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Accurate and efficient pre-exponential factor approximations for the semiclassical initial value representation propagator

Giovanni Di Liberto, Michele Ceotto

Published Jun 8, 2016• 0 citations
OpenAlex

Abstract

Overview• A brief remind about the nature of the semiclassical prefactor.• The problems of the semiclassical prefactor.• Overview of the existing approximations.• Derivation of a new class of prefactor approximations.• Numerical test on chaotic potentials.

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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.
A log-derivative formulation of the prefactor term appearing in the semiclassical Herman−Kluk propagator is presented. The resulting new expression is found in practice to avoid the branch cut problem which has hampered previous formulations. The enhanced performance of the log-derivative version of the prefactor has been confirmed by testing it on several one- and two-dimensional model problems. This log-derivative algorithm is also incorporated in the forward−backward initial value representation and applied to a model of the double-slit diffraction problem.
Three new developments are presented regarding the semiclassical coherent state propagator. First, we present a conceptually different derivation of Huber and Heller's method for identifying complex root trajectories and their equations of motion [D. Huber and E. J. Heller, J. Chem. Phys. 87, 5302 (1987)]. Our method proceeds directly from the time-dependent Schrödinger equation and therefore allows various generalizations of the formalism. Second, we obtain an analytic expression for the semiclassical coherent state propagator. We show that the prefactor can be expressed in a form that requires solving significantly fewer equations of motion than in alternative expressions. Third, the semiclassical coherent state propagator is used to formulate a final value representation of the time-dependent wavefunction that avoids the root search, eliminates problems with caustics and automatically includes interference. We present numerical results for the 1D Morse oscillator showing that the method may become an attractive alternative to existing semiclassical approaches.