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Journal of Physics A Mathematical and General

Comment on Semiclassical approximations in phase space with coherent states

Michel Baranger, Marcus A. M. de Aguiar, H. J. Korsch

Published Sep 2, 2003• 12 citations
Publisher

Abstract

In two recent papers (Baranger et al 2001 J. Phys. A: Math. Gen. 34 7227; 2002 J. Phys. A: Math. Gen. 35 9493) co-authored by us, we mentioned the work of Dr Kenneth G Kay (Kay 1994 J. Chem. Phys. 100(6) 4377; 100 4432) in an unfavourable light. In this comment we correct this impression, as it was based on a misunderstanding of his work. The point at issue has been the derivation of the Herman–Kluk formula (Herman and Kluk 1984 Chem. Phys. 91 27). We find Kay's derivation to be the most solid at the moment.

DOI
10.1088/0305-4470/36/37/314

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Reply to Comment on Semiclassical approximations in phase space with coherent states

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The paper of Ünal [J. Math. Phys. 59, 062104 (2018)], though worthy of attention, contains a conclusion that is in error and may mislead the efforts to extend his results. The aim of the present note is twofold: we provide a correction to such a conclusion and then we emphasize some missing points that are necessary to clarify the content of the paper.

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Generalization of the coherent-state path integrals and systematic derivation of semiclassical propagators

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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.

Physical Review A•Mar 21, 2011•6 citations
In two recent papers (Baranger et al 2001 J. Phys. A: Math. Gen. 34 7227; 2002 J. Phys. A: Math. Gen. 35 9493) co-authored by us, we mentioned the work of Dr Kenneth G Kay (Kay 1994 J. Chem. Phys. 100(6) 4377; 100 4432) in an unfavourable light. In this comment we correct this impression, as it was based on a misunderstanding of his work. The point at issue has been the derivation of the Herman–Kluk formula (Herman and Kluk 1984 Chem. Phys. 91 27). We find Kay's derivation to be the most solid at the moment.
The Herman–Kluk (HK) formula was shown in (Baranger et al J. Phys. A: Math. Gen. 34 7227 ) not to be a correct semiclassical limit of an exact quantum mechanical formula. Two previous attempts to derive it using semiclassical arguments contain serious errors. These statements are left totally untouched by Herman and Grossmann's comment. They argue that the formula which we found to be at fault is not the one that should be called the HK formula. However, the formula we criticized is definitely one of the steps, in fact the main step, in these two published derivations of the HK formula. Very recently, a new derivation was published by Miller. It is interesting, but it is not semiclassical.
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.
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.