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Chemical Physics

Derivation of semiclassical asymptotic initial value representations of the quantum propagator

Eva Martín-Fierro, Jose M. Gomez Llorente

Published Aug 4, 2005• 18 citations
Publisher
DOI
10.1016/j.chemphys.2005.07.022

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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.

The Journal of Chemical Physics•Aug 10, 2004•62 citations
An algebraic approach based on the multimode two-photon Lie algebra and its corresponding Lie group is followed to derive a formal solution to the time-dependent Schrödinger equation. This solution is written as an expansion series whose leading term corresponds to the thawed Gaussian approximation (TGA). Our scheme provides the most general expression reported so far for this approximation. By using the coherent state representation of the formal solution, the correction term to the TGA is analysed in the zero ℏ asymptotic limit. The error is generally found not to vanish in this semiclassical limit. The same approach is followed to study the remainder to the TGA initial value representation (IVR) of the quantum propagator. This correction is found not to vanish either in the zero ℏ limit. Hence, the TGA IVR would not be the correct semiclassical asymptotic form of the quantum propagator. The origin of this behaviour is shown to be in the existence of contributions from unphysical saddle points in the semiclassical limit. These would unveil an incorrect analytic structure of the TGA IVR propagator in that limit.
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.