Communication: Proper treatment of classically forbidden electronic transitions significantly improves detailed balance in surface hopping

Exciton energy transfer inside cavity-A benchmark study of polaritonic dynamics using the surface hopping method

Strong coupling between the molecular system and photon inside the cavity generates polaritons, which can alter reaction rates by orders of magnitude. In this work, we benchmark the surface hopping method to simulate non-adiabatic dynamics in a cavity. The comparison is made against a numerically exact method (the hierarchical equations of motion) for a model system investigating excitonic energy transfer for a broad range of parameters. Surface hopping captures the effects of the radiation mode well, both at resonance and off-resonance. We have further investigated parameters that can increase or decrease the rate of population transfer, and we find that surface hopping in general can capture both effects well. Finally, we show that the dipole self-energy term within our parameter regime does not significantly affect the system's dynamics.

Energy-Conserving Surface Hopping for Auger Processes

Auger-type processes are ubiquitous in nanoscale materials because quantum confinement enhances Coulomb interactions, and there exist large densities of states. Modeling Auger processes requires the modification of nonadiabatic (NA) molecular dynamics algorithms to include transitions caused by both NA and Coulomb couplings. The system is split into quantum and classical subsystems, e.g., electrons and vibrations, and as a result, energy conservation becomes nontrivial. In surface hopping, an electronic transition induced by NA coupling is accompanied by a classical velocity readjustment to ensure conservation of the total quantum-classical energy. A different treatment is needed for Auger transitions driven by Coulomb interactions. We develop a nonadiabatic molecular dynamics methodology that meticulously differentiates the energy redistribution accompanying hops induced by the NA coupling and the Coulomb interaction and correctly conserves the total energy at each transition. If the transition is driven by a Coulomb interaction, the hop energy is redistributed within the quantum electronic subsystem only. If the transition is NA, the energy is redistributed between the quantum and classical subsystems. Properly maintaining energy conservation for both types of transitions is crucial to generate a correct order of events, obtain accurate transition times, maintain a proper statistical distribution of state populations, and reach thermodynamic equilibrium. We test the method with biexciton annihilation and Auger-assisted hot electron relaxation in a CdSe quantum dot. The sequence of Auger and phonon-driven processes and the calculated time scales are in excellent agreement with the experimental results. The developed approach can be coupled with any surface-hopping method and provides a crucial practical advance to study charge-carrier dynamics in the nanoscale and condensed matter systems.

A size-consistent multi-state mapping approach to surface hopping

We develop a multi-state generalization of the recently proposed mapping approach to surface hopping (MASH) for the simulation of electronically nonadiabatic dynamics. This new approach extends the original MASH method to be able to treat systems with more than two electronic states. It differs from previous approaches in that it is size consistent and rigorously recovers the original two-state MASH in the appropriate limits. We demonstrate the accuracy of the method by applying it to a series of model systems for which exact benchmark results are available, and we find that the method is well suited to the simulation of photochemical relaxation processes.

Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients

We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.

Real-Time Dynamics and Detailed Balance in Ring Polymer Surface Hopping: The Impact of Frustrated Hops

Ring polymer surface hopping (RPSH) has been recently introduced as a well-tailored method for incorporating nuclear quantum effects, such as zero-point energy and tunneling, into nonadiabatic molecular dynamics simulations. The practical widespread usage of RPSH demands a comprehensive benchmarking of different reaction regimes and conditions with equal emphasis on demonstrating both the cons and the pros of the method. Here, we investigate the fundamental questions related to the conservation of energy and detailed balance in the context of RPSH. Using Tully's avoided crossing model as well as a 2-state quantum system coupled to a classical bath undergoing Langevin dynamics, we probe the critical problem of the proper treatment of the classically forbidden transitions stemming from the surface hopping algorithm. We show that proper treatment of these frustrated hops is key to the accurate description of real-time dynamics as well as reproducing the correct quantum Boltzmann populations.

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.

Pedagogical Overview of the Fewest Switches Surface Hopping Method

The fewest switches surface hopping method continues to grow in popularity to capture electronic nonadiabaticity and quantum nuclear effects due to its simplicity and accuracy. Knowing the basics of the method is essential for the correct implementation and interpretation of results. This review covers the fundamentals of the fewest switches surface hopping method with a detailed discussion of the nuances such as decoherence schemes and frustrated hops and the correct approach to calculating populations. The consequences of incorrect implementation are further discussed toward calculating kinetic and thermodynamic properties. Some tips for practitioners and a step-by-step algorithm for developers are provided. Finally, some of the finer technicalities of the fewest switches surface hopping method that are buried deep in the literature are pointed out to help graduate students better appreciate this method.

Detailed Balance and Independent Electron Surface-Hopping Method: The Importance of Decoherence and Correct Calculation of Diabatic Populations

We benchmark and improve the independent electron surface-hopping (IESH) method developed by J. C. Tully's group for nonadiabatic simulations near metal surfaces. We have incorporated decoherence within the IESH method as well as implemented a scheme for the accurate calculation of diabatic populations. We benchmark the original IESH method with the above inclusions for a model system to calculate rate constants and long-time populations. The original IESH method fails to capture the detailed balance for some of the parameters, which is corrected with the inclusion of decoherence and accurate calculation of diabatic populations. Total rate constants are well captured both within the original IESH method as well as within our modified IESH.

Benchmarking the Surface Hopping Method to Include Nuclear Quantum Effects

We have benchmarked the surface hopping method to capture nuclear quantum effects in the spin-Boson model in the deep tunneling regime. The thermal populations and the rate constants calculated using the surface hopping method are compared with those calculated using Boltzmann theory and Fermi's golden rule, respectively. Additionally, we have proposed a simple kinetic model that partially includes nuclear quantum effects within Marcus theory, and the results of the surface hopping method are analyzed under the framework of this simple kinetic model. A broad range of parameters are investigated to identify the regimes for the successes and failures of the surface hopping method. This work shows that with the accurate treatment of decoherence and velocity reversal, surface hopping can generally capture the nuclear quantum effects in the deep tunneling and weak diabatic coupling regime.

Mixed quantum-classical approach to model non-adiabatic electron-nuclear dynamics: Detailed balance and improved surface hopping method

We develop a density matrix formalism to describe coupled electron-nuclear dynamics. To this end, we introduce an effective Hamiltonian formalism that describes electronic transitions and small (quantum) nuclear fluctuations along a classical trajectory of the nuclei. Using this Hamiltonian, we derive equations of motion for the electronic occupation numbers and for the nuclear coordinates and momenta. We show that, in the limit, when the number of nuclear degrees of freedom coupled to a given electronic transition is sufficiently high (i.e., the strong decoherence limit), the equations of motion for the electronic occupation numbers become Markovian. Furthermore, the transition rates in these (rate) equations are asymmetric with respect to the lower-to-higher energy transitions and vice versa. In thermal equilibrium, such asymmetry corresponds to the detailed balance condition. We also study the equations for the electronic occupations in the non-Markovian regime and develop a surface hopping algorithm based on our formalism. To treat the decoherence effects, we introduce additional "virtual" nuclear wave packets whose interference with the "real" (physical) wave packets leads to the reduction in coupling between the electronic states (i.e., decoherence) as well as to the phase shifts that improve the accuracy of the numerical approach. Remarkably, the same phase shifts lead to the detailed balance condition in the strong decoherence limit.

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.

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.

Suppressed Carrier Recombination in Janus MoSSe Bilayer Stacks: A Time-Domain Ab Initio Study

Janus transition metal dichalcogenides (TMDs) have recently emerged as a new class of two-dimensional materials with a vertical dipole moment. Here, using time-domain ab initio simulations, we show that electron-hole recombination can be substantially suppressed via different stacking orientations of bilayer MoSSe. Despite having a larger net dipole moment, a S-Se/S-Se oriented MoSSe bilayer has a shorter carrier lifetime due to strong nonadiabatic coupling and a small band gap. The electron-hole recombination is coupled to the interlayer out-of-plane motion. In contrast, the opposite vertical dipoles weaken interlayer interactions in symmetric oriented MoSSe bilayers. Consequently, initial and final states are localized within different layers, and this significantly suppresses carrier recombination, resulting in an order of magnitude longer excited carrier lifetime in Se-S/S-Se oriented MoSSe bilayers. Our simulations provide theoretical insights into the carrier dynamics and suggest a way to enhance the carrier lifetime in Janus TMDs for efficient energy harvesting.

Coexistence of Different Charge-Transfer Mechanisms in the Hot-Carrier Dynamics of Hybrid Plasmonic Nanomaterials

Hot-carrier dynamics at the interfaces of semiconductors and nanoclusters is of significant importance for photovoltaic and photocatalytic applications. Plasmon-driven charge separation processes are considered to be only dependent on the type of donor-acceptor interactions, that is, the conventional hot-electron-transfer mechanism for van der Waals interactions and the plasmon-induced interfacial charge-transfer transition mechanism for chemical bonds. Here, we demonstrate that the two mechanisms can coexist in a nanoparticle-semiconductor hybrid nanomaterial, both leading to faster transfer than carrier relaxation. The origin of the two mechanisms is attributed to the spatial polarization of the excited hot carriers, where the longitudinal state couples to semiconductors more strongly than the transverse state. Our findings provide a new insight into the photoinduced carrier dynamics, which is relevant for many applications in solar energy conversion, including efficient water splitting, photocatalysis, and photovoltaics.

Performance of trajectory surface hopping method in the treatment of ultrafast intersystem crossing dynamics

We carried out extensive studies to examine the performance of the fewest-switches surface hopping method in the description of the ultrafast intersystem crossing dynamic of various singlet-triplet (S-T) models by comparison with the results of the exact full quantum dynamics. Different implementation details and some derivative approaches were examined. As expected, it is better to perform the trajectory surface hopping calculations in the spin-adiabatic representation or by the local diabatization approach, instead of in the spin-diabatic representation. The surface hopping method provides reasonable results for the short-time dynamics in the S-T model with weak spin-orbital coupling (diabatic coupling), although it does not perform well in the models with strong spin-orbital coupling (diabatic coupling). When the system accesses the S-T potential energy crossing with rather high kinetic energy, the trajectory surface hopping method tends to produce a good description of the nonadiabatic intersystem crossing dynamics. The impact of the decoherence correction on the performance of the trajectory surface hopping is system dependent. It improves the result accuracy in many cases, while its influence may also be minor for other cases.

Detailed balance, internal consistency, and energy conservation in fragment orbital-based surface hopping

We have recently introduced an efficient semi-empirical non-adiabatic molecular dynamics method for the simulation of charge transfer/transport in molecules and molecular materials, denoted fragment orbital-based surface hopping (FOB-SH) [J. Spencer et al., J. Chem. Phys. 145, 064102 (2016)]. In this method, the charge carrier wavefunction is expanded in a set of charge localized, diabatic electronic states and propagated in the time-dependent potential due to classical nuclear motion. Here we derive and implement an exact expression for the non-adiabatic coupling vectors between the adiabatic electronic states in terms of nuclear gradients of the diabatic electronic states. With the non-adiabatic coupling vectors (NACVs) available, we investigate how different flavours of fewest switches surface hopping affect detailed balance, internal consistency, and total energy conservation for electron hole transfer in a molecular dimer with two electronic states. We find that FOB-SH satisfies detailed balance across a wide range of diabatic electronic coupling strengths provided that the velocities are adjusted along the direction of the NACV to satisfy total energy conservation upon a surface hop. This criterion produces the right fraction of energy-forbidden (frustrated) hops, which is essential for correct population of excited states, especially when diabatic couplings are on the order of the thermal energy or larger, as in organic semiconductors and DNA. Furthermore, we find that FOB-SH is internally consistent, that is, the electronic surface population matches the average quantum amplitudes, but only in the limit of small diabatic couplings. For large diabatic couplings, inconsistencies are observed as the decrease in excited state population due to frustrated hops is not matched by a corresponding decrease in quantum amplitudes. The derivation provided here for the NACV should be generally applicable to any electronic structure approach where the electronic Hamiltonian is constructed in a diabatic electronic state basis.

Lewis Base Passivation of Hybrid Halide Perovskites Slows Electron-Hole Recombination: Time-Domain Ab Initio Analysis

Nonradiative electron-hole recombination plays a key role in determining photon conversion efficiencies in solar cells. Experiments demonstrate significant reduction in the recombination rate upon passivation of methylammonium lead iodide perovskite with Lewis base molecules. Using nonadiabatic molecular dynamics combined with time-domain density functional theory, we find that the nonradiative charge recombination is decelerated by an order of magnitude upon adsorption of the molecules. Thiophene acts by the traditional passivation mechanism, forcing electron density away from the surface. In contrast, pyridine localizes the electron at the surface while leaving it energetically near the conduction band edge. This is because pyridine creates a stronger coordinative bond with a lead atom of the perovskite and has a lower energy unoccupied orbital compared with thiophene due to the more electronegative nitrogen atom relative to thiophene's sulfur. Both molecules reduce two-fold the nonadiabatic coupling and electronic coherence time. A broad range of vibrational modes couple to the electronic subsystem, arising from inorganic and organic components. The simulations reveal the atomistic mechanisms underlying the enhancement of the excited-state lifetime achieved by the perovskite passivation, rationalize the experimental results, and advance our understanding of charge-phonon dynamics in perovskite solar cells.

Simple Quantum Dynamics with Thermalization

In this paper, we introduce two simple quantum dynamics methods. One is based on the popular surface-hopping method, and the other is based on rescaling of the propagation on the bath ground-state potential surface. The first method is special, as it avoids specific feedback from the simulated quantum system to the bath and can be applied for precalculated classical trajectories. It is based on the equipartition theorem to determine if hops between different potential energy surfaces are allowed. By comparing with the formally exact Hierarchical Equations Of Motion approach for four model systems we find that the method generally approximates the quantum dynamics toward thermal equilibrium very well. The second method is based on rescaling of the nonadiabatic coupling and also neglect the effect of the state of the quantum system on the bath. By the nature of the approximations, they cannot reproduce the effect of bath relaxation following excitation. However, the methods are both computationally more tractable than the conventional fewest switches surface hopping, and we foresee that the methods will be powerful for simulations of quantum dynamics in systems with complex bath dynamics, where the system-bath coupling is not too strong compared to the thermal energy.

Stochastic and Quasi-Stochastic Hamiltonians for Long-Time Nonadiabatic Molecular Dynamics

In the condensed-matter environments, the vibronic Hamiltonian that describes nonadiabatic dynamics often appears as an erratic entity, and one may assume it can be generated stochastically. This property is utilized to formulate novel stochastic and quasi-stochastic vibronic Hamiltonian methodologies, which open a new route to long-time excited state dynamics in atomistic solid-state systems at negligible computational cost. Using a model mimicking a typical solid-state material in noisy environment, general conclusions regarding the simulation of nonadiabatic dynamics are obtained: (1) including bath is critical to complete excited state relaxation; (2) a totally stochastic modulation of energies and couplings has a net effect of no bath and inhibits relaxation; (3) including a single or several dominant electron-phonon modes may be insufficient to complete the excited state relaxation; (4) only the multiple modes, even those that have negligible weights, can represent both the deterministic modulation of system's Hamiltonian and stochastic effects of bath.

An Efficient, Augmented Surface Hopping Algorithm That Includes Decoherence for Use in Large-Scale Simulations

We propose and implement a highly efficient augmented surface hopping algorithm that (i) can be used for large simulations (with many nuclei and many electronic states) and (ii) includes the effects of decoherence without parametrization. Our protocol is based on three key modifications of the surface hopping methodology: (a) a novel separation of classical and quantum degrees of freedom that treats avoided and trivial crossings efficiently, (b) a multidimensional approximation of the time derivative matrix that avoids explicit construction of the derivative coupling at most time steps, and (c) an efficient approximation for the augmented fewest-switches surface hopping decoherence rate. We will show that this protocol can be several orders of magnitude more efficient than the traditional protocol for large multidimensional problems. Furthermore, the marginal cost for including decoherence effects is now negligible.

Confronting surface hopping molecular dynamics with Marcus theory for a molecular donor–acceptor system

We investigate the performance of fewest switches surface hopping (SH) in describing electron transfer (ET) for a molecular donor–acceptor system. Computer simulations are carried out for a wide range of reorganisation energy (λ), electronic coupling strength (H_ab) and driving force using our recently developed fragment orbital-based SH approach augmented with a simple decoherence correction. This methodology allows us to compute SH ET rates over more than four orders of magnitude, from the sub-picosecond to the nanosecond time regime. We find good agreement with semi-classical ET theory in the non-adiabatic ET regime. The correct scaling of the SH ET rate with electronic coupling strength is obtained and the Marcus inverted regime is reproduced, in line with previously reported results for a spin-boson model. Yet, we find that the SH ET rate falls below the semi-classical ET rate in the adiabatic regime, where the free energy barrier is in the order of k_BT in our simulations. We explain this by first signatures of non-exponential population decay of the initial charge state. For even larger electronic couplings (H_ab = λ/2), the free energy barrier vanishes and ET rates are no longer defined. At this point we observe a crossover from ET on the vibronic time scale to charge relaxation on the femtosecond time scale that is well described by thermally averaged Rabi oscillations. The extension of the analysis from the non-adiabatic limit to large electronic couplings and small or even vanishing activation barriers is relevant for our understanding of charge transport in organic semiconductors.