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The Journal of Chemical Physics

Optimization of the semiclassical initial value representation of the exact quantum-mechanical real time propagator

Shesheng Zhang, Eli Pollak

Published Nov 15, 2003• 52 citations
Publisher

Abstract

The recently derived exact representation of the quantum propagator in terms of semiclassical initial value representations (SCIVR) is used to optimize the width parameter in the SCIVR. Minimization of the expectation value of the correction operator related to the SCIVR leads to improved convergence of the representation. A test on a model one-dimensional double-well potential demonstrates that this optimization can give essentially exact results using only the first two terms in the SCIVR expansion of the exact propagator.

DOI
10.1063/1.1622931

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The recently derived exact representation of the quantum propagator in terms of semiclassical initial value representations (SCIVR) is used to optimize the width parameter in the SCIVR. Minimization of the expectation value of the correction operator related to the SCIVR leads to improved convergence of the representation. A test on a model one-dimensional double-well potential demonstrates that this optimization can give essentially exact results using only the first two terms in the SCIVR expansion of the exact propagator.
A new class of prefactor free semiclassical initial value representations (SCIVR) of the quantum propagator is presented. The derivation is based on the physically motivated demand, that on the average in phase space and in time, the propagator obey the exact quantum equation of motion. The resulting SCIVR series representation of the exact quantum propagator is also free of prefactors. When using a constant width parameter, the prefactor free SCIVR propagator is identical to the frozen Gaussian propagator of Heller [J. Chem. Phys. 75, 2923 (1981)]. A numerical study of the prefactor free SCIVR series is presented for scattering through a double slit potential, a system studied extensively previously by Gelabert et al. [J. Chem. Phys. 114, 2572 (2001)]. As a basis for comparison, the SCIVR series is also computed using the optimized Herman-Kluk SCIVR. We find that the sum of the zeroth order and the first order terms in the series suffice for an accurate determination of the diffraction pattern. The same exercise, but using the prefactor free propagator series needs also the second order term in the series, however the numerical effort is not greater than that needed for the Herman-Kluk propagator, since one does not need to compute the monodromy matrix elements at each point in time. The numerical advantage of the prefactor free propagator grows with increasing dimensionality of the problem.
The forward-backward (FB) approximation as applied to semiclassical initial value representations (SCIVR's) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FB-SCIVR has proven elusive. In this paper, we provide a new derivation of a forward-backward phase space SCIVR expression (FBPS-SCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPS-SCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPS-SCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPS-SCIVR term already provides an excellent approximation.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.
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.