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arXiv:2211.10730 · The Journal of Chemical Physics

Overcoming positivity violations for density matrices in surface hopping

Anna S. Bondarenko, Roel Tempelaar

Published Jan 13, 2023• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0135456

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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.
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 paper, we describe a new and fully coherent stochastic surface hopping method for simulating mixed quantum-classical systems. We illustrate the approach on the simple but unforgiving problem of quantum evolution of a two-state quantum system in the limit of unperturbed pure state dynamics and for dissipative evolution in the presence of both stationary and nonstationary random environments. We formulate our approach in the Liouville representation and describe the density matrix elements by ensembles of trajectories. Population dynamics are represented by stochastic surface hops for trajectories representing diagonal density matrix elements. These are combined with an unconventional coherent stochastic hopping algorithm for trajectories representing off-diagonal quantum coherences. The latter generalizes the binary (0,1) "probability" of a trajectory to be associated with a given state to allow integers that can be negative or greater than unity in magnitude. Unlike existing surface hopping methods, the dynamics of the ensembles are fully entangled, correctly capturing the coherent and nonlocal structure of quantum mechanics.
In this contribution quantum/classical surface hopping methodology is applied to vibrational energy relaxation of a quantum oscillator in a classical heat bath. The model of a linearly damped (harmonic) oscillator is chosen which can be mapped onto the Brownian motion (Caldeira-Leggett) Hamiltonian. In the simulations Tully's fewest switches surface hopping scheme is adopted with inclusion of dephasing in the adiabatic basis using a simple decoherence algorithm. The results are compared to the predictions of a Redfield-type quantum master equation modeling using the classical heat bath force correlation function as input. Thereby a link is established between both types of quantum/classical approaches. Viewed from the latter perspective, surface hopping with dephasing may be interpreted as "on-the-fly" stochastic realization of a quantum/classical Pauli master equation.
Methods based on propagation of the one‐body reduced density‐matrix hold much promise for the simulation of correlated many‐electron dynamics far from equilibrium, but difficulties with finding good approximations for the interaction term in its equation of motion have so far impeded their application. These difficulties include the violation of fundamental physical principles such as energy conservation, positivity conditions on the density, or unchanging natural orbital occupation numbers. We review some of the recent efforts to confront these problems, and explore a semiclassical approximation for electron correlation coupled to time‐dependent Hartree‐Fock propagation. We find that this approach captures changing occupation numbers, and excitations to doubly‐excited states, improving over TDHF and adiabatic approximations in density‐matrix propagation. However, it does not guarantee N‐representability of the density‐matrix, consequently resulting sometimes in violation of positivity conditions, even though a purely semiclassical treatment preserves these conditions. © 2016 Wiley Periodicals, Inc.
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.
An algorithm is presented for the exact solution of the evolution of the density matrix of a mixed quantum-classical system in terms of an ensemble of surface hopping trajectories. The system comprises a quantum subsystem coupled to a classical bath whose evolution is governed by a mixed quantum-classical Liouville equation. The integral solution of the evolution equation is formulated in terms of a concatenation of classical evolution segments for the bath phase space coordinates separated by operators that change the quantum state and bath momenta. A hybrid Molecular Dynamics–Monte Carlo scheme which follows a branching tree of trajectories arising from the action of momentum derivatives is constructed to solve the integral equation. We also consider a simpler scheme where changes in the bath momenta are approximated by momentum jumps. These schemes are illustrated by considering the computation of the evolution of the density matrix for a two-level system coupled to a low dimensional classical bath.