Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes

Modeling nonadiabatic dynamics in condensed matter materials: some recent advances and applications

This review focuses on recent developments in the field of nonadiabatic molecular dynamics (NA-MD), with particular attention given to condensed-matter systems. NA-MD simulations for small molecular systems can be performed using high-level electronic structure (ES) calculations, methods accounting for the quantization of nuclear motion, and using fewer approximations in the dynamical methodology itself. Modeling condensed-matter systems imposes many limitations on various aspects of NA-MD computations, requiring approximations at various levels of theory-from the ES, to the ways in which the coupling of electrons and nuclei are accounted for. Nonetheless, the approximate treatment of NA-MD in condensed-phase materials has gained a spin lately in many applied studies. A number of advancements of the methodology and computational tools have been undertaken, including general-purpose methods, as well as those tailored to nanoscale and condensed matter systems. This review summarizes such methodological and software developments, puts them into the broader context of existing approaches, and highlights some of the challenges that remain to be solved.

Non-adiabatic Excited-State Molecular Dynamics: Theory and Applications for Modeling Photophysics in Extended Molecular Materials

Optically active molecular materials, such as organic conjugated polymers and biological systems, are characterized by strong coupling between electronic and vibrational degrees of freedom. Typically, simulations must go beyond the Born-Oppenheimer approximation to account for non-adiabatic coupling between excited states. Indeed, non-adiabatic dynamics is commonly associated with exciton dynamics and photophysics involving charge and energy transfer, as well as exciton dissociation and charge recombination. Understanding the photoinduced dynamics in such materials is vital to providing an accurate description of exciton formation, evolution, and decay. This interdisciplinary field has matured significantly over the past decades. Formulation of new theoretical frameworks, development of more efficient and accurate computational algorithms, and evolution of high-performance computer hardware has extended these simulations to very large molecular systems with hundreds of atoms, including numerous studies of organic semiconductors and biomolecules. In this Review, we will describe recent theoretical advances including treatment of electronic decoherence in surface-hopping methods, the role of solvent effects, trivial unavoided crossings, analysis of data based on transition densities, and efficient computational implementations of these numerical methods. We also emphasize newly developed semiclassical approaches, based on the Gaussian approximation, which retain phase and width information to account for significant decoherence and interference effects while maintaining the high efficiency of surface-hopping approaches. The above developments have been employed to successfully describe photophysics in a variety of molecular materials.

Nonadiabatic Molecular Dynamics at Metal Surfaces

Dynamics at molecule-metal interfaces are a subject of intense current interest and come in many different flavors of experiments: gas-phase scattering, chemisorption, electrochemistry, nanojunction transport, and heterogeneous catalysis, to name a few. These dynamics involve nuclear degrees of freedom entangled with many electronic degrees of freedom (in the metal), and as such there is always the possibility for nonadiabatic phenomena to appear: the nuclei do not necessarily need to move slower than the electrons to break the Born-Oppenheimer (BO) approximation. In this Feature Article, we review a set of dynamical methods developed recently to deal with such nonadiabatic phenomena at a metal surface, methods that serve as alternatives to Tully's independent electron surface hopping (IESH) model. In the weak molecule-metal coupling regime, a classical master equation (CME) can be derived and a simple surface hopping approach is proposed to propagate nuclear and electronic dynamics stochastically. In the strong molecule-metal interaction regime, a Fokker-Planck equation can be derived for the nuclear dynamics, with electronic DoFs incorporated into the overall friction and random force. Lastly, a broadened classical master equation (BCME) can interpolate between the weak and strong molecule-metal interactions. Here, we briefly review these methods and the relevant benchmarking data, showing in particular how the methods can be used to calculate nonequilibrium transport properties. We highlight several open questions and pose several avenues for future study.

Spin-Orbit Interactions in Ultrafast Molecular Processes

We investigate spin-orbit interactions in ultrafast molecular processes employing the exact factorization of the electron-nuclear wave function. We revisit the original derivation by including spin-orbit coupling, and show how the dynamics driven by the time-dependent potential energy surface alleviates inconsistencies arising from different electronic representations. We propose a novel trajectory-based scheme to simulate spin-forbidden non-radiative processes, and we show its performance in the treatment of excited-state dynamics where spin-orbit effects couple different spin multiplets.

Exact Potential Energy Surface for Molecules in Cavities

We find and analyze the exact time-dependent potential energy surface driving the proton motion for a model of cavity-induced suppression of proton-coupled electron transfer. We show how, in contrast to the polaritonic surfaces, its features directly correlate to the proton dynamics and we discuss cavity modifications of its structure responsible for the suppression. The results highlight the interplay between nonadiabatic effects from coupling to photons and coupling to electrons and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces.

Quantum Trajectory Mean-Field Method for Nonadiabatic Dynamics in Photochemistry

The mixed quantum-classical dynamical approaches have been widely used to study nonadiabatic phenomena in photochemistry and photobiology, in which the time evolutions of the electronic and nuclear subsystems are treated based on quantum and classical mechanics, respectively. The key issue is how to deal with coherence and decoherence during the propagation of the two subsystems, which has been the subject of numerous investigations for a few decades. A brief description on Ehrenfest mean-field and surface-hopping (SH) methods is first provided, and then different algorithms for treatment of quantum decoherence are reviewed in the present paper. More attentions were paid to quantum trajectory mean-field (QTMF) method under the picture of quantum measurements, which is able to overcome the overcoherence problem. Furthermore, the combined QTMF and SH algorithm is proposed in the present work, which takes advantages of the QTMF and SH methods. The potential to extend the applicability of the QTMF method was briefly discussed, such as the generalization to other type of nonadiabatic transitions, the combination with multiscale computational models, and possible improvements on its accuracy and efficiency by using machine-learning techniques.

Nonadiabatic Quantum Molecular Dynamics in Dense Manifolds of Electronic States

Most nonadiabatic molecular dynamics methods require the determination of a basis of adiabatic or diabatic electronic states at every time step, but in dense manifolds of electronic states, such approaches become intractable. A notable exception is Ehrenfest molecular dynamics, which can be implemented without explicit determination of such a basis but suffers from unphysical behavior when propagation on a mean-field potential energy surface (PES) does not accurately reflect the true dynamics on multiple electronic states. Here we introduce the multiple cloning for dense manifolds of states (MCDMS) method, a systematically improvable approximation to the multiple cloning method. MCDMS avoids both the mean-field PES problem and the need to compute the full electronic spectrum. This is achieved by reformulating multiple cloning to use a subspace of approximate eigenstates constructed from the time-dependent Ehrenfest electronic wave function. By application to model systems, we show that this approach allows a substantial reduction in the size of the required electronic basis without significant loss in accuracy.

On the numerical solution of the exact factorization equations

The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. Here, we provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a Hamiltonian outside the class of Hermitian Hamiltonians usually encountered in time-propagation, and so the usual numerical methods for time-dependent Schrödinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.

Microscopic calculation of absorption spectra of macromolecules: An analytic approach

We develop a cross-disciplinary approach to analytically compute optical response functions of open macromolecular systems by exploiting the mathematical formalism of quantum field theory (QFT). Indeed, the entries of the density matrix for the electronic excitations interacting with their open dissipative environment are mapped into vacuum-to-vacuum Green's functions in a fictitious relativistic closed quantum system. We show that by re-summing appropriate self-energy diagrams in this dual QFT, it is possible to obtain analytic expressions for the response functions in Mukamel's theory. This yields physical insight into the structure and dynamics of vibronic resonances, since their frequency and width is related to fundamental physical constants and microscopic model parameters. For illustration, we apply this scheme to compute the linear absorption spectrum of the Fenna-Matthews-Olson light harvesting complex, comparing analytic calculations, numerical Monte Carlo simulations, and experimental data.

Improved Ehrenfest Approach to Model Correlated Electron-Nuclear Dynamics

Mixed quantum-classical mechanical descriptions are critical to modeling coupled electron-nuclear dynamics, i.e., nonadiabatic molecular dynamics, relevant to photochemical and photophysical processes. We introduce an efficient description of such dynamics in terms of an effective Hamiltonian that not only properly captures electron-nuclear correlation effects but also helps develop an efficient computational method. In particular, we introduce a coupled Gaussian wavepacket parametrization of the nuclear wave function, which generalizes the Ehrenfest approach to account for electron-nuclei correlations. We test this new approach, Ehrenfest-Plus, on a suite of model problems that probe electron-nuclear correlation in nonadiabatic transitions. The high accuracy of our approach, combined with mixed quantum-classical efficiency, opens a path for improved simulation of nonadiabatic molecular dynamics in realistic molecular systems.

Theoretical modelling of the dynamics of primary photoprocess of cyclopropanone

Photodecomposition of cyclopropanone is investigated by static quantum chemical calculations and non-adiabatic molecular dynamics (NAMD) simulations.

Hot electron relaxation dynamics in semiconductors: assessing the strength of the electron-phonon coupling from the theoretical and experimental viewpoints

The rapid development of the computational methods based on density functional theory, on the one hand, and of time-, energy-, and momentum-resolved spectroscopy, on the other hand, allows today an unprecedently detailed insight into the processes governing hot-electron relaxation dynamics, and, in particular, into the role of the electron-phonon coupling. Instead of focusing on the development of a particular method, theoretical or experimental, this review aims to treat the progress in the understanding of the electron-phonon coupling which can be gained from both, on the basis of recently obtained results. We start by defining several regimes of hot electron relaxation via electron-phonon coupling, with respect to the electron excitation energy. We distinguish between energy and momentum relaxation of hot electrons, and summarize, for several semiconductors of the IV and III-V groups, the orders of magnitude of different relaxation times in different regimes, on the basis of known experimental and numerical data. Momentum relaxation times of hot electrons become very short around 1 eV above the bottom of the conduction band, and such ultrafast relaxation mechanisms are measurable only in the most recent pump-probe experiments. Then, we give an overview of the recent progress in the experimental techniques allowing to obtain detailed information on the hot-electron relaxation dynamics, with the main focus on time-, energy-, and momentum-resolved photoemission experiments. The particularities of the experimental approach developed by one of us, which allows to capture time-, energy-, and momentum-resolved hot-electron distributions, as well as to measure momentum relaxation times of the order of 10 fs, are discussed. We further discuss the main advances in the calculation of the electron-phonon scattering times from first principles over the past ten years, in semiconducting materials. Ab initio techniques and efficient interpolation methods provide the possibility to calculate electron-phonon scattering times with high precision at reasonable numerical cost. We highlight the methods of analysis of the obtained numerical results, which allow to give insight into the details of the electron-phonon scattering mechanisms. Finally, we discuss the concept of hot electron ensemble which has been proposed recently to describe the hot-electron relaxation dynamics in GaAs, the applicability of this concept to other materials, and its limitations. We also mention some open problems.

A walk through the approximations of ab initio multiple spawning

Full multiple spawning offers an in principle exact framework for excited-state dynamics, where nuclear wavefunctions in different electronic states are represented by a set of coupled trajectory basis functions that follow classical trajectories. The couplings between trajectory basis functions can be approximated to treat molecular systems, leading to the ab initio multiple spawning method which has been successfully employed to study the photochemistry and photophysics of several molecules. However, a detailed investigation of its approximations and their consequences is currently missing in the literature. In this work, we simulate the explicit photoexcitation and subsequent excited-state dynamics of a simple system, LiH, and we analyze (i) the effect of the ab initio multiple spawning approximations on different observables and (ii) the convergence of the ab initio multiple spawning results towards numerically exact quantum dynamics upon a progressive relaxation of these approximations. We show that, despite the crude character of the approximations underlying ab initio multiple spawning for this low-dimensional system, the qualitative excited-state dynamics is adequately captured, and affordable corrections can further be applied to ameliorate the coupling between trajectory basis functions.

Surface Hopping Dynamics beyond Nonadiabatic Couplings for Quantum Coherence

Description of correct electron-nuclear couplings is crucial in modeling of nonadiabatic dynamics. Within traditional semiclassical or mixed quantum-classical dynamics, the coupling between quantum electronic states and classical nuclear trajectories is governed by nonadiabatic coupling vectors coupled to classical nuclear momenta. This enables us to develop a very powerful nonadiabatic dynamics algorithm, namely, surface hopping dynamics, which can describe the splitting of nuclear wave packets and detailed balance. Despite its efficiency and practicality, it suffers from the lack of quantum decoherence due to incorrect accounts for the electron-nuclear coupling. Here we present a new surface hopping algorithm based on the exact electron-nuclear correlation from the exact factorization of molecular wave functions. This algorithm demands comparable computational costs to existing surface hopping methods. Numerical simulations with two-state models and a multidimensional multistate realistic molecule show that the electron-nuclear coupling beyond the nonadiabatic coupling terms can describe the quantum coherence properly.

Ab Initio Nonadiabatic Dynamics with Coupled Trajectories: A Rigorous Approach to Quantum (De)Coherence

We report the first nonadiabatic molecular dynamics study based on the exact factorization of the electron-nuclear wave function. Our approach (a coupled-trajectory mixed quantum-classical, CT-MQC, scheme) is based on the quantum-classical limit derived from systematic and controlled approximations to the full quantum-mechanical problem formulated in the exact-factorization framework. Its strength is the ability to correctly capture quantum (de)coherence effects in a trajectory-based approach to excited-state dynamics. We show this by benchmarking CT-MQC dynamics against a revised version of the popular fewest-switches surface-hopping scheme that is able to fix its well-documented overcoherence issue. The CT-MQC approach is successfully applied to investigation of the photochemistry (ring-opening) of oxirane in the gas phase, analyzing in detail the role of decoherence. This work represents a significant step forward in the establishment of the exact factorization as a powerful tool to study excited-state dynamics, not only for interpretation purposes but mainly for nonadiabatic ab initio molecular dynamics simulations.

Partial hydrodynamic representation of quantum molecular dynamics

A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrödinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.

On the Dynamics through a Conical Intersection

Conical intersections represent critical topological features of potential energy surfaces and open ultrafast nonradiative deactivation channels for photoexcited molecules. In the following, we investigate how this funneling picture is transposed in the eyes of the exact factorization formalism for a 2D model system. The exact factorization of the total molecular wave function leads to the fundamental concept of time-dependent potential energy surface and time-dependent vector potential, whose behavior during a dynamics through a conical intersection has up to now remained unexplored. Despite the fact that these quantities might be viewed as time-dependent generalizations of the adiabatic potential energy surfaces and the nonadiabatic coupling vectors, characteristic quantities appearing in the Born-Oppenheimer framework, we observe that they do not exhibit particular topological features in the region of conical intersection but still reflect the complex dynamics of the nuclear wavepacket.

Electronic non-adiabatic dynamics in enhanced ionization of isotopologues of hydrogen molecular ions from the exact factorization perspective

An exact-factorization perspective of enhanced ionization in isotopologues of H₂⁺ demonstrates the concept of the exact potential driving the electrons in non-adiabatic motion of molecules in strong fields, and sets a new platform for introducing various approximations.

Non-Hermitian surface hopping

We present a generalized non-Hermitian equation of motion (nH-EOM) to go beyond standard trajectory surface hopping dynamics. The derivation is based on the Born-Huang expansion of the total wave function and the polar representation of the nuclear factor. The nH-EOM contains two additional terms, a skew symmetry term iΓ with dissipation operator Γ to account for decoherence, and a kinetic-energy renormalization term to account for phase shifts, without destroying the invariance to the choice of representation. Numerically, the nH-EOM can still be solved efficiently using a semiclassical approximation in the framework of Tully's fewest-switches surface hopping (FSSH) algorithm. Applications to model Hamiltonians demonstrate improved performance over the standard FSSH approach, through comparison to exact quantum results.

Drift of charge carriers in crystalline organic semiconductors

We investigate the direct-current response of crystalline organic semiconductors in the presence of finite external electric fields by the quantum-classical Ehrenfest dynamics complemented with instantaneous decoherence corrections (IDC). The IDC is carried out in the real-space representation with the energy-dependent reweighing factors to account for both intermolecular decoherence and energy relaxation by which conduction occurs. In this way, both the diffusion and drift motion of charge carriers are described in a unified framework. Based on an off-diagonal electron-phonon coupling model for pentacene, we find that the drift velocity initially increases with the electric field and then decreases at higher fields due to the Wannier-Stark localization, and a negative electric-field dependence of mobility is observed. The Einstein relation, which is a manifestation of the fluctuation-dissipation theorem, is found to be restored in electric fields up to ∼10(5) V/cm for a wide temperature region studied. Furthermore, we show that the incorporated decoherence and energy relaxation could explain the large discrepancy between the mobilities calculated by the Ehrenfest dynamics and the full quantum methods, which proves the effectiveness of our approach to take back these missing processes.

Coupled wave-packets for non-adiabatic molecular dynamics: a generalization of Gaussian wave-packet dynamics to multiple potential energy surfaces

Coupled wave-packets for non-adiabatic dynamics is a new method for simulation of molecular dynamics on coupled potential energy surfaces, which efficiency and correctly accounts for decoherence and interferences effects.

Accurate simulation of the non-adiabatic dynamics of molecules in excited electronic states is key to understanding molecular photo-physical processes. Here we present a novel method, based on a semi-classical approximation, that is as efficient as the commonly used mean field Ehrenfest or ad hoc surface hopping methods and properly accounts for interference and decoherence effects. This novel method is an extension of Heller's thawed Gaussian wave-packet dynamics that includes coupling between potential energy surfaces. By studying several standard test problems we demonstrate that the accuracy of the method can be systematically improved while maintaining high efficiency. The method is suitable for investigating the role of quantum coherence in the non-adiabatic dynamics of many-atom molecules.

Quantum-Classical Nonadiabatic Dynamics: Coupled- vs Independent-Trajectory Methods

Trajectory-based mixed quantum-classical approaches to coupled electron-nuclear dynamics suffer from well-studied problems such as the lack of (or incorrect account for) decoherence in the trajectory surface hopping method and the inability of reproducing the spatial splitting of a nuclear wave packet in Ehrenfest-like dynamics. In the context of electronic nonadiabatic processes, these problems can result in wrong predictions for quantum populations and in unphysical outcomes for the nuclear dynamics. In this paper, we propose a solution to these issues by approximating the coupled electronic and nuclear equations within the framework of the exact factorization of the electron-nuclear wave function. We present a simple quantum-classical scheme based on coupled classical trajectories and test it against the full quantum mechanical solution from wave packet dynamics for some model situations which represent particularly challenging problems for the above-mentioned traditional methods.

Quantum Mechanical Wave Packet Dynamics at a Conical Intersection with Strong Vibrational Dissipation

We derive a reduced model for the nonadiabatic quantum dynamics of an electronic wave packet moving through a conical intersection in the presence of strong vibrational damping. Starting from the dissipative two-state two-model model, we transform the tuning and the coupling mode to the bath. The resulting quantum two-state model with two highly structured environments is solved numerically exactly in the regime of strong vibrational damping. We find negative cross peaks in the ultrafast optical 2D spectra as clear signatures of the conical intersection. They arise from secondary excitations of the wave packet after having passed through the photophysical energy funnel. This feature is in agreement with recent transient absorption measurements of rhodopsin.

Exact Potential Driving the Electron Dynamics in Enhanced Ionization of H(2)(+)

It was recently shown that the exact factorization of the electron-nuclear wave function allows the construction of a Schrödinger equation for the electronic system, in which the potential contains exactly the effect of coupling to the nuclear degrees of freedom and any external fields. Here we study the exact potential acting on the electron in charge-resonance enhanced ionization in a model one-dimensional H(2)(+) molecule. We show there can be significant differences between the exact potential and that used in the traditional quasistatic analyses, arising from nonadiabatic coupling to the nuclear system, and that these are crucial to include for accurate simulations of time-resolved ionization dynamics and predictions of the ionization yield.

Vibronic and Coherent Effects on Interfacial Electron Transfer Dynamics

This Letter examines fundamental issues for electron transfer (ET) dynamics, such as adiabatic versus nonadiabatic effects during interfacial ET, the influence of vibrational degrees of freedom on the electronic dynamics, the occurrence of electronic coherences and the ensuing dephasing effects. The interplay of these mechanisms during the ultrafast ET is discussed. A theoretical method for the quantum dynamics of electrons in flexible molecular systems is used to study such issues on the interfacial ET from the perylene chromophore to the TiO2 semiconductor surface. By analyzing the Fourier transform of the survival probability curves, it is possible to discern the oscillating features that are caused by electronic coherences and vibronic effects. The vibronic degrees of freedom are treated within the atomistic level of description and their effects identified on the charge transfer dynamics. The insights revealed are general and thus can be useful for the analysis of other ET phenomena.

Laser-induced electron localization in H2+: mixed quantum-classical dynamics based on the exact time-dependent potential energy surface

The exact nuclear time-dependent potential energy surface for laser-induced electron localization is studied with a view to developing a mixed quantum-classical dynamics method for strong-field processes.