Mixed Quantum-classical Methods Research Articles

Nonadiabatic dynamics of molecules interacting with metal surfaces: A quantum-classical approach based on Langevin dynamics and the hierarchical equations of motion.

A novel mixed quantum-classical approach to simulating nonadiabatic dynamics of molecules at metal surfaces is presented. The method combines the numerically exact hierarchical equations of motion approach for the quantum electronic degrees of freedom with Langevin dynamics for the classical degrees of freedom, namely, low-frequency vibrational modes within the molecule. The approach extends previous mixed quantum-classical methods based on Langevin equations to models containing strong electron-electron or quantum electronic-vibrational interactions, while maintaining a nonperturbative and non-Markovian treatment of the molecule-metal coupling. To demonstrate the approach, nonequilibrium transport observables are calculated for a molecular nanojunction containing strong interactions.

Simulating Attochemistry: Which Dynamics Method to Use?

Attochemistry aims to exploit the properties of coherent electronic wavepackets excited via attosecond pulses to control the formation of photoproducts. Such molecular processes can, in principle, be simulated with various nonadiabatic dynamics methods, yet the impact of the approximations underlying the methods is rarely assessed. The performances of widely used mixed quantum-classical approaches, Tully surface hopping, and classical Ehrenfest methods are evaluated against the high-accuracy DD-vMCG quantum dynamics. This comparison is conducted for the valence ionization of fluorobenzene. Analyzing the nuclear motion induced in the branching space of the nearby conical intersection, the results show that the mixed quantum-classical methods reproduce quantitatively the average motion of a quantum wavepacket when initiated on a single electronic state. However, they fail to properly capture the nuclear motion induced by an electronic wavepacket along the derivative coupling, the latter originating from the quantum electronic coherence property, key to attochemistry.

Resonance Enhancement of Vibrational Polariton Chemistry Obtained from the Mixed Quantum-Classical Dynamics Simulations.

We applied a variety of mixed quantum-classical (MQC) approaches to simulate the VSC-influenced reaction rate constant. All of these MQC simulations treat the key vibrational levels associated with the reaction coordinate in the quantum subsystem (as quantum states), whereas all other degrees of freedom (DOFs) are treated inside the classical subsystem. We find that, as long as we have the quantum state descriptions for the vibrational DOFs, one can correctly describe the VSC resonance condition when the cavity frequency matches the bond vibrational frequency. This correct resonance behavior can be obtained regardless of the detailed MQC methods that one uses. The results suggest that the MQC approaches can generate semiquantitative agreement with the exact results for rate constant changes when changing the cavity frequency, the light-matter coupling strength, or the cavity lifetime. The finding of this work suggests that one can use computationally economic MQC approaches to explore the collective coupling scenario when many molecules are collectively coupled to many cavity modes in the future.

Assessing Mixed Quantum-Classical Molecular Dynamics Methods for Nonadiabatic Dynamics of Molecules on Metal Surfaces.

Mixed quantum-classical (MQC) methods for simulating the dynamics of molecules at metal surfaces have the potential to accurately and efficiently provide mechanistic insight into reactive processes. Here, we introduce simple two-dimensional models for the scattering of diatomic molecules at metal surfaces based on recently published electronic structure data. We apply several MQC methods to investigate their ability to capture how nonadiabatic effects influence molecule-metal energy transfer during the scattering process. Specifically, we compare molecular dynamics with electronic friction, Ehrenfest dynamics, independent electron surface hopping, and the broadened classical master equation approach. In the case of independent electron surface hopping, we implement a simple decoherence correction approach and assess its impact on vibrationally inelastic scattering. Our results show that simple, low-dimensional models can be used to qualitatively capture experimentally observed vibrational energy transfer and provide insight into the relative performance of different MQC schemes. We observe that all approaches predict similar kinetic energy dependence but return different vibrational energy distributions. Finally, by varying the molecule-metal coupling, we can assess the coupling regime in which some MQC methods become unsuitable.

Different flavors of exact-factorization-based mixed quantum-classical methods for multistate dynamics.

The exact factorization approach has led to the development of new mixed quantum-classical methods for simulating coupled electron-ion dynamics. We compare their performance for dynamics when more than two electronic states are occupied at a given time, and analyze: (1) the use of coupled versus auxiliary trajectories in evaluating the electron-nuclear correlation terms, (2) the approximation of using these terms within surface-hopping and Ehrenfest frameworks, and (3) the relevance of the exact conditions of zero population transfer away from nonadiabatic coupling regions and total energy conservation. Dynamics through the three-state conical intersection in the uracil radical cation as well as polaritonic models in one dimension are studied.

A unified framework of mixed quantum-classical dynamics with trajectory branching.

As popular mixed quantum-classical dynamics methods, trajectory surface hopping and Ehrenfest mean field have been widely utilized to simulate nonadiabatic dynamics. Recently, we have proposed the branching-corrected surface hopping and the branching-corrected mean field methods, both of which closely reproduce the exact quantum dynamics in a series of standard scattering models. Here, the mixed surface hopping and mean field with branching correction (BCSHMF) is presented as a unified framework of mixed quantum-classical dynamics. As benchmarked in thousands of diverse three-level and four-level scattering models, BCSHMF achieves high reliability and flexibility, implying that surface hopping and mean field are compatible with each other in nature, and trajectory branching is essential for the mixed quantum-classical description of nonadiabatic dynamics.

Modeling Spin-Dependent Nonadiabatic Dynamics with Electronic Degeneracy: A Phase-Space Surface-Hopping Method.

Nuclear Berry curvature effects emerge from electronic spin degeneracy and can lead to nontrivial spin-dependent (nonadiabatic) nuclear dynamics. However, such effects are not captured fully by any current mixed quantum-classical method such as fewest-switches surface hopping. In this work, we present a phase-space surface-hopping (PSSH) approach to simulate singlet-triplet intersystem crossing dynamics. We show that with a simple pseudodiabatic ansatz, a PSSH algorithm can capture the relevant Berry curvature effects and make predictions in agreement with exact quantum dynamics for a simple singlet-triplet model Hamiltonian. Thus, this approach represents an important step toward simulating photochemical and spin processes concomitantly, as relevant to intersystem crossing and spin-lattice relaxation dynamics.

Nonadiabatic Dynamics Algorithms with Only Potential Energies and Gradients: Curvature-Driven Coherent Switching with Decay of Mixing and Curvature-Driven Trajectory Surface Hopping.

Direct dynamics by mixed quantum-classical nonadiabatic methods is an important tool for understanding processes involving multiple electronic states. Very often, the computational bottleneck of such direct simulation comes from electronic structure theory. For example, at every time step of a trajectory, nonadiabatic dynamics requires potential energy surfaces, their gradients, and the matrix elements coupling the surfaces. The need for the couplings can be alleviated by employing the time derivatives of the wave functions, which can be evaluated from overlaps of electronic wave functions at successive time steps. However, evaluation of overlap integrals is still expensive for large systems. In addition, for electronic structure methods for which the wave functions or the coupling matrix elements are not available, nonadiabatic dynamics algorithms become inapplicable. In this work, building on recent work by Baeck and An, we propose new nonadiabatic dynamics algorithms that only require adiabatic potential energies and their gradients. The new methods are named curvature-driven coherent switching with decay of mixing (CSDM) and curvature-driven trajectory surface hopping (TSH). We show how powerful these new methods are in terms of computation time and accuracy as compared to previous mixed quantum-classical nonadiabatic dynamics algorithms. The lowering of the computational cost will allow longer nonadiabatic trajectories and greater ensemble averaging to be affordable, and the ability to calculate the dynamics without electronic structure coupling matrix elements extends the dynamics capability to new classes of electronic structure methods.

An accurate, non-empirical method for incorporating decoherence into Ehrenfest dynamics.

In mixed quantum-classical nonadiabatic molecular dynamics methods, the anchoring of the electronic wave function to a single nuclear geometry results in both quantitative and qualitative errors in the dynamics. In the context of both Ehrenfest and trajectory surface hopping methods, methods for incorporating decoherence are widely used to eliminate these errors. However, the accuracy of these methods often depends strongly on the parameterization of the decoherence time and/or other related quantities. Here, we present a refinement of the recently introduced collapse to a block (TAB) scheme for incorporating decoherence into Ehrenfest dynamics. The proposed approach incorporates an approximation to the history of the population dynamics and treats the coherence decay as Gaussian, rather than exponential. This method uses parameters that can be obtained from first principles, rather than empirical fitting. Application to one-dimensional models indicates excellent agreement with numerically exact simulations. We also introduce a second refinement to the TAB method: a robust linear least-squares algorithm for determining collapse probabilities.

A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation.

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. 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. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

Case studies of the time-dependent potential energy surface for dynamics in cavities.

The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons in cavity settings. Here, we provide a detailed analysis of its structure for exactly solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron-transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately, while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.

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.

Simulation of Excitation by Sunlight in Mixed Quantum-Classical Dynamics.

This paper proposes a method to simulate nonadiabatic dynamics initiated by thermal light, including solar radiation, in the frame of mixed quantum-classical (MQC) methods, like surface hopping. The method is based on the Chenu–Brumer approach, which treats the thermal radiation as an ensemble of coherent pulses. It is composed of three steps: (1) sampling initial conditions from a broad blackbody spectrum, (2) dynamics propagation using conventional methods, and (3) ensemble averaging considering the field and realization time of the pulses. The application of MQC dynamics with pulse ensembles (MQC-PE) to a model system of nucleic acid photophysics showed the emergence of a steady excited-state population. In another test case, modeling retinal photophysics, MQC-PE predicted that although the photoisomerization occurs within 200 fs, it may take tens of microseconds of continuous solar irradiation to photoactivate a single retinal. Such emergent long timescales may impact our understanding of biological and technological phenomena occurring under solar radiation.

Performance of Mixed Quantum-Classical Approaches on Modeling the Crossover from Hopping to Bandlike Charge Transport in Organic Semiconductors.

In the present study, several mixed quantum-classical (MQC) methods are applied to on-the-fly nonadiabatic molecular dynamics simulations of hole transport in molecular organic semiconductors (OSCs). The tested MQC methods contain the mean-field Ehrenfest (MFE), trajectory surface hopping (TSH) approaches based on Tully's fewest switches surface hopping (FSSH) and the global flux surface hopping (GFSH), the latter in the diabatic/adiabatic representation, and a Landau-Zener type trajectory surface hopping (LZSH). We also tested several correction schemes which were proposed to identify trivial crossings and to remove unphysical long-range charge transfers due to decoherence corrections. In addition, several cost-effective approaches for the nuclear velocity adjustment after an energy-allowed/energy-forbidden hop are investigated with respect to detailed balance and internal consistency conditions. To model a broad spectrum of OSCs with different charge transport characteristics, we derived from the anthracene structural model the construction of two additional models by uniformly scaling down the electronic couplings by the factors of 0.1 and 0.5. Anthracene shows a bandlike charge transport mechanism, characterized by slightly delocalized charge carriers 'diffusing' through the crystal. For smaller couplings, the mechanism changes to a hopping type, characteristically differing in the charge delocalization and temperature dependence. The MFE and corrected adiabatic TSH approaches are able to quantitatively reproduce the expected behavior, while the diabatic LZSH method fails for large couplings, as do approaches which are based on the hopping of localized charge between neighboring sites. Moreover, we find that while the hole mobility of the anthracene crystal simulated using the celebrated Marcus theory is in good agreement with the experimental value, its agreement has to be regarded as an accident due to the overestimation of the prefactor in the Marcus rate equation.

Nonequilibrium heat transport in a molecular junction: A mixed quantum-classical approach.

In a recent study [J. Liu et al., J. Chem. Phys. 149, 224104 (2018)], we developed a general mixed quantum-classical framework for studying heat transport through molecular junctions, in which the junction molecule is treated quantum mechanically and the thermal reservoirs to which the molecule is coupled are treated classically. This framework yields expressions for the transferred heat and steady-state heat current, which could be calculated using a variety of mixed quantum-classical dynamics methods. In this work, we use the recently developed "Deterministic Evolution of Coordinates with Initial Decoupled Equations" (DECIDE) method for calculating the steady-state heat current in the nonequilibrium spin-boson model in a variety of parameter regimes. Our results are compared and contrasted with those obtained using the numerically exact multilayer multiconfiguration time-dependent Hartree approach, and using approximate methods, including mean field theory, Redfield theory, and adiabatic mixed quantum-classical dynamics. Despite some quantitative differences, the DECIDE method performs quite well, is capable of capturing the expected trends in the steady-state heat current, and, overall, outperforms the approximate methods. These results hold promise for DECIDE simulations of nonequilibrium heat transport in realistic models of nanoscale systems.

Nonadiabatic dynamics simulation method for transition metal complexes: Theory, developments, and applications

Excited-state related properties of transition metal complexes have received much attention in recent years due to their potential applications in the field of energy, material and biology. To fully understand the underlying mechanism, we always need to resort to ultrafast time-resolved spectroscopies as well as reliable ab initio nonadiabatic dynamics simulations. However, nonadiabatic dynamics simulation methods that are suitable for investigating excited-state dynamics of transition metal complexes still demand further developments to improve simulation efficiency and accuracy. In this short review, we have shortly present some recent developments of non-adiabatic dynamics methods that are suitable for investigating excited state properties of transition metal complexes, mainly including Multi-Configuration Time Dependent Hartree (MCTDH) and mixed quantum-classical surface hopping methods. In addition, we have introduced our recently developed TD-DFT-based generalized trajectory surface hopping method in which an efficient algorithm of computing nonadiabatic couplings is realized. In combination with the classical path approximation, this newly developed method can effectively simulate internal conversion and intersystem crossing processes of transition metal complexes on an equal footing. With this newly developed method, we have systematically investigated the early-time ultrafast excited-state dynamics of three Ir(III) complexes bearing distinct ligands. The ISC rates estimated by our simulations are 65, 81 and 140 fs, which agree quite well with experimentally measured ca. 80, 80 and 110 fs. These Ir(III) complexes show similar macroscopic phenomena, i.e., ultrafast IC and ISC, their microscopic excited-state relaxation dynamics are individually different, for example, electron and hole transfer dynamics. These new insights for excited-state dynamics of Ir(III) complexes could be helpful for rationally designing Ir-containing compounds with excellent photoluminescence performance. In addition, we have also investigated the ultrafast intersystem crossing process of Au(I) naphthalene derivatives. We have found that (1) ultrafast and sub-picosecond intersystem crossing processes are mainly caused by small energy gaps and large spin-orbit couplings between S1 and T n . (2) Adding the second gold(I)-phosphine group does not increase spin-orbit couplings between S1 and T n but decrease their values remarkably, which implies that heavy-atom effects are state-specific, not-universal. (3) The position at which the second gold(I)-phosphine group is attached has remarkable influence on electronic structures of S1 and T n and their relative energies, which affect energy gaps and spin-orbit couplings between S1 and T n and eventually modulate intersystem crossing rates from S1 to T n . This work exemplifies that different isomers of a compound could have distinct excited-state relaxation dynamics. Above simulation results not only agree pretty well with experiments, but also provide valuable insights into excited-state relaxation dynamics for these transition-metal-containing complexes. In one word, our present developments have established an efficient and accurate simulation tool for investigating excited-state dynamics of transition metal complexes.

Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics.

Nonadiabatic mixed quantum-classical (NA-MQC) dynamics methods form a class of computational theoretical approaches in quantum chemistry tailored to investigate the time evolution of nonadiabatic phenomena in molecules and supramolecular assemblies. NA-MQC is characterized by a partition of the molecular system into two subsystems: one to be treated quantum mechanically (usually but not restricted to electrons) and another to be dealt with classically (nuclei). The two subsystems are connected through nonadiabatic couplings terms to enforce self-consistency. A local approximation underlies the classical subsystem, implying that direct dynamics can be simulated, without needing precomputed potential energy surfaces. The NA-MQC split allows reducing computational costs, enabling the treatment of realistic molecular systems in diverse fields. Starting from the three most well-established methods-mean-field Ehrenfest, trajectory surface hopping, and multiple spawning-this review focuses on the NA-MQC dynamics methods and programs developed in the last 10 years. It stresses the relations between approaches and their domains of application. The electronic structure methods most commonly used together with NA-MQC dynamics are reviewed as well. The accuracy and precision of NA-MQC simulations are critically discussed, and general guidelines to choose an adequate method for each application are delivered.

Theories and Applications of Mixed Quantum-Classical Non-adiabatic Dynamics

Electronically non-adiabatic processes are essential parts of photochemical process, collisions of excited species, electron transfer processes, and quantum information processing. Various non-adiabatic dynamics methods and their numerical implementation have been developed in the last decades. This review summarizes the most significant development of mixed quantum-classical methods and their applications which mainly include the Liouville equation, Ehrenfest mean-field, trajectory surface hopping, and multiple spawning methods. The recently developed quantum trajectory mean-field method that accounts for the decoherence corrections in a parameter-free fashion is discussed in more detail.

Vibrational Energy Relaxation: A Benchmark for Mixed Quantum-Classical Methods.

We investigate the ability of mixed quantum-classical methods to capture the dynamics of vibrational energy relaxation. Several methods, including surface hopping, and Ehrenfest and symmetrical quasiclassical (SQC) dynamics, are benchmarked for the exactly solvable model problem of a harmonic oscillator bilinearly coupled to a bath of harmonic oscillators. Results show that, very often, one can recover accurate vibrational relaxation rates and detailed balance using simple mixed quantum-classical approaches. A few anomalous results do appear, however, especially regarding Ehrenfest and SQC dynamics.

Path-integral isomorphic Hamiltonian for including nuclear quantum effects in non-adiabatic dynamics.

We describe a path-integral approach for including nuclear quantum effects in non-adiabatic chemical dynamics simulations. For a general physical system with multiple electronic energy levels, a corresponding isomorphic Hamiltonian is introduced such that Boltzmann sampling of the isomorphic Hamiltonian with classical nuclear degrees of freedom yields the exact quantum Boltzmann distribution for the original physical system. In the limit of a single electronic energy level, the isomorphic Hamiltonian reduces to the familiar cases of either ring polymer molecular dynamics (RPMD) or centroid molecular dynamics Hamiltonians, depending on the implementation. An advantage of the isomorphic Hamiltonian is that it can easily be combined with existing mixed quantum-classical dynamics methods, such as surface hopping or Ehrenfest dynamics, to enable the simulation of electronically non-adiabatic processes with nuclear quantum effects. We present numerical applications of the isomorphic Hamiltonian to model two- and three-level systems, with encouraging results that include improvement upon a previously reported combination of RPMD with surface hopping in the deep-tunneling regime.