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The Journal of Chemical Physics

Assessment of trajectory surface hopping methods in long-time nonadiabatic dynamics

Mohammad Shakiba, Daeho Han, Saikat Mukherjee, Alexey V. Akimov

Published Mar 11, 2026• 1 citation
Publisher

Abstract

We present an assessment of an extensive set of trajectory surface hopping methodologies for modeling long-time population dynamics in a two-level spin-boson model. The considered methodological recipes involve combinations of three surface hopping approaches with five decoherence correction methods, several decoherence-time computation schemes, and two initial-condition sampling methods. In addition to these combinations, the phase correction approach of Shenvi, Subotnik, and Yang (SSY) is considered. By exploring a wide range of meta-parameters controlling decoherence and dephasing times, we determine the optimal performance of such methods and provide a ranking of the best-performing approaches. We find that inclusion of the SSY correction generally over-accelerates the population relaxation dynamics. We also find a strong dependence of the trajectory surface hopping (TSH) calculations on the type of initial-condition sampling: Wigner sampling leads to overestimated population transfer rates and requires decoherence corrections. The simplified decay of mixing approach is found to perform well in this situation. In contrast, Boltzmann sampling leads to a surprisingly remarkable performance of bare TSH schemes without decoherence, which may be a consequence of error cancellation. We provide a critical discussion of the observed trends in different methods' performance and suggest possible avenues for their further improvement.

DOI
10.1063/5.0314503

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We present an assessment of an extensive set of trajectory surface hopping methodologies for modeling long-time population dynamics in a two-level spin-boson model. The considered methodological recipes involve combinations of three surface hopping approaches with five decoherence correction methods, several decoherence-time computation schemes, and two initial-condition sampling methods. In addition to these combinations, the phase correction approach of Shenvi, Subotnik, and Yang (SSY) is considered. By exploring a wide range of meta-parameters controlling decoherence and dephasing times, we determine the optimal performance of such methods and provide a ranking of the best-performing approaches. We find that inclusion of the SSY correction generally over-accelerates the population relaxation dynamics. We also find a strong dependence of the trajectory surface hopping (TSH) calculations on the type of initial-condition sampling: Wigner sampling leads to overestimated population transfer rates and requires decoherence corrections. The simplified decay of mixing approach is found to perform well in this situation. In contrast, Boltzmann sampling leads to a surprisingly remarkable performance of bare TSH schemes without decoherence, which may be a consequence of error cancellation. We provide a critical discussion of the observed trends in different methods' performance and suggest possible avenues for their further improvement.
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We derive a partially linearized version of the mapping approach to surface hopping (MASH) that provides a new method for simulating nonadiabatic dynamics in molecular systems. As in the original MASH formalism, trajectories travel on adiabatic surfaces with hops determined by the dynamics of a spin vector on the Bloch sphere. However, because the new approach is based on the partially linearized density matrix (PLDM), it has two spin vectors, which may be in the same or different hemispheres of the Bloch sphere. The former scenario describes an electronic population moving on a single adiabatic surface, whereas the latter describes a coherence moving on the average surface. We show that this new method, called MASH-PLDM, leads to improved results over the original MASH dynamics for a variety of scattering problems and spin-boson models, especially in cases where coherences play an important role. A further benefit of the new methodology is that it can treat multi-time correlation functions such as those required for nonlinear spectroscopy. We demonstrate this capability with a proof-of-principle simulation of time-resolved pump-probe spectroscopy.
Fewest switches surface hopping (FSSH) is a well benchmarked dynamical method for simulating nonadiabatic systems. In particular, the literature shows that for the spin-Boson model Hamiltonian, FSSH with appropriate corrections usually captures the detailed balance well and obtains rate constants within a factor of 2 compared to numerically exact results. In this study, we show that in the deep inverted Marcus regime, the augmented-FSSH (AFSSH, one version that includes decoherence) yields reasonably accurate rate constants but incorrect thermal populations over a broad range of parameters. We present an analytical derivation to understand the AFSSH behavior, and therefore, show that AFSSH obtains correct rate constants owing to the resonance of the time derivative coupling with the exothermicity, but obtains an incorrect thermal population owing to the self-consistency issue. The presented derivation provides an analytical expression for the quantum correction factor for AFSSH simulations in the Marcus inverted regime.
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.
The spin-boson model is solved within the framework of quantum-classical dynamics using our recently-developed surface-hopping scheme. The quantum-classical equation of motion is expressed in an adiabatic basis and its solution is constructed from an ensemble of trajectories which undergo nonadiabatic transitions and evolve coherently on the adiabatic surfaces. Details of the algorithm for the simulation of the dynamics are presented and the method of simple Monte Carlo sampling used to evaluate the expectation values of observables is discussed. The simulation method is applied to a spin-boson system with a harmonic bath composed of ten oscillators with an Ohmic spectral density. For the spin-boson model the present implementation of quantum-classical dynamics is exact and the results of our surface-hopping simulations are in accord with previous numerically exact results for this model.
In open quantum systems, decoherence occurs through interaction of a quantum subsystem with its environment. The computation of expectation values requires a knowledge of the quantum dynamics of operators and sampling from initial states of the density matrix describing the subsystem and bath. We consider situations where the quantum evolution can be approximated by quantum-classical Liouville dynamics and examine the circumstances under which the evolution can be reduced to surface-hopping dynamics, where the evolution consists of trajectory segments exclusively evolving on single adiabatic surfaces, with probabilistic hops between these surfaces. The justification for the reduction depends on the validity of a Markovian approximation on a bath averaged memory kernel that accounts for quantum coherence in the system. We show that such a reduction is often possible when initial sampling is from either the quantum or classical bath initial distributions. If the average is taken only over the quantum dispersion that broadens the classical distribution, then such a reduction is not always possible.