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Physical Review Letters

Semiclassical Description of Nonadiabatic Quantum Dynamics

Gerhard Stock, Michael Thoss

Published Jan 27, 1997• 585 citations
Publisher

Abstract

A semiclassical approach is presented that allows us to extend the usual Van Vleck--Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. Based on Schwinger's theory of angular momentum, the formulation employs an exact mapping of the discrete quantum variables onto continuous degrees of freedom. The resulting dynamical problem is evaluated through a semiclassical initial-value representation of the time-dependent propagator. As a first application we have performed semiclassical simulations for a spin-boson model, which reproduce the exact quantum-mechanical results quite accurately.

DOI
10.1103/physrevlett.78.578

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A semiclassical approach is presented that allows us to extend the usual Van Vleck--Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. Based on Schwinger's theory of angular momentum, the formulation employs an exact mapping of the discrete quantum variables onto continuous degrees of freedom. The resulting dynamical problem is evaluated through a semiclassical initial-value representation of the time-dependent propagator. As a first application we have performed semiclassical simulations for a spin-boson model, which reproduce the exact quantum-mechanical results quite accurately.
We present a semiclassical method for simulating the dynamics of nonadiabatic transitions in a discrete-state quantum system coupled to a bath of explicit continuous coordinates. This method employs a coherent-state formulation of the path integrals for the discrete system whose dynamics is described by spin operators. This spin coherent-state formulation allows the discrete system to be mapped onto a continuous coordinate. Stationary approximations of the resulting coherent-state path integrals of the system plus bath lead to quasiclassical equations of motion which can be solved numerically by direct integration. This algorithm reduces the problem to a number of simple classical trajectory calculations and does not require calculating any fluctuation determinants.
Dynamics based on quantum trajectories with approximate quantum potential is generalized to nonadiabatic systems and its semiclassical properties are discussed. The formulation uses the mixed polar-coordinate space representation of a wave function. The polar part describes the overall time evolution of the wave-function components semiclassically using the single-surface approximate quantum potential. The coordinate part represents a complex``population'' amplitude, which in case of localized coupling can be solved for quantum mechanically in an efficient manner. In the high-energy regime this is accomplished by using a small basis determined by the coupling between surfaces. An illustration is given for a typical curve-crossing problem. The energy-resolved probabilities obtained from the time evolution of two wave packets for a wide range of energies are in excellent agreement with exact results for energies above the threshold of the diabatic reaction, including the case of total nonadiabatic transition.
A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.
We study the phase-space representation of dynamics of bosons in the semiclassical regime where the occupation number of the modes is large. To this end, we employ the van Vleck-Gutzwiller propagator to obtain an approximation for the Green's function of the Wigner distribution. The semiclassical analysis incorporates interference of classical paths and reduces to the truncated Wigner approximation (TWA) when the interference is ignored. Furthermore, we identify the Ehrenfest time after which the TWA fails. As a case study, we consider a single-mode quantum nonlinear oscillator, which displays collapse and revival of observables. We analytically show that the interference of classical paths leads to revivals, an effect that is not reproduced by the TWA or a perturbative analysis.
We show a general initial value form of the semiclassical propagator. Similar to cellular dynamics, this formulation involves only the nearby orbits approximation: the evolution of nearby orbits is approximated by linearized dynamics. This phase space smearing formulation keeps the accuracy of the original Van Vleck--Gutzwiller propagator. As an illustration, we present a simple initial value form of the semiclassical propagator. It is nonsingular everywhere and is efficient for numeric implementation.