Quantum-classical Liouville Equation Research Articles

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.

Floquet-driven frictional effects

When the coupled electron-nuclear dynamics are subjected to strong Floquet driving, there is a strong breakdown of the Born-Oppenheimer approximation. In this article, we derive a Fokker-Planck equation to describe non-adiabatic molecular dynamics with electronic friction for Floquet driven systems. We first provide a new derivation of the Floquet quantum-classical Liouville equation (QCLE) for driven electron-nuclear dynamics. We then transform the Floquet QCLE into a Fokker-Planck equation with explicit forms of frictional force and random force. We recast the electronic friction in terms of Floquet Green's functions such that we can evaluate the electronic friction explicitly. We show that the Floquet electronic friction tensor exhibits antisymmetric terms even at equilibrium for real-valued Hamiltonian, suggesting that there is a Lorentz-like force in Floquet driven non-Born Oppenheimer dynamics even without any spin-orbit couplings.

A mapping approach to surface hopping.

We present a nonadiabatic classical-trajectory approach that offers the best of both worlds between fewest-switches surface hopping (FSSH) and quasiclassical mapping dynamics. This mapping approach to surface hopping (MASH) propagates the nuclei on the active adiabatic potential-energy surface, such as in FSSH. However, unlike in FSSH, transitions between active surfaces are deterministic and occur when the electronic mapping variables evolve between specified regions of the electronic phase space. This guarantees internal consistency between the active surface and the electronic degrees of freedom throughout the dynamics. MASH is rigorously derivable from exact quantum mechanics as a limit of the quantum-classical Liouville equation (QCLE), leading to a unique prescription for momentum rescaling and frustrated hops. Hence, a quantum-jump procedure can, in principle, be used to systematically converge the accuracy of the results to that of the QCLE. This jump procedure also provides a rigorous framework for deriving approximate decoherence corrections similar to those proposed for FSSH. We apply MASH to simulate the nonadiabatic dynamics in various model systems and show that it consistently produces more accurate results than FSSH at a comparable computational cost.

A quantum-classical Liouville formalism in a preconditioned basis and its connection with phase-space surface hopping.

We revisit a recent proposal to model nonadiabatic problems with a complex-valued Hamiltonian through a phase-space surface hopping (PSSH) algorithm employing a pseudo-diabatic basis. Here, we show that such a pseudo-diabatic PSSH (PD-PSSH) ansatz is consistent with a quantum-classical Liouville equation (QCLE) that can be derived following a preconditioning process, and we demonstrate that a proper PD-PSSH algorithm is able to capture some geometric magnetic effects (whereas the standard fewest switches surface hopping approach cannot capture such effects). We also find that a preconditioned QCLE can outperform the standard QCLE in certain cases, highlighting the fact that there is no unique QCLE. Finally, we also point out that one can construct a mean-field Ehrenfest algorithm using a phase-space representation similar to what is done for PSSH. These findings would appear extremely helpful as far as understanding and simulating nonadiabatic dynamics with complex-valued Hamiltonians and/or spin degeneracy.

Low-temperature vibronic spectroscopy of condensed chromophore exhibiting inhomogeneous distribution of vibrational frequencies in a mixed quantum-classical environment.

This work has been motivated by the recent paper by the author [M. Toutounji, Phys. Chem. Chem. Phys., 2021, 23, 21981] whereby a mixed quantum-classical Liouville equation was used to probe the spectroscopy and dynamics of a spin-boson system. A mixed quantum-classical Liouville equation treats the system of interest quantum mechanically, the bath classically, and the coupling term mixed quantum-classical mechanically. This paper offers a two-fold advantage: correcting the treatment of the electronic transition decay (width in frequency domain) and assessing the local heterogeneous vibrational structure. The homogeneous linear absorption spectrum of a chromophore embedded in a mixed quantum-classical environment at low temperature is composed of a sharp peak called a zero-phonon line (ZPL) and a broad phonon sideband (PSB), whereby the ZPL and the PSB are assimilated by a Lorentzian function and Voigt profiles, respectively. The PSB, in this case, is characterized by a local heterogeneous structure due to a dispersive medium of vibrations, modeled by vibrational Gaussian distributions to represent the arising inhomogeneous broadening and Lorentzians to model the homogeneous vibrations. This description seems to model proteins and amorphous solids exhibiting a local heterogeneous structure as both electronic and vibrational inhomogeneous broadening seems to be large in these media. This work provides a derivation of linear absorption lineshape and vibronic transition dipole moment time correlation functions, both of which account for pure electronic dephasing (ZPL width) the Voigt profile description of the phonon profiles (PSB) in dispersive media.

Quantum-Classical Approach for Calculations of Absorption and Fluorescence: Principles and Applications.

Absorption and fluorescence spectroscopy techniques provide a wealth of information on molecular systems. The simulations of such experiments remain challenging, however, despite the efforts put into developing the underlying theory. An attractive method of simulating the behavior of molecular systems is provided by the quantum–classical theory—it enables one to keep track of the state of the bath explicitly, which is needed for accurate calculations of fluorescence spectra. Unfortunately, until now there have been relatively few works that apply quantum–classical methods for modeling spectroscopic data. In this work, we seek to provide a framework for the calculations of absorption and fluorescence lineshapes of molecular systems using the methods based on the quantum–classical Liouville equation, namely, the forward–backward trajectory solution (FBTS) and the non-Hamiltonian variant of the Poisson bracket mapping equation (PBME-nH). We perform calculations on a molecular dimer and the photosynthetic Fenna–Matthews–Olson complex. We find that in the case of absorption, the FBTS outperforms PBME-nH, consistently yielding highly accurate results. We next demonstrate that for fluorescence calculations, the method of choice is a hybrid approach, which we call PBME-nH-Jeff, that utilizes the effective coupling theory [GelzinisA.;J. Chem. Phys.2020, 152, 05110332035455] to estimate the excited state equilibrium density operator. Thus, we find that FBTS and PBME-nH-Jeff are excellent candidates for simulating, respectively, absorption and fluorescence spectra of real molecular systems.

Electronic dephasing in mixed quantum–classical molecular systems using the spin-boson model

Pure electronic dephasing is investigated using the spin-boson Hamiltonian in mixed quantum–classical environment. The spin-boson model used here is a composite system made up of a quantum subsystem, an electronic 2-level subsystem linearly coupled to harmonic vibrations, interacting with a classical bath. Experimental results for a multitude of molecular systems indicate that the zero-phonon line (ZPL) profile is determined by electronic dephasing, which is not accounted for in the multimode Brownian oscillator (MBO) model due to the unphysical contribution from the MBO bath modes to the ZPL profile. Mixed quantum–classical dynamics formalism of non-equilibrium systems is employed to assess the contribution of the bath modes to pure electronic dephasing by probing the ZPL profile when coupled to a classical bath in the mixed quantum–classical condensed systems. Pure electronic dephasing is discussed in the context of mixed quantum–classical dynamics formalism which starts with mixed quantum–classical Liouville equation in a mixed quantum–classical environment. It is noteworthy, however, that the fundamental difference between the fully quantum MBO model and the mixed quantum–classical Brownian oscillator, is that the zero-phonon line calculated by the former shows unphysical asymmetry on the low-energy side as it has not been observed in real systems, whereas the ZPL reported herein eliminates this asymmetry. A systematic approach using matrix mechanics is developed to treat this phenomenon. To this end, a closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in a dissipative media are derived. Linear absorption spectra and 4-wave mixing signals at various temperatures showing a sound thermal broadening, temporal decay, and accurate pure dephasing further ratify the applicability and correctness of the mixed quantum–classical dynamics approach to spectroscopy and dynamics are computed.

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.

A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes.

The hierarchical equations of motion (HEOMs) have developed into an important tool in simulating quantum dynamics in condensed phases. Yet, it has recently been found that the HEOM may become numerically unstable in simulations using discrete harmonic oscillator modes [I. S. Dunn, et al., J. Chem. Phys. 150, 184109 (2019)]. In this paper, a new set of equations of motion are obtained based on the equivalence between the HEOM for discrete harmonic oscillator modes and the mixed quantum-classical Liouville equation. The new set of equations can thus be regarded as the expansion of the same phase space partial differential equation using different basis sets. It is shown that they have similar structures as the original HEOM but are free from the problem of numerical instability. The new set of equations are also incorporated into the matrix product state method, where it is found that the trace of the reduced density operator is not well conserved during the propagation. A modified time-dependent variational principle is then proposed to achieve better trace conservation.

On the proper derivation of the Floquet-based quantum classical Liouville equation and surface hopping describing a molecule or material subject to an external field.

We investigate different approaches to derive the proper Floquet-based quantum-classical Liouville equation (F-QCLE) for laser-driven electron-nuclear dynamics. The first approach projects the operator form of the standard QCLE onto the diabatic Floquet basis and then transforms to the adiabatic representation. The second approach directly projects the QCLE onto the Floquet adiabatic basis. Both approaches yield a form that is similar to the usual QCLE with two modifications: (1) The electronic degrees of freedom are expanded to infinite dimension and (2) the nuclear motion follows Floquet quasi-energy surfaces. However, the second approach includes an additional cross derivative force due to the dual dependence on time and nuclear motion of the Floquet adiabatic states. Our analysis and numerical tests indicate that this cross derivative force is a fictitious artifact, suggesting that one cannot safely exchange the order of Floquet state projection with adiabatic transformation. Our results are in accord with similar findings by Izmaylov et al., [J. Chem. Phys. 140, 084104 (2014)] who found that transforming to the adiabatic representation must always be the last operation applied, although now we have extended this result to a time-dependent Hamiltonian. This paper and the proper derivation of the F-QCLE should lay the basis for further improvements of Floquet surface hopping.

Modeling voltammetry curves for proton coupled electron transfer: The importance of nuclear quantum effects.

We investigate rates of proton-coupled electron transfer (PCET) in potential sweep experiments for a generalized Anderson-Holstein model with the inclusion of a quantized proton coordinate. To model this system, we utilize a quantum classical Liouville equation embedded inside of a classical master equation, which can be solved approximately with a recently developed algorithm combining diffusional effects and surface hopping between electronic states. We find that the addition of nuclear quantum effects through the proton coordinate can yield quantitatively (but not qualitatively) different IV curves under a potential sweep compared to electron transfer (ET). Additionally, we find that kinetic isotope effects give rise to a shift in the peak potential, but not the peak current, which would allow for quantification of whether an electrochemical ET event is proton-coupled or not. These findings suggest that it will be very difficult to completely understand coupled nuclear-electronic effects in electrochemical voltammetry experiments using only IV curves, and new experimental techniques will be needed to draw inferences about the nature of electrochemical PCET.

Benchmarking the forward-backward trajectory solution of the quantum-classical Liouville equation.

Various quantum-classical approaches to the simulation of processes taking place in real molecular systems have been shown to provide quantitatively correct results in a number of scenarios. However, it is not immediately clear how strongly the approximations related to the classical treatment of the system's environment compromise the accuracy of these methods. In this work, we present the analysis of the accuracy of the forward-backward trajectory solution (FBTS) of the quantum-classical Liouville equation. To this end, we simulate the excitation dynamics in a molecular dimer using the FBTS and the exact hierarchical equations of motion approach. To facilitate the understanding of the possible benefits of the FBTS, the simulations are also performed using a closely related quantum-classical Poisson Bracket Mapping Equation (PBME) method, as well as the well-known Förster and Redfield theories. We conclude that the FBTS is considerably more accurate than the PBME and the perturbative approaches for most realistic parameter sets and is, therefore, more versatile. We investigate the impact each parameter has on the accuracy of the FBTS. Our results can be used to predict whether the FBTS may be expected to yield satisfactory results when calculating system dynamics for the given system parameters.

Exciton dissociation and charge separation at donor-acceptor interfaces from quantum-classical dynamics simulations.

In organic photovoltaic (OPV) systems, exciton dissociation and ultrafast charge separation at donor-acceptor heterojunctions both play a key role in controlling the efficiency of the conversion of excitation energy into free charge carriers. In this work, nonadiabatic dynamics simulations based on the quantum-classical Liouville equation are employed to study the real-time dynamics of exciton dissociation and charge separation at a model donor-acceptor interface. Benchmark comparisons for a variety of low dimensional donor-acceptor chain models are performed to assess the accuracy of the quantum classical dynamics technique referred to as the forward-backward trajectory solution (FBTS). Although not always quantitative, the FBTS approach offers a reasonable balance between accuracy and computational cost. The short-time dynamics of exciton dissociation in related higher-dimensional lattice models for the interface are also investigated to assess the effect of the dimensionality on the first steps in the mechanism of charge carrier generation.

Absorption matrix of multi-site systems calculated by a hybrid quantum-classical Liouville equation.

The linear absorption spectrum of a multisite system can be written as a weighted accumulation of elements of an absorption matrix. In the framework of the quantum-classical Liouville equation (QCLE), a mean-field approximation is introduced to simplify the calculation of the absorption matrix. The classical bath oscillators are propagated under partially shifted potentials, which reflects an averaged behavior after considering quantum jumps of the system states. For a specified initial condition, the time-dependent shifting possibility of each bath potential is given by the time evolution of site population estimated by the Redfield equation or the noninteracting blip approximation. The two hybrid QCLE approaches are tested in various models, including biased and unbiased two-site models, a subnetwork and the whole monomer of Fenna-Matthews-Olson, and harmonic and anharmonic baths. With numerically excellent results, the numerical studies show reliability and flexibility of the hybrid QCLE in calculating the absorption matrix and spectrum.

A demonstration of consistency between the quantum classical Liouville equation and Berry's phase and curvature for the case of complex Hamiltonians.

Although the quantum classical Liouville equation (QCLE) arises by cutting off the exact equation of motion for a coupled nuclear-electronic system at order 1 (1 = ℏ0), we show that the QCLE does include Berry's phase effects and Berry's forces (which are proportional to a higher order, ℏ = ℏ1). Thus, the fundamental equation underlying mixed quantum-classical dynamics does not need a correction for Berry's phase effects and is valid for the case of complex (i.e., not just real) Hamiltonians, where exotic features can arise in the course of electronic relaxation. Furthermore, we also show that, even though Tully's surface hopping model ignores Berry's phase, Berry's phase effects are included automatically within Ehrenfest dynamics. These findings should be of great importance if we seek to model coupled nuclear-electronic dynamics for systems with odd numbers of electrons and spin-orbit coupling, where the complex nature of the Hamiltonian is paramount.

Heat transfer statistics in mixed quantum-classical systems.

The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.

Efficient and Deterministic Propagation of Mixed Quantum-Classical Liouville Dynamics.

We propose a highly efficient mixed quantum-classical molecular dynamics scheme based on a solution of the quantum-classical Liouville equation (QCLE). By casting the equations of motion for the quantum subsystem and classical bath degrees of freedom onto an approximate set of coupled first-order differential equations for c-numbers, this scheme propagates the composite system in time deterministically in terms of independent classical-like trajectories. To demonstrate its performance, we apply the method to the spin-boson model, a photoinduced electron transfer model, and a Fenna-Matthews-Olsen complex model, and find excellent agreement out to long times with the numerically exact results, using several orders of magnitude fewer trajectories than surface-hopping solutions of the QCLE. Owing to its accuracy and efficiency, this method promises to be very useful for studying the dynamics of mixed quantum-classical systems.

Quantum kinetic expansion in the spin-boson model: Implemented by the quantum-classical Liouville equation in an anharmonic bath.

In the framework of the quantum-classical Liouville equation (QCLE), the quantum kinetic expansion (QKE) of the spin-boson model is extended to an arbitrary combination of the bath potential and the system-bath interaction. The mixed quantum-classical estimation of the QKE rate kernels and modification functions are transformed into averages of deterministic classical trajectories over the Wigner initial distribution. For the standard spin-boson model, the QCLE-QKE method produces exactly the same result as that from full quantum dynamics and the numerical applicability of the approximate action-angle initial distribution is verified. For an anharmonic bath with the quartic potential, the QCLE-QKE calculation under the action-angle initial distribution illustrates the influence of this specific anharmonicity. With the increase of the quartic parameter, the fourth order QKE corrections are suppressed and the short-time population transfer is accelerated together with an enhanced quantum oscillation.

Universality of electronic friction. II. Equivalence of the quantum-classical Liouville equation approach with von Oppen's nonequilibrium Green's function methods out of equilibrium

In a recent publication [W. Dou et al., Phys. Rev. Lett. 119, 046001 (2017)], using the quantum-classical Liouville equation (QCLE), we derived a very general form for the electronic friction felt by a molecule moving near one or many metal surfaces. Moreover, we have already proved the equivalence of the QCLE electronic friction with the Head-Gordon--Tully model as well as a generalized version of von Oppen's nonequilibrium Green's function (NEGF) method at equilibrium [W. Dou and J. E. Subotnik, Phys. Rev. B 96, 104305 (2017)]. In the present paper, we now further prove the equivalence between the QCLE friction and the NEGF friction for the case of multiple metal surfaces and an out-of-equilibrium electronic current without electron-electron interactions. The present results reinforce our recent claim that there is only one universal electronic friction tensor arising from the Born-Oppenheimer approximation.

Born-Oppenheimer Dynamics, Electronic Friction, and the Inclusion of Electron-Electron Interactions.

We present a universal expression for the electronic friction as felt by a set of classical nuclear degrees of freedom (DOFs) coupled to a manifold of quantum electronic DOFs; no assumptions are made regarding the nature of the electronic Hamiltonian and electron-electron repulsions are allowed. Our derivation is based on a quantum-classical Liouville equation for the coupled electronic-nuclear motion, followed by an adiabatic approximation whereby electronic transitions are assumed to equilibrate faster than nuclear movement. The resulting form of friction is completely general, but does reduce to previously published expressions for the quadratic Hamiltonian (i.e., Hamiltonians without electronic correlation). At equilibrium, the second fluctuation-dissipation theorem is satisfied and the frictional matrix is symmetric. To demonstrate the importance of electron-electron correlation, we study electronic friction within the Anderson-Holstein model, where a proper treatment of electron-electron interactions shows signatures of a Kondo resonance and a mean-field treatment is completely inadequate.