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The Journal of Chemical Physics

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

Wenjun Ouyang, Joseph Eli Subotnik

Published May 22, 2014• 14 citations

Abstract

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.

DOI: 10.1063/1.4876491

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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.

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

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.

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.

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.

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.