximate methods based on their accuracy better than previously used benchmark models. In the three systems studied, two of the LSC methods are found to provide the most accurate description of electronic transitions through conical intersections.

The modular decomposition of the path integral is a linear-scaling, numerically exact algorithm for calculating dynamical properties of extended systems composed of multilevel units with local couplings. In a recent article, we generalized the method to wavefunction propagation in aggregates characterized by non-diagonal couplings between adjacent units. Here, we extend the method to the calculation of reduced density matrices in aggregates where each unit includes an arbitrary number of coupled harmonic bath modes, which may describe intramolecular normal mode vibrations, at finite temperature. The effects of harmonic modes are included through influence functional factors, which involve analytical expressions that we derive. Representative applications to spin arrays described by the Heisenberg Hamiltonian with dissipative interactions and to J-aggregates of perylene bisimide, where all coupled normal modes are treated explicitly, are presented.

We present a new methodology for simulating multidimensional electronic spectra of complex multiexcitonic molecular systems within the framework of quasiclassical mapping Hamiltonian (QC/MH) methods. The methodology is meant to be cost-effective for molecular systems with a large number of nuclear degrees of freedom undergoing nonequilibrium nonadiabatic dynamics on multiple coupled anharmonic electronic potential energy surfaces, for which quantum-mechanically exact methods are not feasible. The methodology is based on a nonperturbative approach to field-matter interaction, which mimics the experimental measurement of those nonlinear time-resolved spectra via phase cycling and can accommodate laser pulses of arbitrary shape and intensity. The ability of different QC/MH methods to accurately simulate two-dimensional and pump-probe electronic spectra within the proposed methodology is compared in the context of a biexcitonic benchmark model that includes both the singly-excited and doubly-excited electronic states. The QC/MH methods compared include five variations of the linearized semiclassical (LSC) method and the mean-field (Ehrenfest) method. The results show that LSC-based methods are significantly more accurate than the mean-eld method and can yield quantitatively accurate two-dimensional and pump-probe spectra when nuclear degrees of freedom can be treated as classical-like.

In this paper, we compare the ability of different quasi-classical mapping Hamiltonian methods to accurately simulate the absorption spectra of multi-excitonic molecular systems. Two distinctly different approaches for simulating absorption spectra are considered: (1) A perturbative approach, which relies on first-order perturbation theory with respect to the field-matter interaction; (2) A nonperturbative approach, which mimics the experimental measurement of absorption spectra from the free-induction decay that follows a short laser pulse. The methods compared are several variations of the linearized semiclassical (LSC) method, the symmetrical quasi-classical (SQC) method and the mean-field (Ehrenfest) method. The comparison is performed in the context of a biexcitonic model and a 7 excitonic model of the Fenna-Matthews-Olson (FMO) complex. The accuracy of the various methods is tested by comparing their predictions to quantum-mechanically exact results obtained via the HEOM method, as well as to results based on the Redfield quantum Master equation. The results show that LSC-based quasi-classical mapping Hamiltonian methods can yield accurate and robust absorption spectra in the high temperature and/or slow bath limit, where the nuclear degrees of freedom can be treated as classical.

We present a new partially linearized mapping-based approach for approximating real-time quantum correlation functions in condensed-phase nonadiabatic systems, called the spin partially linearized density matrix (spin-PLDM) approach. Within a classical trajectory picture, partially linearized methods treat the electronic dynamics along forward and backward paths separately by explicitly evolving two sets of mapping variables. Unlike previously derived partially linearized methods based on the Meyer–Miller–Stock–Thoss mapping, spin-PLDM uses the Stratonovich–Weyl transform to describe the electronic dynamics for each path within the spin-mapping space; this automatically restricts the Cartesian mapping variables to lie on a hypersphere and means that the classical equations of motion can no longer propagate the mapping variables out of the physical subspace. The presence of a rigorously derived zero-point energy parameter also distinguishes spin-PLDM from other partially linearized approaches. These new features appear to give the method superior accuracy for computing dynamical observables of interest when compared with other methods within the same class. The superior accuracy of spin-PLDM is demonstrated in this paper through application of the method to a wide range of spin-boson models as well as to the Fenna–Matthews–Olsen complex.

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 employed 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 two 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. The 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 Frank-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.

Semiclassical approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature we introduce Modified Filinov filtration as a powerful tool for mixed quantum-classical simulations in a uniform semiclassical framework.

Mapping Hamiltonian methods for simulating electronically nonadiabatic molecular dynamics are based on representing the electronic population and coherence operators in terms of isomorphic mapping operators, which are given in terms of the auxiliary position and momentum operators. Adding a quasiclassical approximation then makes it possible to treat those auxiliary coordinates and momenta, as well as the nuclear coordinates and momenta, as classical-like phase-space variables. Within such quasiclassical mapping Hamiltonian methods, the initial sampling of the auxiliary coordinates and momenta and the calculation of expectation values of electronic observables at a later time are based on window functions whose functional form differ from one method to another. However, different methods also differ with respect to the way in which they treat the window width. More specifically, while the window width is treated as an adjustable parameter within the symmetrical quasiclassical (SQC) method, this has not been the case for methods based on the linearized semiclasscial (LSC) approximation. In the present study, we investigate the effect that turning the window width into an adjustable parameter within LSC-based methods has on their accuracy compared to SQC. The analysis is performed in the context of the spin-boson and Fenna-Matthews-Olson (FMO) complex benchmark models. We find that treating the window width in LSC-based methods as an adjustable parameter can make their accuracy comparable to that of the SQC method.

We report recent progress on the phase space formulation of quantum mechanics with coordinate‐momentum variables, focusing more on new theory of (weighted) constraint coordinate‐momentum phase space for discrete‐variable quantum systems. This leads to a general coordinate‐momentum phase space formulation of composite quantum systems, where conventional representations on infinite phase space are employed for continuous variables. It is convenient to utilize (weighted) constraint coordinate‐momentum phase space for representing the quantum state and describing nonclassical features. Various numerical tests demonstrate that new trajectory‐based quantum dynamics approaches derived from the (weighted) constraint phase space representation are useful and practical for describing dynamical processes of composite quantum systems in the gas phase as well as in the condensed phase. This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics Schematic representation of dynamics of composite quantum systems via phase space formulations.

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.

The generalized quantum master equation (GQME) provides a general and exact approach for simulating the reduced dynamics in open quantum systems where a quantum system is embedded in a quantum environment. Dynamics of open quantum systems is important in excitation energy, charge, and quantum coherence transfer as well as reactive photochemistry. The system is usually chosen to be the interested degrees of freedom such as the electronic states in light-harvesting molecules or tagged vibrational modes in a condensed-phase system. The environment is also called the bath, whose influence on the system has to be considered, and for instance can be described by the GQME formalisms using the projection operator technique. In this review, we provide a heuristic description of the development of two canonical forms of GQME, namely the time-convoluted Nakajima-Zwanzig form (NZ-GQME) and the time-convolutionless form (TCL-GQME). In the more popular NZ-GQME form, the memory kernel serves as the essential part that reflects the non-Markovian and non-perturbative effects, which gives formally exact dynamics of the reduced density matrix. We summarize several schemes to express the projection-based memory kernel of NZ-GQME in terms of projection-free time correlation function inputs that contain molecular information. In particular, the recently proposed modified GQME approach based on NZ-GQME partitions the Hamiltonian into a more general diagonal and off-diagonal parts. The projection-free inputs in the above-mentioned schemes expressed in terms of different system-dependent time correlation functions can be calculated via numerically exact or approximate dynamical methods. We hope this contribution would help lower the barrier of understanding the theoretical pillars for GQME-based quantum dynamics methods and also envisage that their combination with the quantum computing techniques will pave the way for solving complex problems related to quantum dynamics and quantum information that are currently intractable even with today's state-of-the-art classical supercomputers.

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 a 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 Schrodinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrodinger-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 ODQMEs.The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar (GOA) 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 Schrodinger-picture coherences can be obtained from interaction-picture electronic inputs.

We present a partially linearized method based on spin mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin-PLDM method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is indeed the case by calculating the nonlinear optical response functions relevant for the pump--probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna--Matthews--Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiment.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) A full density matrix GQME; (2) A single-population scalar GQME; (3) A populations-only GQME; and (4) A subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations are comparable to that obtained via the full density matrix GQMEs.

We report recent progress on the phase space formulation of quantum mechanics with coordinate-momentum variables, focusing more on new theory of (weighted) constraint coordinate-momentum phase space for discrete-variable quantum systems. This leads to a general coordinate-momentum phase space formulation of composite quantum systems, where conventional representations on infinite phase space are employed for continuous variables. It is convenient to utilize (weighted) constraint coordinate-momentum phase space for representing the quantum state and describing nonclassical features. Various numerical tests demonstrate that new trajectory-based quantum dynamics approaches derived from the (weighted) constraint phase space representation are useful and practical for describing dynamical processes of composite quantum systems in gas phase as well as in condensed phase.

We investigate the use of accurate path integral methods, namely the quasi-adiabatic propagator path integral (QuAPI) and the quantum-classical path integral (QCPI), for generating the memory kernel entering generalized quantum master equations (GQME). Our calculations indicate that the length of the memory kernel in system-bath models is equal to the full length of time nonlocality encoded in the Feynman-Vernon influence functional and that the solution of the GQME with a QuAPI kernel is identical to that obtained through an iterative QuAPI calculation with the same memory length. Further, we show that the memory length in iterative QCPI calculations is always shorter than the GQME kernel memory length. This stems from the ability of the QCPI methodology to pretreat all memory effects of a classical nature (i.e., those associated with phonon absorption and stimulated emission), as well as some of the quantum memory contributions (arising from spontaneous phonon emission). Furthermore, trajectory-based iterative QCPI simulations can fully account for important structural/conformational changes that may occur on very long time scales and that cannot be captured via master equation treatments.

We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics.

We investigate the incorporation of the surface-leaking (SL) algorithm into Tully's fewest-switches surface hopping (FSSH) algorithm to simulate some electronic relaxation induced by an electronic bath in conjunction with some electronic transitions between discrete states. The resulting SL-FSSH algorithm is benchmarked against exact quantum scattering calculations for three one-dimensional model problems. The results show excellent agreement between SL-FSSH and exact quantum dynamics in the wide band limit, suggesting the potential for a SL-FSSH algorithm. Discrepancies and failures are investigated in detail to understand the factors that will limit the reliability of SL-FSSH, especially the wide band approximation. Considering the easiness of implementation and the low computational cost, we expect this method to be useful in studying processes involving both a continuum of electronic states (where electronic dynamics are probabilistic) and processes involving only a few electronic states (where non-adiabatic processes cannot ignore short-time coherence).

We address the issue of quantum decoherence in mixed quantum-classical simulations. We demonstrate that restricting the classical paths to a single path among all the quantum paths affects a coarse graining of the quantum paths. Such coarse graining causes the quantum paths to lose coherence as the various possible classical paths associated with each quantum state diverge. This defines a reduction mapping of the quantum density matrix, and we derive a quantum master equation suitable for mixed quantum-classical systems. The equation includes two terms: first, the ordinary quantum Liouvillian which is parametrized by a single classical path, and second, a quantum decoherence term that includes both a coherence time and length scale which are determined by the dynamics of the classical paths. Model calculations for electronic coherence loss in nonadiabatic mixed quantum-classical dynamics are presented as examples. For a model charge transfer chemical reaction with nonadiabatic transitions, application of the present formulation reveals that nonadiabaticity is diminished as the decoherence timescale becomes shorter and adiabatic dynamics are recovered in the limit of rapid decoherence.

In this paper, we present a new approach to treating many-body molecular dynamics on coupled electronic surfaces. The method is based on a semiclassical limit of the quantum Liouville equation. The formal result is a set of coupled classical-like partial differential equations for generalized distribution functions which describe both the nuclear probability densities on the coupled surfaces and the coherences between the electronic states. The Hamiltonian dynamics underlying the evolution of these distributions is augmented by nonclassical source and sink terms, which allow the flow of probability between the coupled surfaces and the corresponding formation and decay of electronic coherences. The formal results are shown analytically to reproduce the well-known Rabi and Landau–Zener results in appropriate limits. In addition, a direct numerical solution of the phase space partial differential equations is performed, and the results compared with exact quantum solutions for a model curve-crossing problem, yielding excellent agreement. Future trajectory-based implementation of the method in molecular dynamics simulations is also discussed.

An algorithm is presented for the exact solution of the evolution of the density matrix of a mixed quantum-classical system in terms of an ensemble of surface hopping trajectories. The system comprises a quantum subsystem coupled to a classical bath whose evolution is governed by a mixed quantum-classical Liouville equation. The integral solution of the evolution equation is formulated in terms of a concatenation of classical evolution segments for the bath phase space coordinates separated by operators that change the quantum state and bath momenta. A hybrid Molecular Dynamics-Monte Carlo scheme which follows a branching tree of trajectories arising from the action of momentum derivatives is constructed to solve the integral equation. We also consider a simpler scheme where changes in the bath momenta are approximated by momentum jumps. These schemes are illustrated by considering the computation of the evolution of the density matrix for a two-level system coupled to a low dimensional classical bath.

A simple surface hopping method for nonadiabatic molecular dynamics is developed. The method derives from a stochastic modeling of the time-dependent Schrödinger and master equations for open systems and accounts simultaneously for quantum mechanical branching in the otherwise classical (nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem due to coupling to nuclei. Electronic dynamics in the Hilbert space takes the form of a unitary evolution, intermittent with stochastic decoherence events that are manifested as a localization toward (adia-batic) basis states. Classical particles evolve along a single potential energy surface and can switch surfaces only at the decoherence events. Thus, decoherence provides physical justification of sur-face hopping, obviating the need for ad hoc surface hopping rules. The method is tested with model problems, showing good agreement with the exact quantum mechanical results and providing an improvement over the most popular surface hopping technique. The method is implemented within real-time time-dependent density functional theory formulated in the Kohn-Sham representation and is applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiative quenching of luminescence in these systems agree with the experimental data and earlier calculations.

Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ = (m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories. © 1999 American Institute of Physics.

It is shown how a formally exact classical analog can be defined for a finite dimensional (in Hilbert space) quantum mechanical system. This approach is then used to obtain a classical model for the electronic degrees of freedom in a molecular collision system, and the combination of this with the usual classical description of the heavy particle (i.e., nuclear) motion provides a completely classical model for the electronic and heavy particle degrees of freedom. The resulting equations of motion are shown to be equivalent to describing the electronic degrees of freedom by the time‐dependent Schrödinger equation, the time dependence arising from the classical motion of the nuclei, the trajectory of which is determined by the quantum mechanical average (i.e., Ehrenfest) force on the nuclei. Quantizing the system via classical S‐matrix theory is shown to provide a dynamically consistent description of nonadiabatic collision processes; i.e., different electronic transitions have different heavy particle trajectories and, for example, the total energy of the electronic and heavy particle degrees of freedom is conserved. Application of this classical model for the electronic degrees of freedom (plus classical S‐matrix theory) to the two‐state model problem shows that the approach provides a good description of the electronic dynamics.

This text provides a uniform and consistent approach to diversified problems encountered in the study of dynamical processes in condensed phase molecular systems. Given the broad interdisciplinary aspect of this subject, the book focuses on three themes: coverage of needed background material, in-depth introduction of methodologies, and analysis of several key applications. The uniform approach and common language used in all discussions help to develop general understanding and insight on condensed phases chemical dynamics. The applications discussed are among the most fundamental processes that underlie physical, chemical and biological phenomena in complex systems. The first part of the book starts with a general review of basic mathematical and physical methods (Chapter 1) and a few introductory chapters on quantum dynamics (Chapter 2), interaction of radiation and matter (Chapter 3) and basic properties of solids (chapter 4) and liquids (Chapter 5). In the second part the text embarks on a broad coverage of the main methodological approaches. The central role of classical and quantum time correlation functions is emphasized in Chapter 6. The presentation of dynamical phenomena in complex systems as stochastic processes is discussed in Chapters 7 and 8. The basic theory of quantum relaxation phenomena is developed in Chapter 9, and carried on in Chapter 10 which introduces the density operator, its quantum evolution in Liouville space, and the concept of reduced equation of motions. The methodological part concludes with a discussion of linear response theory in Chapter 11, and of the spin-boson model in chapter 12. The third part of the book applies the methodologies introduced earlier to several fundamental processes that underlie much of the dynamical behaviour of condensed phase molecular systems. Vibrational relaxation and vibrational energy transfer (Chapter 13), Barrier crossing and diffusion controlled reactions (Chapter 14), solvation dynamics (Chapter 15), electron transfer in bulk solvents (Chapter 16) and at electrodes/electrolyte and metal/molecule/metal junctions (Chapter 17), and several processes pertaining to molecular spectroscopy in condensed phases (Chapter 18) are the main subjects discussed in this part.

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.

We present a modified approach for simulating electronically nonadiabatic dynamics based on the Nakajima-Zwanzig generalized quantum master equation (GQME). The modified approach utilizes the fact that the Nakajima-Zwanzig formalism does not require casting the overall Hamiltonian in system-bath form, which is arguably neither natural nor convenient in the case of the Hamiltonian that governs nonadiabatic dynamics. Within the modified approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density operator is fully captured by a memory kernel super-operator. A methodology for calculating the memory kernel from projection-free inputs is developed. Simulating the electronic dynamics via the modified approach, with a memory kernel obtained using exact or approximate methods, can be more cost effective and/or lead to more accurate results than direct application of those methods. The modified approach is compared to previously proposed GQME-based approaches, and its robustness and accuracy are demonstrated on a benchmark spin-boson model with a memory kernel which is calculated within the Ehrenfest method.

The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics—as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model—can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation—because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation—it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton’s equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrödinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton’s equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton’s equations (and presumably also in the Schrödinger equation) can cause very significant errors.

We introduce the `tensor-train split-operator Fourier transform' (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wavefunctions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wavefunction in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S1/S2 interconversion dynamics of pyrazine after UV photoexcitation to the S2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations.

We propose a new unified theoretical framework to construct equivalent representations of the multi-state Hamiltonian operator and present several approaches for the mapping onto the Cartesian phase space. After mapping an F-dimensional Hamiltonian onto an F+1- dimensional space, creation and annihilation operators are defined such that the F+1 dimensional space is complete for any combined excitations. Commutation and anti-commutation relations are then naturally derived, which show that the underlying degrees of freedom are neither bosons nor fermions. This sets the scene for developing equivalent expressions of the Hamiltonian operator in quantum mechanics and their classical/semiclassical counterparts. Six mapping models are presented as examples. The framework also offers a novel way to derive such as the well-known Meyer-Miller model.

In this letter, we combine the recently introduced transfer tensor method with the mixed quantum-classical Liouville method. The resulting protocol provides an accurate, general, flexible and robust new route for simulating the reduced dynamics of the quantum subsystem for arbitrarily long times, starting with computationally feasible short-time mixed quantum-classical Liouville dynamical maps. The accuracy and feasibility of the methodology are demonstrated on a spin-boson benchmark model.

Previous work has shown how a symmetrical quasi-classical (SQC) windowing procedure can be used to quantize the initial and final electronic degrees of freedom in the Meyer-Miller (MM) classical vibronic (i.e, nuclear + electronic) Hamiltonian, and that the approach provides a very good description of electronically non-adiabatic processes within a standard classical molecular dynamics framework for a number of benchmark problems. This paper explores application of the SQC/MM approach to the case of very weak non-adiabatic coupling between the electronic states, showing (as anticipated) how the standard SQC/MM approach used to date fails in this limit, and then devises a new SQC windowing scheme to deal with it. Application of this new SQC model to a variety of realistic benchmark systems shows that the new model not only treats the weak coupling case extremely well, but it is also seen to describe the “normal” regime (of electronic transition probabilities ≳ 0.1) even more accurately than the previous “standard” model.

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory—e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states—and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

Both classical and quantum mechanics (as well as hybrids thereof, i.e., semiclassical approaches) find widespread use in simulating dynamical processes in molecular systems. For large chemical systems, however, which involve potential energy surfaces (PES) of general/arbitrary form, it is usually the case that only classical molecular dynamics (MD) approaches are feasible, and their use is thus ubiquitous nowadays, at least for chemical processes involving dynamics on a single PES (i.e., within a single Born-Oppenheimer electronic state). This paper reviews recent developments in an approach which extends standard classical MD methods to the treatment of electronically non-adiabatic processes, i.e., those that involve transitions between different electronic states. The approach treats nuclear and electronic degrees of freedom (DOF) equivalently (i.e., by classical mechanics, thereby retaining the simplicity of standard MD), and provides "quantization" of the electronic states through a symmetrical quasi-classical (SQC) windowing model. The approach is seen to be capable of treating extreme regimes of strong and weak coupling between the electronic states, as well as accurately describing coherence effects in the electronic DOF (including the de-coherence of such effects caused by coupling to the nuclear DOF). A survey of recent applications is presented to illustrate the performance of the approach. Also described is a newly developed variation on the original SQC model (found universally superior to the original) and a general extension of the SQC model to obtain the full electronic density matrix (at no additional cost/complexity).

The nonequilibrium Fermi’s golden rule describes the transition between a photoexcited bright donor electronic state and a dark acceptor electronic state, when the nuclear degrees of freedom start out in a nonequilibrium state. In a previous paper [X. Sun and E. Geva, J. Chem. Theory Comput. 12, 2926 (2016)], we proposed a new expression for the nonequilibrium Fermi’s golden rule within the framework of the linearized semiclassical approximation and based on the Condon approximation, according to which the electronic coupling between donor and acceptor is assumed constant. In this paper we propose a more general expression, which is applicable to the case of non-Condon electronic coupling. We test the accuracy of the new non-Condon nonequilibrium Fermi’s golden rule linearized semiclassical expression on a model where the donor and acceptor potential energy surfaces are parabolic and identical except for shifts in the equilibrium energy and geometry, and the coupling between them is linear in the nuclear coordinates. Since non-Condon effects may or may not give rise to conical intersections, both possibilities are examined by considering the following: (1) A modified Garg-Onuchic-Ambegaokar model for charge transfer in the condensed phase, where the donor-acceptor coupling is linear in the primary-mode coordinate, and for which non-Condon effects do not give rise to a conical intersection; (2) the linear vibronic coupling model for electronic transitions in gas phase molecules, where non-Condon effects give rise to conical intersections. We also present a comprehensive comparison between the linearized semiclassical expression and a progression of more approximate expressions, in both normal and inverted regions, and over a wide range of initial nonequilibrium states, temperatures, and frictions.

In this paper, we test the accuracy of the linearized semiclassical (LSC) expression for the equilibrium Fermi’s golden rule rate constant for electronic transitions in the presence of non-Condon effects. We do so by performing a comparison with the exact quantum-mechanical result for a model where the donor and acceptor potential energy surfaces are parabolic and identical except for shifts in the equilibrium energy and geometry, and the coupling between them is linear in the nuclear coordinates. Since non-Condon effects may or may not give rise to conical intersections, both possibilities are examined by considering: (1) A modified Garg-Onuchic-Ambegaokar model for charge transfer in the condensed phase, where the donor-acceptor coupling is linear in the primary mode coordinate, and for which non-Condon effects do not give rise to a conical intersection; (2) the linear vibronic coupling model for electronic transitions in gas phase molecules, where non-Condon effects give rise to conical intersections. We also present a comprehensive comparison between the linearized semiclassical expression and a progression of more approximate expressions. The comparison is performed over a wide range of frictions and temperatures for model (1) and over a wide range of temperatures for model (2). The linearized semiclassical method is found to reproduce the exact quantum-mechanical result remarkably well for both models over the entire range of parameters under consideration. In contrast, more approximate expressions are observed to deviate considerably from the exact result in some regions of parameter space.

We present a current, up-to-date review of the surface hopping methodology for solving nonadiabatic problems, 25 years after Tully published the fewest switches surface hopping algorithm. After reviewing the original motivation for and failures of the algorithm, we give a detailed examination of modern advances, focusing on both theoretical and practical issues. We highlight how one can partially derive surface hopping from the Schrödinger equation in the adiabatic basis, how one can change basis within the surface hopping algorithm, and how one should understand and apply the notions of decoherence and wavepacket bifurcation. The question of time reversibility and detailed balance is also examined at length. Recent applications to photoexcited conjugated polymers are discussed briefly.

Developed twenty-five years ago, Tully's fewest switches surface hopping (FSSH) has proven to be the most popular approach for simulating quantum-classical dynamics in a broad variety of systems, ranging from the gas phase, to the liquid and solid phases, to biological and nanoscale materials. FSSH is widely adopted as the fundamental platform to introduce modifications as needed. Significant progress has been made recently to enhance the accuracy and efficiency of the surface hopping technique. Various limitations of the standard FSSH, associated with quantum nuclear effects, interference and decoherence, trivial or “unavoided” crossings, super-exchange, and representation dependence, have been lifted. These advances are needed to allow one to treat many important phenomena in chemistry, physics, materials and related disciplines. Examples include charge transport in extended systems such as organic solids, singlet fission in molecular aggregates, Auger-type exciton multiplication, recombination and relaxation in quantum dots and other nanoscale materials, Auger-assisted charge transfer, non-radiative luminescence quenching and electron-hole recombination. This perspective summarizes recent advances in the surface hopping formulation of nonadiabatic dynamics and provides an outlook on the future of surface hopping.

The nonequilibrium Fermi's golden rule describes the transition between a photoexcited bright donor electronic state and a dark acceptor electronic state, when the nuclear degrees of freedom start out in a nonequilibrium state. In this paper, we derive a new expression for the nonequilibrium Fermi's golden rule within the framework of the linearized semiclassical approximation. The new expression opens the door to applications of the nonequilibrium Fermi's golden rule to complex condensed-phase molecular systems described in terms of anharmonic force fields. We show that the linearized semiclassical expression for the nonequilibrium Fermi's golden rule yields the exact fully quantum-mechanical result for the canonical Marcus model, where the coupling between donor and acceptor is assumed constant (the Condon approximation) and the donor and acceptor potential energy surfaces are parabolic and identical except for a shift in the equilibrium energy and geometry. For this model, we also present a comprehensive comparison between the linearized semiclassical expression and a hierarchy of more approximate expressions, in both normal and inverted regions, and over a wide range of initial nonequilibrium states, temperatures and frictions.

In a recent series of papers it has been illustrated that a symmetrical quasi-classical (SQC) windowing model applied to the Meyer-Miller (MM) classical vibronic Hamiltonian provides an excellent description of a variety of electronically non-adiabatic benchmark model systems for which exact quantum results are available for comparison. In this paper, the SQC/MM approach is used to treat energy transfer dynamics in site-exciton models of light-harvesting complexes, and in particular, the well-known 7-state Fenna-Mathews-Olson (FMO) complex. Again, numerically "exact" results are available for comparison---here via the hierarchical equation of motion (HEOM) approach of Ishizaki & Fleming---and it is seen that the simple SQC/MM approach provides very reasonable agreement with the previous HEOM results. It is noted, however, that unlike most (if not all) simple approaches for treating these systems, because the SQC/MM approach presents a fully atomistic simulation based on classical trajectory simulation, it places no restrictions on the characteristics of the thermal baths coupled to each two-level site---e.g., bath spectral densities (SD) of any analytic functional form may be employed, as well as discrete SD determined experimentally or from MD simulation (nor is there any restriction that the baths be harmonic), opening up the possibility of simulating more realistic variations on the basic site-exciton framework for describing the non-adiabatic dynamics of photosynthetic pigment complexes.

The generalized quantum master equation provides a powerful tool to describe
the dynamics in quantum impurity models driven away from equilibrium. Two
complementary approaches, one based on Nakajima--Zwanzig--Mori time-convolution
(TC) and the other on the Tokuyama--Mori time-convolutionless (TCL)
formulations provide a starting point to describe the time-evolution of the
reduced density matrix. A key in both approaches is to obtain the so called
"memory kernel" or "generator", going beyond second or fourth order
perturbation techniques. While numerically converged techniques are available
for the TC memory kernel, the canonical approach to obtain the TCL generator is
based on inverting a super-operator in the \emph{full} Hilbert space, which is
difficult to perform and thus, all applications of the TCL approach rely on a
perturbative scheme of some sort. Here, the TCL generator is expressed using a
reduced system propagator which can be obtained from system observables alone
and requires the calculation of super-operators and their inverse in the
\emph{reduced }Hilbert space rather than the full one. This makes the
formulation amenable to quantum impurity solvers or to diagrammatic techniques,
such as the nonequilibrium Green's function. We implement the TCL approach for
the resonant level model driven away from equilibrium and compare the time
scales for the decay of the generator with that of the memory kernel in the TC
approach. Furthermore, the effects of temperature, source-drain bias, and gate
potential on the TCL/TC generators are discussed.

In this paper, we present a comprehensive comparison between the linearized semiclassical expression for the equilibrium Fermi's golden rule rate constant and the progression of more approximate expressions that lead to the classical Marcus expression. We do so within the context of the canonical Marcus model, where the donor and acceptor potential energy surface are parabolic and identical except for a shift in both the free energies and equilibrium geometries, and within the Condon region. The comparison is performed for two different spectral densities and over a wide range of frictions and temperatures, thereby providing a clear test for the validity, or lack thereof, of the more approximate expressions. We also comment on the computational cost and scaling associated with numerically calculating the linearized semiclassical expression for the rate constant and its dependence on the spectral density, temperature and friction.

A recent series of papers have shown that a symmetrical quasi-classical (SQC) windowing model applied to the Meyer-Miller (MM) classical vibronic Hamiltonian provides a very good treatment of electronically non-adiabatic processes in a variety of benchmark model systems, including systems exhibiting strong "quantum" coherence effects and systems that other simple purely-classical approaches are known to have difficulty in describing correctly. In this paper, a different classical electronic Hamiltonian for the treatment of electronically non-adiabatic processes is proposed and "quantized" within the context of the SQC windowing model, which maps the dynamics of F coupled electronic states to a set of F spin-1/2 degrees of freedom (DOF), similar to the fermionic spin model described by Miller and White [J. Chem. Phys. 84, 5059 (1986)]. It is shown that this spin-mapping (SM) Hamiltonian is an exact Hamiltonian if treated as a quantum mechanical (QM) operator---and thus QM'ly equivalent to the MM Hamiltonian---but that an analytically distinct classical analogue is obtained by replacing the QM spin-operators with their classical counterparts. Due to their analytic differences, a practical comparison is then made between the MM and SM Hamiltonians (when quantized with the SQC technique) by applying the latter to many of the same benchmark test problems successfully treated in our recent work with the SQC/MM model. Surprisingly, we find for every benchmark problem that the MM model provides a (slightly) superior description of the true electronically non-adiabatic quantum dynamics versus the new SM model. This is despite the fact that one might expect, a priori, a more natural description of electronic state populations (occupied versus unoccupied) to be provided by DOF with only two states, i.e., spin-1/2 DOF, rather than by harmonic oscillator DOF which have an infinite manifold of states (though only two of these are ever occupied).

In this article, we show how Ehrenfest mean field theory can be made both a more accurate and efficient method to treat nonadiabatic quantum dynamics by combining it with the generalized quantum master equation framework. The resulting mean field generalized quantum master equation (MF-GQME) approach is a non-perturbative and non-Markovian theory to treat open quantum systems without any restrictions on the form of the Hamiltonian that it can be applied to. By studying relaxation dynamics in a wide range of dynamical regimes, typical of charge and energy transfer, we show that MF-GQME provides a much higher accuracy than a direct application of mean field theory. In addition, these increases in accuracy are accompanied by computational speed-ups of between one and two orders of magnitude that become larger as the system becomes more nonadiabatic. This combination of quantum-classical theory and master equation techniques thus makes it possible to obtain the accuracy of much more computationally expensive approaches at a cost lower than even mean field
dynamics, providing the ability to treat the quantum dynamics of atomistic condensed phase systems for long times.

Charge transfer (CT) states formed at the donor/acceptor heterointerface are key for photocurrent generation in organic photovoltaics (OPV). Our calculations show that interfacial donor-to-donor CT states in the phthalocyanine–fullerene OPV system may be more stable than donor-to-acceptor CT states and that they may rapidly recombine, thereby constituting a potentially critical and thus far overlooked loss mechanism. Our results provide new insight into processes that may compete with charge separation, and suggest that the efficiency for charge separation may be improved by destabilizing donor-to-donor CT states or decoupling them from other states.

A novel global flux surface hopping (GFSH) approach is proposed. In this method, the surface hopping probabilities rely on the gross population flow between states, rather than the state-to-state flux as in the standard fewest switches surface hopping (FSSH). GFSH captures the superexchange mechanism of population transfer, while FSSH lacks this capability. In other aspects, including minimization of the number of hops, internal consistency, velocity rescaling, and detailed balance, the GFSH algorithm is similar to FSSH. The advantages of GFSH are demonstrated with a model 3-level system and an Auger process in a semiconductor quantum dot. Current studies indicate that GFSH can replace FSSH, but further tests are needed.

The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long time scales. The non-Markovian transfer tensor method (TTM) demonstrates the coherent-to-incoherent transition as a function of the strength of quantum dissipation and predicts the noncanonical equilibrium distribution due to the system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig equation and, therefore, can be used to reconstruct the dynamical operators (the system Hamiltonian and memory kernel) from quantum trajectories obtained in simulations or experiments. The concept underlying the approach can be generalized to physical observables with the goal of learning and manipulating the trajectories of an open quantum system.

A recently described symmetrical windowing methodology [S. J. Cotton and W. H. Miller, J. Phys. Chem. A 117, 7190 (2013)] for quasi-classical trajectory simulations is applied here to the Meyer-Miller [H.-D. Meyer and W. H. Miller, J. Chem. Phys. 70, 3214 (1979)] model for the electronic degrees of freedom in electronically non-adiabatic dynamics. Results generated using this classical approach are observed to be in very good agreement with accurate quantum mechanical results for a variety of test applications, including problems where coherence effects are significant such as the challenging asymmetric spin-boson system.

The feasibility of calculating photoinduced intramolecular electron transfer rate constants in realistic molecular donor−acceptor systems via Fermi's golden rule, using inputs obtained from state-of-the-art electronic structure techniques, is demonstrated and tested. To this end, calculations of photoinduced electron transfer rate constants were performed on two benchmark systems: (1) phenyl-acetylene-bridged carbazole-naphthalimide (meta and para) and (2) C 60 -(N,N-dimethylaniline). Intramolecular input parameters such as normal-mode frequencies, Huang−Rhys factors, and electronic coupling coefficients were obtained via ground state, time-dependent, and constrained density func-tional theory. Good agreement between the intramolecular Fermi's golden rule rate constants and the experimental rate constants is found for both systems without accounting for the solvent reorganization. The relative roles of intramolecular vs intermolecular modes at promoting electron transfer and the validity of several limits of Fermi's golden rule for describing intramolecular electron transfer are discussed.

A variational method is described for finding approximate solutions of the time-dependent Schrodinger equation. It is closely related to Frenkel's, but better in some respects because it always leads to a true minimum of the error. For several interacting systems it leads to the time-dependent Hartree equations; for a system in an oscillating electric or magnetic field it gives a variational principle for calculating the susceptibility at any frequency.

An extension of the classical trajectory approach is proposed that may be useful in treating many types of nonadiabatic molecular collisions. Nuclei are assumed to move classically on a single potential energy surface until an avoided surface crossing or other region of large nonadiabatic coupling is reached. At such points the trajectory is split into two branches, each of which follows a different potential surface. The validity of this model as applied to the HD2+ system is assessed by numerical integration of the appropriate semiclassical equations. A 3d “trajectory surface hopping” treatment of the reaction of H+ with D2 at a collision energy of 4 eV is reported. The excellent agreement with experiment is an encouraging indication of the potential usefulness of this approach.

The quantum-classical Liouville equation offers a rigorous approach to nonadiabatic quantum dynamics based on surface hopping type trajectories. However, in practice the applicability of this approach has been limited to short times owing to unfavorable numerical scaling. In this paper we show that this problem can be alleviated by combining it with a formally exact generalized quantum master equation treatment. This allows dramatic improvements in the efficiency of the approach in nonadiabatic regimes, making it computationally tractable to treat the quantum dynamics of complex systems for long times. We demonstrate our approach by applying it to a model of condensed phase charge transfer where our method is shown to be numerically exact in regimes where fewest-switches surface hopping and mean field approaches fail to obtain either the correct rates or long-time populations.

A semiclassical approach is presented that allows us to extend the usual Van Vleck-Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. Based on Schwinger's theory of angular momentum, the formulation employs an exact mapping of the discrete quantum variables onto continuous degrees of freedom. The resulting dynamical problem is evaluated through a semiclassical initial-value representation of the time-dependent propagator. As a first application we have performed semiclassical simulations for a spin-boson model, which reproduce the exact quantum-mechanical results quite accurately.

A general formulation is given to the quantum theory of steady diffusion. In seeking for a steady solution of Liouville's
equation, the boundary condition is taken into account by requiring that the solution should lead to a given distribution
of average density. The distribution is to be determined by macroscopic law of diffusion and macroscopic boundary condition.
The basic equation thus obtained has a form similar to Bloch's kinetic equation and reduces to the latter in the limit of
a system of weakly interacting particles. This is shown by generalizing a demping theoretical expansion of Kohn and Luttinger.
It is found that the Einstein relation is valid only for the symmetric part of diffusion- and electric conductivity tensors,
in agreement with Kasuya's suggestion.

The impact of quantum decoherence and zero point motion on non-adiabatic transition rates in condensed matter systems is studied in relation to non-adiabatic (NA) molecular dynamics (MD) techniques. Both effects, and decoherence in particular, strongly influence the transition rate, while neither is accounted for by straightforward quantum-classical approaches. Quantum corrections to the quantum-classical results are rigorously introduced based on Kubo’s generating function formulation of Fermi’s Golden rule and the frozen Gaussian approximation for the nuclear wave functions. The development provides a one-to-one correspondence between the decoherence function and the Franck–Condon factor. The decoherence function defined in this paper corrects an error in our previous work [J. Chem. Phys. 104, 5942 (1996)]. The relationship between the short time approach and the real time NA MD is investigated and a specific prescription for incorporating quantum decoherence into NA simulations is given. The proposed scheme is applied to the hydrated electron. The rate of excited state non-radiative relaxation is found to be very sensitive to the decoherence time. Quantum coherence decays about 50% faster in H2O than in D2O, providing a theoretical rationalization for the lack of experimentally observed solvent isotope effect on the relaxation rate. Microscopic analysis of solvent mode contributions to the coherence decay shows that librational degrees of freedom are primarily responsible, due to the strong coupling between the electron and molecular rotations and to the small widths of the wave packets describing these modes. Zero point motion of the O–H bonds decreases the life time of the excited state of the hydrated electron by a factor of 1.3–1.5. The implications of the use of short time approximations for the NA transition rate and for the evolution of the nuclear wave functions are considered.

A formula for computing approximate leakage of population from an initially prepared electronic state with a nonequilibrium nuclear distribution to a second nonadiabatically coupled electronic state is derived and applied. The formula is a nonequilibrium generalization of the familiar golden rule, which applies when the initial nuclear state is a rovibrational eigenstate of the potential energy surface associated with the initially populated electronic state. Here, more general initial nuclear states are considered. The resultant prescription, termed the nonequilibrium golden rule formula, can be evaluated via semiclassical procedures and hence applied to multidimensional, e.g., condensed phase systems. To illustrate its accuracy, application is made to a spin–boson model of ‘‘inner sphere’’ electron transfer. This model, introduced by Garg et al. [J. Chem. Phys. 83, 4491 (1985)] for the nonadiabatic transition out of a thermal distribution of states in the initial (donor) electronic level, is extended to include nonequilibrium, nonstationary initial nuclear states on the donor surface. The predictions of the nonequilibrium golden rule are found to agree well with numerically exact path integral results for a wide range of initial distortions of the initial nuclear wave packet from its equilibrium configuration.

A microscopically-reversible approach toward computing reaction probabilities via classical trajectory simulation has been developed which bins trajectories symmetrically based upon their initial and final classical actions. The symmetrical quasi-classical (SQT) approach involves defining a classical action window function centered at integer quantum values of the action, choosing a width parameter which is less than unit quantum width, and applying the window function to both initial reactant and final product vibrational states. Calculations were performed using flat histogram windows and Gaussian windows over a range of width parameters. Use of the Wigner distribution function was also investigated as a possible choice. It was demonstrated for collinear H + H2 reactive scattering on the BKMP2 potential energy surface that reaction probabilities computed via the SQT methodology using a Gaussian window function of ½ unit width produces good agreement with quantum mechanical results over the 0.4-0.6 eV energy range relevant to the ground-vibrational state to ground-vibrational state reactive transition.

A theoretical formulation is outlined that allows us to extend the semiclassical Van Vleck–Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. In this formulation the problem of a classical treatment of discrete quantum degrees of freedom (DoF) such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. The mapping approach thus consists of two steps: an exact quantum-mechanical transformation of discrete onto continuous DoF (the “mapping”) and a standard semiclassical treatment of the resulting dynamical problem. Extending previous work [G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997)], various possibilities for obtaining a mapping from discrete to continuous DoF are investigated, in particular the Holstein-Primakoff transformation, Schwinger’s theory of angular momentum [in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam (Academic, New York, 1965)], and the spin-coherent-state representation. Although all these representations are exact on the quantum-mechanical level, the accuracy of their semiclassical evaluation is shown to differ considerably. In particular, it is shown that a generalization of Schwinger’s theory appears to be the only transformation that provides an exact description of a general N-level system within a standard semiclassical evaluation. Exploiting the connection between spin-coherent states and Schwinger’s representation for a two-level system, furthermore, a semiclassical initial-value representation of the corresponding spin-coherent-state propagator is derived. Although this propagator represents an approximation, its appealing numerical features make it a promising candidate for the semiclassical description of large molecular systems with many DoF. Adopting various spin-boson-type models (i.e., a two-level system coupled to a single or many DoF), computational studies are presented for Schwinger’s and the spin-coherent-state representation, respectively. The performance of the semiclassical approximation in the case of regular and chaotic classical dynamics as well as for multimode electronic relaxation dynamics is discussed in some detail.

Redfield theory is applied to investigate the photoinduced dynamics at a conical intersection (the so-called system) which is weakly coupled to a thermal environment (the so-called bath). The dynamics of the system is described by a two-state three-mode model Hamiltonian, chosen to represent the S1(nπ∗)–S2(ππ∗) conical intersection in pyrazine. Dissipative effects are introduced through a bilinear coupling of the system vibrational modes with a harmonic bath, which represents the remaining vibrational degrees of freedom of the molecule and/or interactions with a condensed-phase environment. The Redfield equations for the reduced density matrix are solved numerically without further approximations. From the reduced density matrix the time evolutions of electronic-state populations and vibrational coherences are obtained, as well as time-dependent probability densities of individual vibrational modes. The results provide a visualization of the essential features of the ultrafast (time scale of 10 fs) internal-conversion process at the conical intersection and the ensuing vibrational cooling process on the lower adiabatic potential-energy surface. The effect of vibrational damping on the linear optical absorption spectrum is also investigated. © 2002 American Institute of Physics.

We present a new method for solving the Redfield equation, which describes the evolution of the reduced density matrix of a multilevel quantum‐mechanical system interacting with a thermal bath. The method is based on a new decomposition of the Redfield relaxation tensor that makes possible its direct application to the density matrix without explicit construction of the full tensor. In the resulting expressions, only ordinary matrices are involved and so any quantum system whose Hamiltonian can be diagonalized can be treated with the full Redfield theory. To efficiently solve the equation of motion for the density matrix, we introduce a generalization of the short‐iterative‐Lanczos propagator. Together, these contributions allow the complete Redfield theory to be applied to significantly larger systems than was previously possible. Several model calculations are presented to illustrate the methodology, including one example with 172 quantum states.

We describe a new formulation of methods introduced in the theory of irreversibility by Van Hove and Prigogine, with the purpose of making their ideas easier to understand and to apply. The main tool in this reformulation is the use of projection operators in the Hilbert space of Gibbsian ensemble densities. Projection operators are used to separate an ensemble density into a ``relevant'' part, needed for the calculation of mean values of specified observables, and the remaining ``irrelevant'' part. The relevant part is shown to satisfy a kinetic equation which is a generalization of Van Hove's ``master equation to general order.'' Diagram summation methods are not used. The formalism is illustrated by a new derivation of the Prigogine‐Brout master equation for a classical weakly interacting system.

A method is proposed for carrying out molecular dynamics simulations of processes that involve electronic transitions. The time dependent electronic Schrödinger equation is solved self‐consistently with the classical mechanical equations of motion of the atoms. At each integration time step a decision is made whether to switch electronic states, according to probabilistic ‘‘fewest switches’’ algorithm. If a switch occurs, the component of velocity in the direction of the nonadiabatic coupling vector is adjusted to conserve energy. The procedure allows electronic transitions to occur anywhere among any number of coupled states, governed by the quantum mechanical probabilities. The method is tested against accurate quantal calculations for three one‐dimensional, two‐state models, two of which have been specifically designed to challenge any such mixed classical–quantal dynamical theory. Although there are some discrepancies, initial indications are encouraging. The model should be applicable to a wide variety of gas‐phase and condensed‐phase phenomena occurring even down to thermal energies.

Necessary conditions under which a classical description will give the correct quantum relaxation behavior are analyzed. Assuming a nonequilibrium preparation, it is shown that the long-time mean values of observables can be expressed in terms of the spectral density and state-specific level densities of the system. Any approximation that reproduces these quantities therefore yields the correct expectation values at long times. Apart from this rigorous condition, a weaker but more practical criterion is established, that is, to require that the total level density is well approximated in the energy range defined by the spectral density. Since the integral level density is directly proportional to the phase-space volume that is energetically accessible to the system, the latter condition means that an appropriate classical approximation should explore the same phase-space volume as the quantum description. In general, however, this is not the case. A well-known example is the unrestricted flow of zero-point energy in classical mechanics. To correct for this flaw of classical mechanics, quantum corrections are derived which result in a restriction of the classically accessible phase space. At the simplest level of the theory, these corrections are shown to correspond to the inclusion of only a fraction of the full zero-point energy into the classical calculation. Based on these considerations, a general strategy for the classical simulation of quantum relaxation dynamics is suggested. The method is (i) dynamically consistent in that it refers to the behavior of the ensemble rather than to the behavior of individual trajectories, (ii) systematic in that it provides (rigorous as well as minimal) criteria which can be checked in a practical calculation, and (iii) practical in that it retains the conceptional and computational simplicity of a standard quasiclassical calculation. Employing various model problems which allow for an analytical evaluation of the quantities of interest, the virtues and limitations of the approach are discussed. © 1999 American Institute of Physics.

In a recent letter [Chem. Phys. Lett. 291, 101 (1998)] we presented a semiclassical methodology for calculating influence functionals arising from many-body anharmonic environments in the path integral formulation of quantum dynamics. Taking advantage of the trace operation associated with the unobservable medium, we express the influence functional in terms of a single propagator along a combined forward–backward system path. This propagator is evaluated according to time-dependent semiclassical theory in a coherent state initial value representation. Because the action associated with propagation in combined forward and backward time is governed by the net force experienced by the environment due to its interaction with the system, the resulting propagator is generally a smooth function of coordinates and thus amenable to Monte Carlo sampling; yet, the interference between forward and reverse propagators is fully accounted for. In the present paper we present a more elaborate version of the semiclassical influence functional formalism, along with algorithms for evaluating the coherent state transform of the Boltzmann operator that enters the influence functional. This factor is evaluated by performing an imaginary time path integral, and various approximations of the resulting expression as well as sampling schemes are discussed. The feasibility of the approach is demonstrated via numerous test calculations involving a two-level system coupled to (a) a dissipative harmonic bath and (b) a ten-dimensional bath of coupled anharmonic oscillators. © 1999 American Institute of Physics.