Fewest-switches Surface Hopping Algorithm Research Articles

Nonadiabatic Simulation of Exciton Dynamics in Organic Semiconductors Using Neural Network-Based Frenkel Hamiltonian and Gradients.

In this study, we present a multiscale method to simulate the propagation of Frenkel singlet excitons in organic semiconductors (OSCs). The approach uses neural network models to train a Frenkel-type Hamiltonian and its gradient, obtained by the long-range correction version of density functional tight-binding with self-consistent charges. Our models accurately predict site energies, excitonic couplings, and corresponding gradients, essential for the nonadiabatic molecular dynamics simulations. Combined with the fewest switches surface hopping algorithm, the method was applied to four representative OSCs: anthracene, pentacene, perylenediimide, and diindenoperylene. The simulated exciton diffusion constants align well with experimental and reported theoretical values and offer valuable insights into exciton dynamics in OSCs.

UV-spectrum and photodecomposition of peroxynitrous acid in the troposphere.

The UV spectrum of peroxynitrous acid, HOONO, was computed at the B3LYP/AVTZ and MCSCF/AVTZ levels using the fewest switches surface hopping algorithm. Due to large-amplitude vibrational motions of this molecule, the maxima in the simulated spectra are displaced from the positions of vertical excitations. The three lowest excited electronic singlet states, which are all repulsive, can be reached by UV absorption. The photolysis products are determined, and the photolysis rate constant is provided for the first time. We found that near the tropopause the photolysis rate constant J ≈ 6 × 10-4 s-1, exceeds that for thermal decomposition by two orders of magnitude. The photolysis lifetime is about 30 minutes. Thus, photolysis is an important process and should be included in atmospheric models.

A first principles derivation of energy-conserving momentum jumps in surface hopping simulations.

The fewest switches surface hopping (FSSH) method proposed by Tully in 1990 [Tully, J. Chem. Phys. 93, 1061 (1990)]-along with its many later variations-forms the basis for most practical simulations of molecular dynamics with electronic transitions in realistic systems. Despite its popularity, a rigorous formal derivation of the algorithm has yet to be achieved. In this paper, we derive the energy-conserving momentum jumps employed by FSSH from the perspective of quantum trajectory surface hopping (QTSH) [Martens, J. Phys. Chem. A 123, 1110 (2019)]. In the limit of localized nonadiabatic transitions, simple mathematical and physical arguments allow the FSSH algorithm to be derived from first principles. For general processes, the quantum forces characterizing the QTSH method provide accurate results for nonadiabatic dynamics with rigorous energy conservation, at the ensemble level, within the consistency of the underlying stochastic surface hopping without resorting to the artificial momentum rescaling of FSSH.

Incorporating Berry Force Effects into the Fewest Switches Surface-Hopping Algorithm: Intersystem Crossing and the Case of Electronic Degeneracy.

We present a preliminary surface-hopping approach for modeling intersystem crossing (ISC) dynamics between four electronic states: one singlet and one (triply degenerate) triplet. In order to incorporate all Berry force effects, the algorithm requires that, when moving along an adiabatic surface associated with the triplet manifold, one must also keep track of a quasi-diabatic index (akin to a "ms" quantum number) for each trajectory. For a simple model problem, we find that a great deal of new physics can be captured by our algorithm, setting the stage for larger, more realistic (or perhaps even ab initio) simulations in the future.

Semiclassical description of nuclear dynamics moving through complex-valued single avoided crossings of two electronic states.

The standard fewest-switches surface hopping (FSSH) approach fails to model nonadiabatic dynamics when the electronic Hamiltonian is complex-valued and there are multiple nuclear dimensions; FSSH does not include geometric magnetic effects and does not have access to a gauge independent direction for momentum rescaling. In this paper, for the case of a Hamiltonian with two electronic states, we propose an extension of Tully's FSSH algorithm, which includes geometric magnetic forces and, through diabatization, establishes a well-defined rescaling direction. When combined with a decoherence correction, our new algorithm shows satisfying results for a model set of two-dimensional single avoided crossings.

An efficient implementation of spin–orbit coupling within the framework of semiempirical orthogonalization-corrected methods for ultrafast intersystem crossing dynamics

We implement spin–orbit coupling (SOC) within the framework of the OMx/CIS method and then incorporate it into Tully's fewest switches surface hopping algorithm to enable excited-state nonadiabatic dynamics simulations.

A "backtracking" correction for the fewest switches surface hopping algorithm.

We propose a "backtracking" mechanism within Tully's fewest switches surface hopping (FSSH) algorithm, whereby whenever one detects consecutive (double) hops during a short period of time, one simply rewinds the dynamics backward in time. In doing so, one reduces the number of hopping events and comes closer to a truly fewest switches surface hopping approach with independent trajectories. With this algorithmic change, we demonstrate that surface hopping can be reasonably accurate for nuclear dynamics in a multidimensional configuration space with a complex-valued (i.e., not real-valued) electronic Hamiltonian; without this adjustment, surface hopping often fails. The added computational cost is marginal. Future research will be needed to assess whether or not this backtracking correction can improve the accuracy of a typical FSSH calculation with a real-valued electronic Hamiltonian (that ignores spin).

Theoretical investigation of the electronic relaxation in highly excited chrysene and tetracene: The effect of armchair vs zigzag edge.

Non-adiabatic molecular dynamics of neutral chrysene and tetracene molecules is investigated using Tully's fewest switches surface hopping algorithm coupled to the time-dependent density functional based tight-binding (TD-DFTB) method for electronic structure calculations. We first assess the performance of two DFTB parameter sets based on the computed TD-DFTB absorption spectra. The main focus is given to the analysis of the electronic relaxation from the brightest excited state following absorption of a UV photon. We determine the dynamical relaxation times and discuss the underlying mechanisms. Our results show that the electronic population of the brightest excited singlet state in armchair-edge chrysene decays an order-of-magnitude faster than the one in zigzag-edge tetracene. This is correlated with a qualitatively similar difference of energy gaps between the brightest state and the state lying just below in energy, which is also consistent with our previous study on polyacenes.

An extension of the fewest switches surface hopping algorithm to complex Hamiltonians and photophysics in magnetic fields: Berry curvature and "magnetic" forces.

We present a preliminary extension of the fewest switches surface hopping (FSSH) algorithm to the case of complex Hamiltonians as appropriate for modeling the dynamics of photoexcited molecules in magnetic fields. We make ansätze for the direction of momentum rescaling, and we account for Berry's phase effects through "magnetic" forces as applicable in the adiabatic limit. Because Berry's phase is a nonlocal, topological characteristic of a set of entangled potential energy surfaces, we find that Tully's local FSSH algorithm can only partially capture the correct physics.

Nonadiabatic quantum molecular dynamics with detailed balance

We present an approach for carrying out nonadiabatic molecular dynamics simulations of systems in which nonadiabatic transitions arise from the coupling between the classical atomic motions and a quasicontinuum of electronic quantum states. Such conditions occur in many research areas, including chemistry at metal surfaces, radiation damage of materials, and warm-dense-matter physics. The classical atomic motions are governed by stochastic Langevin-like equations, while the quantum electron dynamics is described by a master equation for the populations of the electronic states. These working equations are obtained from a first-principles derivation. Remarkably, unlike the widely used Ehrenfest and surface-hopping methods, the approach naturally satisfies the principle of detailed balance at equilibrium and therefore can describe the evolution to thermal equilibrium from an arbitrary initial state. A practical algorithm is cast in the form of the widely used fewest-switches surface-hopping algorithm but with switching probabilities that are not specified ad hoc like in the standard algorithm but are instead derived.

An efficient solution to the decoherence enhanced trivial crossing problem in surface hopping.

We provide an in-depth investigation of the time interval convergence when both trivial crossing and decoherence corrections are applied to Tully's fewest switches surface hopping (FSSH) algorithm. Using one force-based and one energy-based decoherence strategies as examples, we show decoherence corrections intrinsically enhance the trivial crossing problem. We propose a restricted decoherence (RD) strategy and incorporate it into the self-consistent (SC) fewest switches surface hopping algorithm [L. Wang and O. V. Prezhdo, J. Phys. Chem. Lett. 5, 713 (2014)]. The resulting SC-FSSH-RD approach is applied to general Hamiltonians with different electronic couplings and electron-phonon couplings to mimic charge transport in tens to hundreds of molecules. In all cases, SC-FSSH-RD allows us to use a large time interval of 0.1 fs for convergence and the simulation time is reduced by over one order of magnitude. Both the band and hopping mechanisms of charge transport have been captured perfectly. SC-FSSH-RD makes surface hops in the adiabatic representation and can be implemented in both diabatic and locally diabatic representations for wave function propagation. SC-FSSH-RD can potentially describe general nonadiabatic dynamics of electrons and excitons in organics and other materials.

Generalization of fewest-switches surface hopping for coherences.

Fewest-switches surface hopping (FSSH) is perhaps the most widely used mixed quantum-classical approach for the modeling of non-adiabatic processes, but its original formulation is restricted to (adiabatic) population terms of the quantum density matrix, leaving its implementations with an inconsistency in the treatment of populations and coherences. In this article, we propose a generalization of FSSH that treats both coherence and population terms on equal footing and which formally reduces to the conventional FSSH algorithm for the case of populations. This approach, coherent fewest-switches surface hopping (C-FSSH), employs a decoupling of population relaxation and pure dephasing and involves two replicas of the classical trajectories interacting with two active surfaces. Through extensive benchmark calculations of a spin-boson model involving a Debye spectral density, we demonstrate the potential of C-FSSH to deliver highly accurate results for a large region of parameter space. Its uniform description of populations and coherences is found to resolve incorrect behavior observed for conventional FSSH in various cases, in particular at low temperature, while the parameter space regions where it breaks down are shown to be quite limited. Its computational expenses are virtually identical to conventional FSSH.

Ring Polymer Surface Hopping: Incorporating Nuclear Quantum Effects into Nonadiabatic Molecular Dynamics Simulations.

We apply a recently proposed ring polymer surface hopping (RPSH) approach to investigate the real-time nonadiabatic dynamics with explicit nuclear quantum effects. The nonadibatic electronic transitions are described through Tully's fewest-switches surface hopping algorithm and the motion of the nuclei are quantized through the ring polymer Hamiltonian in the extended phase space. Applying the RPSH method to simulate Tully's avoided crossing models, we demonstrate the critical role of the nuclear tunneling effect and zero-point energy for accurately describing the transmission and reflection probabilities with low initial momenta. In addition, in Tully's extended coupling model, we show that the ring polymer quantization effectively captures decoherence, yielding more accurate reflection probabilities. These promising results demonstrate the potential of using RPSH as an accurate and efficient method to incorporate nuclear quantum effects into nonadiabatic dynamics simulations.

Understanding the Surface Hopping View of Electronic Transitions and Decoherence

We present a current, up-to-date review of the surface hopping methodology for solving nonadiabatic problems, 25 years after Tully published the fewest switches surface hopping algorithm. After reviewing the original motivation for and failures of the algorithm, we give a detailed examination of modern advances, focusing on both theoretical and practical issues. We highlight how one can partially derive surface hopping from the Schrödinger equation in the adiabatic basis, how one can change basis within the surface hopping algorithm, and how one should understand and apply the notions of decoherence and wavepacket bifurcation. The question of time reversibility and detailed balance is also examined at length. Recent applications to photoexcited conjugated polymers are discussed briefly.

On surface hopping and time-reversal.

While individual Tully-style fewest switches surface hopping (FSSH) trajectories are stochastic and cannot be inverted in time, it is possible to reverse in time the dynamics of a swarm of FSSH trajectories. Here we show exactly how to invert such dynamics, and we investigate the stability of such time-reversed surface hopping dynamics. We demonstrate that FSSH trajectories can be inverted successfully for short time periods, but the time-reversed dynamics become unstable for long times with multiple hopping events. We argue that this instability of FSSH going backward in time can be correlated with the stability of the FSSH algorithm going forward in time.

Estimating the entropy and quantifying the impurity of a swarm of surface-hopping trajectories: a new perspective on decoherence.

In this article, we consider the intrinsic entropy of Tully's fewest switches surface hopping (FSSH) algorithm (as estimated by the impurity of the density matrix) [J. Chem. Phys. 93, 1061 (1990)]. We show that, even for a closed system, the total impurity of a FSSH calculation increases in time (rather than stays constant). This apparent failure of the FSSH algorithm can be traced back to an incorrect, approximate treatment of the electronic coherence between wavepackets moving along different potential energy surfaces. This incorrect treatment of electronic coherence also prevents the FSSH algorithm from correctly describing wavepacket recoherences (which is a well established limitation of the FSSH method). Nevertheless, despite these limitations, the FSSH algorithm often predicts accurate observables because the electronic coherence density is modulated by a phase factor which varies rapidly in phase space and which often integrates to almost zero. Adding "decoherence" events on top of a FSSH calculation completely destroys the incorrect FSSH electronic coherence and effectively sets the Poincaré recurrence time for wavepacket recoherence to infinity; this modification usually increases FSSH accuracy (assuming there are no recoherences) while also offering long-time stability for trajectories. In practice, we show that introducing "decoherence" events does not change the total FSSH impurity significantly, but does lead to more accurate evaluations of the impurity of the electronic subsystem.

Can we derive Tully's surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence

In this article, we demonstrate that Tully's fewest-switches surface hopping (FSSH) algorithm approximately obeys the mixed quantum-classical Liouville equation (QCLE), provided that several conditions are satisfied--some major conditions, and some minor. The major conditions are: (1) nuclei must be moving quickly with large momenta; (2) there cannot be explicit recoherences or interference effects between nuclear wave packets; (3) force-based decoherence must be added to the FSSH algorithm, and the trajectories can no longer rigorously be independent (though approximations for independent trajectories are possible). We furthermore expect that FSSH (with decoherence) will be most robust when nonadiabatic transitions in an adiabatic basis are dictated primarily by derivative couplings that are presumably localized to crossing regions, rather than by small but pervasive off-diagonal force matrix elements. In the end, our results emphasize the strengths of and possibilities for the FSSH algorithm when decoherence is included, while also demonstrating the limitations of the FSSH algorithm and its inherent inability to follow the QCLE exactly.

Communication: The correct interpretation of surface hopping trajectories: How to calculate electronic properties

In a recent paper, we presented a road map for how Tully's fewest switches surface hopping (FSSH) algorithm can be derived, under certain circumstances, from the mixed quantum-classical Liouville equation. In this communication, we now demonstrate how this new interpretation of surface hopping can yield significantly enhanced results for electronic properties in nonadiabatic calculations. Specifically, we calculate diabatic populations for the spin-boson problem using FSSH trajectories. We show that, for some Hamiltonians, without changing the FSSH algorithm at all but rather simply reinterpreting the ensemble of surface hopping trajectories, we recover excellent results and remove any and all ambiguity about the initial condition problem.

Nonadiabatic excited-state molecular dynamics: Treatment of electronic decoherence

Within the fewest switches surface hopping (FSSH) formulation, a swarm of independent trajectories is propagated and the equations of motion for the quantum coefficients are evolved coherently along each independent nuclear trajectory. That is, the phase factors, or quantum amplitudes, are retained. At a region of strong coupling, a trajectory can branch into multiple wavepackets. Directly following a hop, the two wavepackets remain in a region of nonadiabatic coupling and continue exchanging population. After these wavepackets have sufficiently separated in phase space, they should begin to evolve independently from one another, the process known as decoherence. Decoherence is not accounted for in the standard surface hopping algorithm and leads to internal inconsistency. FSSH is designed to ensure that at any time, the fraction of classical trajectories evolving on each quantum state is equal to the average quantum probability for that state. However, in many systems this internal consistency requirement is violated. Treating decoherence is an inherent problem that can be addressed by implementing some form of decoherence correction to the standard FSSH algorithm. In this study, we have implemented two forms of the instantaneous decoherence procedure where coefficients are reinitialized following hops. We also test the energy-based decoherence correction (EDC) scheme proposed by Granucci et al. and a related version where the form of the decoherence time is taken from Truhlar's Coherent Switching with Decay of Mixing method. The sensitivity of the EDC results to changes in parameters is also evaluated. The application of these computationally inexpensive ad hoc methods is demonstrated in the simulation of nonradiative relaxation in two conjugated oligomer systems, specifically poly-phenylene vinylene and poly-phenylene ethynylene. We find that methods that have been used successfully for treating small systems do not necessarily translate to large polyatomic systems and their success depends on the particular system under study.

Surface hopping simulation of vibrational predissociation of methanol dimer

The mixed quantum-classical surface hopping method is applied to the vibrational predissociation of methanol dimer, and the results are compared to more exact quantum calculations. Utilizing the vibrational SCF basis, the predissociation problem is cast into a curve crossing problem between dissociative and quasibound surfaces with different vibrational character. The varied features of the dissociative surfaces, arising from the large amplitude OH torsion, generate rich predissociation dynamics. The fewest switches surface hopping algorithm of Tully [J. Chem. Phys. 93, 1061 (1990)] is applied to both diabatic and adiabatic representations. The comparison affords new insight into the criterion for selecting the suitable representation. The adiabatic method's difficulty with low energy trajectories is highlighted. In the normal crossing case, the diabatic calculations yield good results, albeit showing its limitation in situations where tunneling is important. The quadratic scaling of the rates on coupling strength is confirmed. An interesting resonance behavior is identified and is dealt with using a simple decoherence scheme. For low lying dissociative surfaces that do not cross the quasibound surface, the diabatic method tends to overestimate the predissociation rate whereas the adiabatic method is qualitatively correct. Analysis reveals the major culprits involve Rabi-like oscillation, treatment of classically forbidden hops, and overcoherence. Improvements of the surface hopping results are achieved by adopting a few changes to the original surface hopping algorithms.