Analysis of the quantum-classical Liouville equation in the mapping basis

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

General framework for quantifying dissipation pathways in open quantum systems. II. Numerical validation and the role of non-Markovianity

In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111-1-214111-13 (2024)], we developed a theory called MQME-D, which allows us to decompose the overall energy dissipation process in open quantum system dynamics into contributions by individual components of the bath when the subsystem dynamics is governed by a Markovian quantum master equation (MQME). Here, we contrast the predictions of MQME-D against the numerically exact results obtained by combining hierarchical equations of motion (HEOM) with a recently reported protocol for monitoring the statistics of the bath. Overall, MQME-D accurately captures the contributions of specific bath components to the overall dissipation while greatly reducing the computational cost compared to exact computations using HEOM. The computations show that MQME-D exhibits errors originating from its inherent Markov approximation. We demonstrate that its accuracy can be significantly increased by incorporating non-Markovianity by exploiting time scale separations (TSS) in different components of the bath. Our work demonstrates that MQME-D combined with TSS can be reliably used to understand how energy is dissipated in realistic open quantum system dynamics.

A first principles derivation of energy-conserving momentum jumps in surface hopping simulations

The fewest switches surface hopping (FSSH) method proposed by Tully in 1990 [Tully, J. Chem. Phys. 93, 1061 (1990)]-along with its many later variations-forms the basis for most practical simulations of molecular dynamics with electronic transitions in realistic systems. Despite its popularity, a rigorous formal derivation of the algorithm has yet to be achieved. In this paper, we derive the energy-conserving momentum jumps employed by FSSH from the perspective of quantum trajectory surface hopping (QTSH) [Martens, J. Phys. Chem. A 123, 1110 (2019)]. In the limit of localized nonadiabatic transitions, simple mathematical and physical arguments allow the FSSH algorithm to be derived from first principles. For general processes, the quantum forces characterizing the QTSH method provide accurate results for nonadiabatic dynamics with rigorous energy conservation, at the ensemble level, within the consistency of the underlying stochastic surface hopping without resorting to the artificial momentum rescaling of FSSH.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Mixed Quantum-Classical Liouville Equation Treatment of Electronic Spectroscopy of Condensed Systems: Harmonic and Anharmonic Electron-Phonon Coupling

This Review integrates the use of electronic optical response function theory and the mixed quantum-classical (MQC) Liouville equation (MQCLE), thereby leading to electronic spectroscopy in MQC media. It further sheds light on the applicability, utility, and efficiency of the mixed quantum-classical dynamics (MQCD) formalism, which starts off with the MQCLE, in probing spectroscopy and dynamics of condensed systems, whereby quantum mechanics and classical mechanics are combined systematically. The author has been exploring and implementing MQCD to investigate electron-phonon coupling effects on electronic dephasing in harmonic and anharmonic systems by calculating linear and nonlinear optical transition analytically and numerically dipole moment time correlation functions in an MQC environment, thereby presenting an in depth spectral profile analysis and their shape and symmetry. The distinctive capability of the MQC time correlation functions is that ergodicity and stationarity properties are inherently satisfied as part of the mixed quantum-classical dynamics (MQCD) framework, unlike classical correlation functions. While some research groups have applied MQCLE to calculate vibrational spectra to study hydrogen-bonded complexes in a MQC environment and other groups calculated Optical response function to probe electron transfer dynamics using the basis mapping technique, the approach, purpose, rigor, applications, and path to the end results reported herein are different. Finally, the same framework is employed to study dissipative systems in the MQC limit, whereby the zero-phonon line adopts the correct width and eliminates its asymmetry. While the full quantum mechanical model, like the multimode Brownian oscillator (MBO) model, yields the correct width and inaccurate shape in the low-temperature limit, the MQCD formalism seems to produce an accurate zero-phonon profile. Nonlinear optical signals are also reviewed in MQC media to show the applicability and utility of this approach. The vibronic optical response functions developed here will account for geometry change, frequency change, and anharmonicity upon electronic excitation to accurately probe electronic dephasing, electron-phonon coupling, shape, and symmetry of profiles and present differences and similarities to the MBO model on pure electronic dephasing. Frequency change and anharmonicity are vitally crucial for accurately assessing electron-phonon coupling upon electronic excitation. This is an additional unique result obtained by the author to further demonstrate the applicability and utility of this approach over other approximation schemes in probing electronic dephasing including that of the MBO model.

Non-adiabatic Mapping Dynamics in the Phase Space of the ${SU}(N)$ Lie Group

We present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based up a new mapping formalism recently introduced by Runeson and Richardson in [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich-Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom, and derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this Linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connect the previous methods in a clear and concise manner.

DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach

Mixed quantum-classical dynamics provides an efficient way of simulating the dynamics of quantum subsystems coupled to many-body environments. Many processes, including proton-transfer reactions, electron-transfer reactions, and vibrational energy transport, for example, take place in such open systems. The most accurate algorithms for performing mixed quantum-classical simulations require very large ensembles of trajectories to obtain converged expectation values, which is computationally prohibitive for quantum subsystems containing even a few degrees of freedom. The recently developed “Deterministic evolution of coordinates with initial decoupled equations” (DECIDE) method has demonstrated high accuracy and low computational cost for a host of model systems; however, these applications relied on representing the equations of motion in subsystem and adiabatic energy bases. While these representations are convenient for certain systems, the position representation is convenient for many other systems, including systems undergoing proton- and electron-transfer reactions. Thus, in this review, we provide a step-by-step derivation of the DECIDE approach and demonstrate how to cast the DECIDE equations in a quantum harmonic oscillator position basis for a simple one-dimensional (1D) hydrogen bond model. After integrating the DECIDE equations of motion on this basis, we show that the total energy of the system is conserved for this model and calculate various quantities of interest. Limitations of casting the equations in an incomplete basis are also discussed.

Commutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics

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

Non-adiabatic Matsubara dynamics and non-adiabatic ring-polymer molecular dynamics

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.

Two-oscillator mapping modification of the Poisson bracket mapping equation formulation of the quantum-classical Liouville equation

Mapping basis solutions provide efficient ways for simulating mixed quantum-classical (MQC) dynamics in complex systems by matching multiple quantum states of interest to some fictitious physical states. Recently, various MQC methods were devised such that two harmonic oscillators are employed to represent each electronic state, showing improvements over one-oscillator-based methods. Here, we introduce and analyze newly modified mapping approximations of the quantum-classical Liouville equation (QCLE) using two oscillators for each electronic state. We design two separate mapping relations that we can adopt toward simulating dynamics and computing expectation values. Through the process, two MQC methods can be constructed, one of which actually reproduces the population dynamics of the forward and backward trajectory solution of QCLE. By applying the methods to spin-boson systems with a range of parameters, we find out that the choice of mapping relations greatly affects the simulation results. We also show that further improvement is possible through using modified identity operator formulations. Our findings may be helpful in constructing improved MQC methods in the future.

Toward monitoring the dissipative vibrational energy flows in open quantum systems by mixed quantum-classical simulations

In open quantum system dynamics, rich information about the major energy relaxation channels and corresponding relaxation rates can be elucidated by monitoring the vibrational energy flow among individual bath modes. However, such calculations often become tremendously difficult as the complexity of the subsystem-bath coupling increases. In this paper, we attempt to make this task feasible by using a mixed quantum-classical method, the Poisson-bracket mapping equation with non-Hamiltonian modification (PBME-nH) [H. W. Kim and Y. M. Rhee, J. Chem. Phys. 140, 184106 (2014)]. For a quantum subsystem bilinearly coupled to harmonic bath modes, we derive an expression for the mode energy in terms of the classical positions and momenta of the nuclei, while keeping consistency with the energy of the quantum subsystem. The accuracy of the resulting expression is then benchmarked against a numerically exact method by using relatively simple models. Although our expression predicts a qualitatively correct dissipation rate for a range of situations, cases involving a strong vibronic resonance are quite challenging. This is attributed to the inherent lack of quantum back reaction in PBME-nH, which becomes significant when the subsystem strongly interacts with a small number of bath modes. A rigorous treatment of such an effect will be crucial for developing quantitative simulation methods that can handle generic subsystem-bath coupling.

Benchmarking semiclassical and perturbative methods for real-time simulations of cavity-bound emission and interference

We benchmark a selection of semiclassical and perturbative dynamics techniques by investigating the correlated evolution of a cavity-bound atomic system to assess their applicability to study problems involving strong light-matter interactions in quantum cavities. The model system of interest features spontaneous emission, interference, and strong coupling behavior and necessitates the consideration of vacuum fluctuations and correlated light-matter dynamics. We compare a selection of approximate dynamics approaches including fewest switches surface hopping (FSSH), multitrajectory Ehrenfest dynamics, linearized semiclassical dynamics, and partially linearized semiclassical dynamics. Furthermore, investigating self-consistent perturbative methods, we apply the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the second Born approximation. With the exception of fewest switches surface hopping, all methods provide a reasonable level of accuracy for the correlated light-matter dynamics, with most methods lacking the capacity to fully capture interference effects.

Nonadiabatic Dynamics via the Symmetrical Quasi-Classical Method in the Presence of Anharmonicity

The symmetrical quasi-classical (SQC) method recently proposed by Miller and Cotton allows one to simulate nonadiabatic dynamics based on an algorithm with classical-like scaling with respect to system size. This is made possible by casting the electronic degrees of freedom in terms of mapping variables that can be propagated in a classical-like manner. While SQC was shown to be rather accurate when applied to benchmark models with harmonic electronic potential energy surfaces, it was also found to become inaccurate and to suffer numerical instabilities when applied to anharmonic systems. In this paper, we propose an extended SQC (E-SQC) methodology for overcoming those discrepancies by describing the anharmonic nuclear modes, which are coupled to the electronic degrees of freedom, in terms of classical-like mapping variables. The accuracy of E-SQC relative to standard SQC is demonstrated on benchmark models with quartic and Morse potential energy surfaces.

Accurate Long-Time Mixed Quantum-Classical Liouville Dynamics via the Transfer Tensor Method

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.

Surface hopping with a manifold of electronic states. I. Incorporating surface-leaking to capture lifetimes

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).

Quantum dynamics in open quantum-classical systems

Often quantum systems are not isolated and interactions with their environments must be taken into account. In such open quantum systems these environmental interactions can lead to decoherence and dissipation, which have a marked influence on the properties of the quantum system. In many instances the environment is well-approximated by classical mechanics, so that one is led to consider the dynamics of open quantum-classical systems. Since a full quantum dynamical description of large many-body systems is not currently feasible, mixed quantum-classical methods can provide accurate and computationally tractable ways to follow the dynamics of both the system and its environment. This review focuses on quantum-classical Liouville dynamics, one of several quantum-classical descriptions, and discusses the problems that arise when one attempts to combine quantum and classical mechanics, coherence and decoherence in quantum-classical systems, nonadiabatic dynamics, surface-hopping and mean-field theories and their relation to quantum-classical Liouville dynamics, as well as methods for simulating the dynamics.

On surface hopping and time-reversal

While individual Tully-style fewest switches surface hopping (FSSH) trajectories are stochastic and cannot be inverted in time, it is possible to reverse in time the dynamics of a swarm of FSSH trajectories. Here we show exactly how to invert such dynamics, and we investigate the stability of such time-reversed surface hopping dynamics. We demonstrate that FSSH trajectories can be inverted successfully for short time periods, but the time-reversed dynamics become unstable for long times with multiple hopping events. We argue that this instability of FSSH going backward in time can be correlated with the stability of the FSSH algorithm going forward in time.

Non-adiabatic molecular dynamics with complex quantum trajectories. I. The diabatic representation

We extend a recently developed quantum trajectory method [Y. Goldfarb, I. Degani, and D. J. Tannor, J. Chem. Phys. 125, 231103 (2006)] to treat non-adiabatic transitions. Each trajectory evolves on a single surface according to Newton's laws with complex positions and momenta. The transfer of amplitude between surfaces stems naturally from the equations of motion, without the need for surface hopping. In this paper we derive the equations of motion and show results in the diabatic representation, which is rarely used in trajectory methods for calculating non-adiabatic dynamics. We apply our method to the first two benchmark models introduced by Tully [J. Chem. Phys. 93, 1061 (1990)]. Besides giving the probability branching ratios between the surfaces, the method also allows the reconstruction of the time-dependent wavepacket. Our results are in quantitative agreement with converged quantum mechanical calculations.

Non-adiabatic molecular dynamics with complex quantum trajectories. II. The adiabatic representation

We present a complex quantum trajectory method for treating non-adiabatic dynamics. Each trajectory evolves classically on a single electronic surface but with complex position and momentum. The equations of motion are derived directly from the time-dependent Schrödinger equation, and the population exchange arises naturally from amplitude-transfer terms. In this paper the equations of motion are derived in the adiabatic representation to complement our work in the diabatic representation [N. Zamstein and D. J. Tannor, J. Chem. Phys. 137, 22A517 (2012)]. We apply our method to two benchmark models introduced by John Tully [J. Chem. Phys. 93, 1061 (1990)], and get very good agreement with converged quantum-mechanical calculations. Specifically, we show that decoherence (spatial separation of wavepackets on different surfaces) is already contained in the equations of motion and does not require ad hoc augmentation.