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The Journal of Chemical Physics

Complex trajectory method in semiclassical propagation of wave packets

Marc Boiron, M. Lombardi

Published Mar 1, 1998• 32 citations
Publisher

Abstract

We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.

DOI
10.1063/1.475743

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The Journal of Chemical Physics•Jul 2, 2015•8 citations
We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.
The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
Path-integral derivations are presented for two recently developed complex trajectory techniques for the propagation of wave packets: complex WKB and Bohmian mechanics with complex action (BOMCA). The complex WKB technique is derived using a standard saddle-point approximation of the path integral, but taking into account the $\ensuremath{\hbar}$ dependence of both the amplitude and the phase of the initial wave function, thus giving rise to the need for complex classical trajectories. The BOMCA technique is derived using a modification of the saddle-point technique, in which the path integral is approximated by expanding around a near-classical path, chosen so that up to some predetermined order there is no need to add any correction terms to the leading-order approximation. Both complex WKB and BOMCA techniques give the same leading-order approximation; in the complex WKB technique higher accuracy is achieved by adding correction terms, while in the BOMCA technique no additional terms are ever added: higher accuracy is achieved by changing the path along which the original approximation is computed. The path-integral derivation of the methods explains the need to incorporate contributions from more than one trajectory, as observed in previous numerical work. On the other hand, it emerges that the methods provide efficient schemes for computing the higher-order terms in the asymptotic evaluation of path integrals. The understanding we develop of the BOMCA technique suggests that there should exist near-classical trajectories that give exact quantum dynamical results when used in the computation of the path integral keeping just the leading-order term. We also apply our path-integral techniques to give a compact derivation of the semiclassical approximation to the coherent-state propagator.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.
Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.