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Physical Review A

Semiclassical corrections to the Herman-Kluk propagator

Gili Hochman, Kenneth G. Kay

Published Jun 27, 2006• 14 citations
Publisher

Abstract

The approximate expression for the propagator of Herman and Kluk is systematically improved by application of semiclassical corrections for a double-well system. The results confirm the semiclassical nature of the Herman-Kluk approximation. The influence of caustics on the effectiveness of the corrections is discussed.
DOI
10.1103/physreva.73.064102

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Comment on Semiclassical approximations in phase space with coherent states

Frank Großmann, Michael F. Herman

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.

Journal of Physics A Mathematical and General•Oct 21, 2002•38 citations

On the derivation of the Herman–Kluk propagator

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Following an earlier derivation by Miller, we give a concise derivation of the Herman–Kluk propagator that fully exploits the symplectic structure of the problem. We also show directly that the version of the thawed Gaussian semiclassical propagator obtained by Baranger and co-workers is equivalent to the linearized Herman–Kluk propagator.

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Tunneling by a semiclassical initial value method with higher order corrections

Gili Hochman, Kenneth G. Kay

Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ 1/3 , including those that are O (ℏ 0 ) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.

Journal of Physics A Mathematical and Theoretical•Aug 22, 2008•11 citations

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

Michel Baranger, Marcus A. M. de Aguiar, F Keck +2 more

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.

Journal of Physics A Mathematical and General•Oct 21, 2002•29 citations

Comment on Semiclassical approximations in phase space with coherent states

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

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.

Journal of Physics A Mathematical and General•Sep 2, 2003•12 citations

Integral Representation of Heisenberg Operators and the Semiclassical Approximation

Jiushu Shao

Abstract A simplified formulation of the semiclassical Heisenberg operator based on the forward‐backward evolution methodology is worked out. A new representation related to the Husimi form is suggested to remove its problem, which may also offer a new way to improve the Herman‐Kluk semiclassical propagator.

Journal of the Chinese Chemical Society•Jun 1, 2003•1 citation
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.
Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ 1/3 , including those that are O (ℏ 0 ) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.
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.
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.