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arXiv:2511.18062

Fewest switches surface hopping with decoherence in the Marcus inverted regime: correct rates but wrong thermal populations

Manas Nagda, Priyam Kumar De, Amber Jain

quant-phphysics.chem-phSubmitted Nov 22, 2025• Announced Nov 25, 2025
arXivHTMLPDF

Abstract

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.

DOI
10.1021/acs.jpclett.5c02886

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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.
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.
Fewest-switches surface hopping (FSSH) has emerged as one of the leading methods for modeling the quantum dynamics of molecular systems. While its original formulation was limited to adiabatic populations, the growing interest in the application of FSSH to coherent phenomena prompts the question of how one should construct a complete density matrix based on FSSH trajectories. A straightforward solution is to define adiabatic coherences based on wavefunction coefficients. In this paper, we demonstrate that inconsistencies introduced in the density matrix through such treatment may lead to a violation of positivity. We furthermore show that a recently proposed coherent generalization of FSSH results in density matrices that satisfy positivity while yielding improved accuracy throughout much (but not all) of parameter space.
We present a generalized non-Hermitian equation of motion (nH-EOM) to go beyond standard trajectory surface hopping dynamics. The derivation is based on the Born-Huang expansion of the total wave function and the polar representation of the nuclear factor. The nH-EOM contains two additional terms, a skew symmetry term iΓ with dissipation operator Γ to account for decoherence, and a kinetic-energy renormalization term to account for phase shifts, without destroying the invariance to the choice of representation. Numerically, the nH-EOM can still be solved efficiently using a semiclassical approximation in the framework of Tully's fewest-switches surface hopping (FSSH) algorithm. Applications to model Hamiltonians demonstrate improved performance over the standard FSSH approach, through comparison to exact quantum results.
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.
The Davydov D1 ansatz, which assigns individual bosonic trajectories to each spin state, is an efficient, yet extremely accurate trial state for time-dependent variation of the sub-Ohmic spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. A surface hopping algorithm is developed employing the Davydov D1 ansatz to study the spin dynamics with a sub-Ohmic bosonic bath. The algorithm takes into account both coherent and incoherent dynamics of the population evolution in a unified manner, and compared with semiclassical surface hopping algorithms, hopping rates calculated in this work follow more closely the Marcus formula.
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.