Subspace Surface Hopping with Size-Independent Dynamics

Multiconfigurational Surface Hopping: a Time-Dependent Variational Approach with Momentum-Jump Trajectories

The Ehrenfest mean field dynamics and trajectory surface hopping have been widely used in nonadiabatic dynamics simulations. Based on the time-dependent variational principle (TDVP), the multiconfigurational Ehrenfest (MCE) method has also been developed and can be regarded as a multiconfigurational extension of the traditional Ehrenfest dynamics. However, it is not straightforward to apply the TDVP to surface hopping trajectories because there exists momentum jump during surface hops. To solve this problem, we here propose a multiconfigurational surface hopping (MCSH) method, where continuous momenta are obtained by linear interpolation and the interpolated trajectories are used to construct the basis functions for TDVP in a postprocessing manner. As demonstrated in a series of representative spin-boson models, MCSH achieves high accuracy with only several hundred trajectory bases and can uniformly improve the performance of surface hopping. In principle, MCSH can be combined with all kinds of mixed quantum-classical trajectories and thus has the potential to properly describe general nonadiabatic dynamics.

Consistent Construction of the Density Matrix from Surface Hopping Trajectories

Proper construction of the density matrix based on surface hopping trajectories remains a difficult problem. Due to the well-known overcoherence in traditional surface hopping simulations, the electronic wave function cannot be used directly. In this work, we propose a consistent density matrix construction method, which takes the advantage of occupation of active states to rescale the coherence calculated by wave functions and ensures the intrinsic consistency of the density matrix. This new trajectory analysis method can be used for both Tully's fewest switches surface hopping (FSSH) and our recently proposed branching corrected surface hopping (BCSH). As benchmarked in both one- and two-dimensional standard scattering models, the new approach combined with BCSH trajectories achieves highly accurate time-dependent spatial distributions of adiabatic populations and coherence compared to exact quantum results.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born-Huang formalism, covering different initial states, which either start on a single Born-Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

Surface hopping dynamics in periodic solid-state materials with a linear vibronic coupling model

We report a surface hopping approach in which the implemented linear vibronic coupling Hamiltonian is constructed and the electronic wavefunction is propagated in the reciprocal space. The parameters of the linear vibronic coupling model, including onsite energies, phonon frequencies, and electron-phonon couplings, are calculated with density-functional theory and density-functional perturbation theory and interpolated in fine sampling points of the Brillouin zone with maximally localized Wannier functions. Using this approach, we studied the relaxation dynamics of the photo-excited hot carrier in a one-dimensional periodic carbon chain. The results show that the completeness of the number of Hilbert space k points and the number of phonon q points plays an important role in the hot carrier relaxation processes. By calculating the relaxation times of hot carriers under different reciprocal space sampling and extrapolating with the stretched-compressed exponential function, the relaxation times of hot electrons and holes in the quasi-continuous energy band are obtained. By considering the feedback effect in the hopping processes and analyzing the time-dependent phonon energy in different normal modes, we found that the long-wave longitudinal optical phonons play a major role in the relaxation dynamics of hot electrons and holes. We, therefore, provided herein an efficient and accurate approach for modeling the photophysical processes in periodic solid-state material systems.

Multilayer Subsystem Surface Hopping Method for Large-Scale Nonadiabatic Dynamics Simulation with Hundreds of Thousands of States

We present a multilayer subsystem surface hopping (MSSH) method to deal with nonadiabatic dynamics in large-scale systems. A small subsystem instead of the full system is adopted for surface hopping and is updated on-the-fly to achieve a reliable description of important adiabatic states and the wave function evolution. Additional subsystems for molecular dynamics and statistical description are introduced to further improve the simulation reliability. The global flux hopping probabilities with optimal state assignments are utilized to treat the complex surface crossings. As demonstrated in a series of one- and two-dimensional Holstein models with up to hundreds of thousands of states, MSSH shows weak parameter dependence in all investigated systems. Especially, the computational costs are reduced by 2-6 orders of magnitude compared to traditional surface hopping simulations in full systems, and size-independent results are achieved with a large time-step size of 2-5 fs. The new method is compatible with different decoherence correction strategies and achieves a much better balance between efficiency and reliability, thus promising for applications in general charge and exciton dynamics simulations.

Thermally activated intra-chain charge transport in high charge-carrier mobility copolymers

Disordered or even seemingly amorphous, donor–acceptor type, conjugated copolymers with high charge-carrier mobility have emerged as a new class of functional materials, where transport along the conjugated backbone is key. Here, we report on non-adiabatic molecular dynamics simulations of charge-carrier transport along chains of poly (indacenodithiophene-co-benzothiadiazole), within a model Hamiltonian parameterized against first-principles calculations. We predict thermally activated charge transport associated with a slightly twisted ground-state conformation, on par with experimental results. Our results also demonstrate that the energy mismatch between the hole on the donor vs the acceptor units of the copolymer drives localization of the charge carriers and limits the intra-chain charge-carrier mobility. We predict that room-temperature mobility values in excess of 10 cm2 V−1 s−1 can be achieved through proper chemical tuning of the component monomer units.

All-Atom Nonadiabatic Dynamics Simulation of Hybrid Graphene Nanoribbons Based on Wannier Analysis and Machine Learning

Trajectory surface hopping combined with ab initio electronic structure calculations is a popular and powerful approach for on-the-fly nonadiabatic dynamics simulations. For large systems, however, this remains a significant challenge because of the unaffordable computational cost of large-scale electronic structure calculations. Here, we present an efficient divide-and-conquer approach to construct the system Hamiltonian based on Wannier analysis and machine learning. In detail, the large system under investigation is first decomposed into small building blocks, and then all possible segments formed by building blocks within a cutoff distance are found out. Ab initio molecular dynamics is carried out to generate a sequence of geometries for each equivalent segment with periodicity. The Hamiltonian matrices in the maximum localized Wannier function (MLWF) basis are obtained for all geometries and utilized to train artificial neural networks (ANNs) for the structure-dependent Hamiltonian elements. Taking advantage of the orthogonality and spatial locality of MLWFs, the one-electron Hamiltonian of a large system at arbitrary geometry can be directly constructed by the trained ANNs. As demonstrations, we study charge transport in a zigzag graphene nanoribbon (GNR), a coved GNR, and a series of hybrid GNRs with a state-of-the-art surface hopping method. The interplay between delocalized and localized states is found to determine the electron dynamics in hybrid GNRs. Our approach has successfully studied GNRs with >10 000 atoms, paving the way for efficient and reliable all-atom nonadiabatic dynamics simulation of general systems.

Machine Learning Approach to Calculate Electronic Couplings between Quasi-diabatic Molecular Orbitals: The Case of DNA

Diabatization of one-electron states in flexible molecular aggregates is a great challenge due to the presence of surface crossings between molecular orbital (MO) levels and the complex interaction between MOs of neighboring molecules. In this work, we present an efficient machine learning approach to calculate electronic couplings between quasi-diabatic MOs without the need of nonadiabatic coupling calculations. Using MOs of rigid molecules as references, the MOs that can be directly regarded to be quasi-diabatic in molecular dynamics are selected out, state tracked, and phase corrected. On the basis of this information, artificial neural networks are trained to characterize the structure-dependent onsite energies of quasi-diabatic MOs and the intermolecular electronic couplings. A representative sequence of DNA is systematically studied as an illustration. Smooth time evolution of electronic couplings in all base pairs is obtained with quasi-diabatic MOs. In particular, our method can calculate electronic couplings between different quasi-diabatic MOs independently, and thus, this possesses unique advantages in many applications.

Long-Range Interactions Boost Singlet Exciton Diffusion in Nanofibers of π-Extended Polymer Chains

Raising the distance covered by singlet excitons during their lifetimes to values maximizing light absorption (a few hundred nm) would solve the exciton diffusion bottleneck issue and lift the constraint for fine (∼10 nm) phase segregation in bulk heterojunction organic solar cells. In that context, the recent report of highly ordered conjugated polymer nanofibers featuring singlet exciton diffusion length, L_D, in excess of 300 nm is both appealing and intriguing [Jin, X.; et al. Science 2018, 360 (6391), 897-900]. Here, on the basis of nonadiabatic molecular dynamics simulations, we demonstrate that singlet exciton diffusion in poly(3-hexylthiophene) (P3HT) fibers is highly sensitive to the interplay between delocalization along the polymer chains and long-range interactions along the stacks. Remarkably, the diffusion coefficient is predicted to rocket by 3 orders of magnitude when going beyond nearest-neighbor intermolecular interactions in fibers of extended (30-mer) polymer chains and to be resilient to interchain energetic and positional disorders.

Efficient energy transport in an organic semiconductor mediated by transient exciton delocalization

Efficient energy transport is desirable in organic semiconductor (OSC) devices. However, photogenerated excitons in OSC films mostly occupy highly localized states, limiting exciton diffusion coefficients to below ~10^-2 cm²/s and diffusion lengths below ~50 nm. We use ultrafast optical microscopy and nonadiabatic molecular dynamics simulations to study well-ordered poly(3-hexylthiophene) nanofiber films prepared using living crystallization-driven self-assembly, and reveal a highly efficient energy transport regime: transient exciton delocalization, where energy exchange with vibrational modes allows excitons to temporarily re-access spatially extended states under equilibrium conditions. We show that this enables exciton diffusion constants up to 1.1 ± 0.1 cm²/s and diffusion lengths of 300 ± 50 nm. Our results reveal the dynamic interplay between localized and delocalized exciton configurations at equilibrium conditions, calling for a re-evaluation of exciton dynamics and suggesting design rules to engineer efficient energy transport in OSC device architectures not based on restrictive bulk heterojunctions.

Branching and phase corrected surface hopping: A benchmark of nonadiabatic dynamics in multilevel systems

Since the seminal work of Tully [J. Chem. Phys. 93, 1061 (1990)], two-level scattering models have been extensively adopted as the standard benchmark systems to assess the performance of different trajectory surface hopping methods for nonadiabatic dynamics simulations. Here, we extend the branching and phase corrections to multilevel systems and combine them with both the traditional fewest switches surface hopping (FSSH) and its variant global flux surface hopping (GFSH) algorithms. To get a comprehensive evaluation of the proposed methods, we construct a series of more challenging and diverse three-level and four-level scattering models and use exact quantum solutions as references. Encouragingly, both FSSH and GFSH with the branching and phase corrections produce excellent and nearly identical results in all investigated systems, indicating that the new surface hopping methods are robust to describe multilevel problems and the reliability is insensitive to the definition of self-consistent hopping probabilities in the adiabatic representation. Furthermore, the branching correction is found to be especially important when dealing with strongly repulsive potential energy surfaces, which are common in realistic systems, thus promising for general applications.

A Simple Solution to Trivial Crossings: A Stochastic State Tracking Approach

We present a new state tracking algorithm based on a stochastic state reassignment that reflects the quantum mechanical interpretation of the state time-overlaps. We assess the new method with a range of model Hamiltonians and demonstrate that it yields the results generally consistent with the deterministic min-cost algorithm. However, the stochastic state tracking algorithm reduces magnitudes of the state population fluctuations as the quantum system evolves toward its equilibrium. The new algorithm facilitates the thermalization of quantum state populations and suppresses the population revivals and oscillations near the equilibrium in many-state systems. The new stochastic algorithm has a favorable computational scaling, is easy to implement due to its conceptual transparency, and treats various types of state identity changes (trivial or avoided crossings and any intermediate cases) on equal footing.

Branching Corrected Mean Field Method for Nonadiabatic Dynamics

When nonadiabatic dynamics are described on the basis of trajectories, severe trajectory branching occurs when the nuclear wave packets on some potential energy surfaces are reflected while those on the remaining surfaces are not. As a result, the traditional Ehrenfest mean field (EMF) approximation breaks down. In this study, two versions of the branching corrected mean field (BCMF) method are proposed. Namely, when trajectory branching is identified, BCMF stochastically selects either the reflected or the nonreflected group to build the new mean field trajectory or splits the mean field trajectory into two new trajectories with the corresponding weights. As benchmarked in six standard model systems and an extensive model base with two hundred diverse scattering models, BCMF significantly improves the accuracy while retaining the high efficiency of the traditional EMF. In fact, BCMF closely reproduces the exact quantum dynamics in all investigated systems, thus highlighting the essential role of branching correction in nonadiabatic dynamics simulations of general systems.

Applying Marcus theory to describe the carrier transports in organic semiconductors: Limitations and beyond

Marcus theory has been successfully applied to molecular design for organic semiconductors with the aid of quantum chemistry calculations for the molecular parameters: the intermolecular electronic coupling V and the intramolecular charge reorganization energy λ. The assumption behind this is the localized nature of the electronic state for representing the charge carriers, being holes or electrons. As far as the quantitative description of carrier mobility is concerned, the direct application of Marcus semiclassical theory usually led to underestimation of the experimental data. A number of effects going beyond such a semiclassical description will be introduced here, including the quantum nuclear effect, dynamic disorder, and delocalization effects. The recently developed quantum dynamics simulation at the time-dependent density matrix renormalization group theory is briefly discussed. The latter was shown to be a quickly emerging efficient quantum dynamics method for the complex system.

Design Rules to Maximize Charge-Carrier Mobility along Conjugated Polymer Chains

The emergence of polymeric materials displaying high charge-carrier mobility values despite poor interchain structural order has spawned a renewal of interest in the identification of structure-property relationships pertaining to the transport of charges along conjugated polymer chains and the subsequent design of optimized architectures. Here, we present the results of intrachain charge transport simulations obtained by applying a robust surface hopping algorithm to a phenomenological Hamiltonian parametrized against first-principles simulations. Conformational effects are shown to provide a clear signature in the temperature-dependent charge-carrier mobility that complies with recent experimental observations. We further contrast against molecular crystals the evolution with electronic bandwidth and electron-phonon interactions of the room-temperature mobility in polymers, showing that intrachain charge-carrier mobility values in excess of 100 cm²/(V s) can be achieved through a proper chemical engineering of the backbones.

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.

How to calculate charge mobility in molecular materials from surface hopping non-adiabatic molecular dynamics - beyond the hopping/band paradigm

Charge transport in high mobility organic semiconductors is in an intermediate regime between small polaron hopping and band transport limits. We have recently shown that surface hopping non-adiabatic molecular dynamics is a powerful method for prediction of charge transport mechanisms in organic materials and for near-quantitative prediction of charge mobilities at room temperature where the effects of nuclear zero-point motion and tunneling are still relatively small [S. Giannini et al., Nat. Commun., 2019, 10, 3843]. Here we assess and critically discuss the extensions to Tully's original method that have led to this success: (i) correction for missing electronic decoherence, (ii) detection of trivial crossings and (iii) removal of decoherence correction-induced spurious charge transfer. If any one of these corrections is not included, the charge mobility diverges with system size, each for different physical reasons. Yet if they are included, convergence with system size, detailed balance and good internal consistency are achieved.

Solving the Trivial Crossing Problem While Preserving the Nodal Symmetry of the Wave Function

In an adiabatic mixed quantum-classical simulation, the avoided crossing of weakly coupled eigenstates can lead to unphysical discontinuities in wave function dynamics, otherwise known as the trivial crossing problem. A standard solution to the trivial crossing problem eliminates spatial discontinuities in wave function dynamics by imposing changes to the eigenstate of the wave function. In this paper, we show that this solution has the side effect of introducing transient discontinuities in the nodal symmetry of the wave function. We present an alternative solution to the trivial crossing problem that preserves both the spatial and nodal structure of the adiabatic wave function. By considering a model of exciton dynamics on conjugated polymer systems, we show that failure to preserve wave function symmetry yields exciton dynamics that depends unphysically on polymer system size. We demonstrate that our symmetry-preserving solution to the trivial crossing problem yields more realistic dynamics and can thus improve the accuracy of simulations of larger systems that are prone to the trivial crossing problem.

Influence of Defects on Excited-State Dynamics in Lead Halide Perovskites: Time-Domain ab Initio Studies

This Perspective summarizes recent research into the excited-state dynamics in lead halide perovskites that are of paramount importance for photovoltaic and photocatalytic applications. Nonadiabatic molecular dynamics combined with time-domain ab initio density functional theory allows one to mimic time-resolved spectroscopy experiments at the atomistic level of detail. The focus is placed on realistic aspects of perovskite materials, including point defects, surfaces, grain boundaries, mixed stoichiometries, dopants, and interfaces. The atomistic description of the quantum dynamics of electron and hole trapping and recombination, provided by the time-domain ab initio simulations, generates important insights into the mechanisms of charge and energy losses and guides the development of high-performance perovskite solar cell devices.