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The Journal of Chemical Physics

Locally coupled coherent states and Herman–Kluk dynamics

Mark Sheard Child, Dmitrii V. Shalashilin

Published Jan 16, 2003• 36 citations
Publisher

Abstract

An exact analysis of coupled coherent state (CCS) theory in the moving locally quadratic Hamiltonian approximation is shown to reproduce both the linearized coherent state matrix element of the Herman–Kluk propagator and the coherent state overlap with Heller’s thawed Gaussian wave function. The derivation is applicable to anharmonic as well as harmonic systems, because the quadratic approximation is taken to apply only in the vicinity of a particular classical trajectory. New compact expressions for the linearized Herman–Kluk coherent state matrix element are given, and improvements for the practical application of CCS theory are discussed.

DOI
10.1063/1.1531997

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An exact analysis of coupled coherent state (CCS) theory in the moving locally quadratic Hamiltonian approximation is shown to reproduce both the linearized coherent state matrix element of the Herman–Kluk propagator and the coherent state overlap with Heller’s thawed Gaussian wave function. The derivation is applicable to anharmonic as well as harmonic systems, because the quadratic approximation is taken to apply only in the vicinity of a particular classical trajectory. New compact expressions for the linearized Herman–Kluk coherent state matrix element are given, and improvements for the practical application of CCS theory are discussed.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
Abstract A wavepacket spreading under quadratic potentials is shown to have an exact classical analogue in the spreading of a classical ensemble taken from the Wigner distribution. The wavepacket motion under anharmonic potentials is complicated by greater spreading combined with additional interferences. Both are shown to be well handled by Heller’s frozen Gaussian swarm technique. Applications are outlined to spectral quantization and Franck–Condon transitions by Fourier analysis of the time correlation function. The chapter concludes with an introduction to the popular unitary Herman–Kluk coherent state propagation scheme. Advantages over semiclassical Van Vleck propagation are discussed, and a Filinov–based scheme for handling weakly chaotic systems is described.
A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.
The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.
This paper presents a semiclassical theory for the computation of matrix elements of the type 〈u‖v〉 when either one or both ‖u〉 and ‖v〉 are coherent states (in different representations). Our results can be considered as an extension of Miller’s semiclassical theory [Adv. Chem. Phys. 25, 69 (1974)]. Such an extension has been presented also by Heller [J. Chem. Phys. 66, 5777 (1977)]. We were able to simplify considerably some of Heller’s results by exploiting the canonical properties of the classical coherent variables. This enabled us to relate the elements 〈u‖v〉 to certain generalized, complex generator functions in a manner that is very similar to the relations that appear in the original Miller’s theory. The advantages that are inherent in the coherent states representation are illustrated in a few elementary examples. We were able to derive an excellent approximation to the eigenstates of the harmonic oscillator which is valid even for the ground state. Furthermore, we have demonstrated that it is possible to describe semiclassically dynamical processes that are classically forbidden by real time trajectories in a certain generalized phase space.
The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the ``time-variable'' exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller's thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.