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Physical review. E

Non-Hermitian surface hopping

Xing Bao Gao, Walter Thiel

Published Jan 25, 2017• 41 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.95.013308

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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.
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.
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 analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 115, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different nonadiabatic terms in these equations, including how they are computed in practice. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of nonadiabatic Tully model systems to illustrate our analysis.
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 analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 115, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different nonadiabatic terms in these equations, including how they are computed in practice. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of nonadiabatic Tully model systems to illustrate our analysis.
In this thesis quantum-classical dynamics is applied to the study of quantum condensed phase processes. This approach is based on the quantum-classical Liouville equation where the dynamics of a small subset of the degrees of freedom are treated quantum mechanically while the remaining degrees of freedom are treated by classical mechanics to a good approximation. We use this approach as it is computationally tractable, and the resulting equation of motion accurately accounts for the quantum and classical dynamics, as well as the coupling between these two components of the system. \n\nBy recasting the quantum-classical Liouville equation into the form of a generalized master equation we investigate connections to surface-hopping. The link between these approaches is decoherence arising from interaction of the subsystem with the environment. We derive an evolution equation for the subsystem which contains terms accounting for the effects of the environment. One of these terms involves a memory kernel that accounts for the coherent dynamics. If this term decays rapidly, a Markovian approximation can be made. By lifting the resulting subsystem master equation into the full phase space, we obtain a Markovian master equation that prescribes surface-hopping-like dynamics. Our analysis outlines the conditions under which such a description is valid.\n\nNext, we consider the calculation of the rate constant for a quantum mechanical barrier crossing process. Starting from the reactive-flux autocorrelation function, we derive a quantum-classical expression for the rate kernel. This expression involves quantum-classical evolution of a species operator averaged over the initial quantum equilibrium structure of the system making it possible to compute the rate constant via computer simulation. Using a simple model for a proton transfer reaction we compare the results of the rate calculation obtained by quantum-classical Liouville dynamics with that of master equation dynamics. The master equation provides a good approximation to the full quantum-classical Liouville calculation for our model and a more stable algorithm results due to the elimination of oscillating phase factors in the simulation. \n\nFinally, we make use of the theoretical framework established in this thesis to analyze some aspects of decoherence used in popular surface-hopping techniques.