Semiclassical corrections to the Herman-Kluk propagator

Hochman, Gili; Kay, Kenneth G.

doi:10.1103/physreva.73.064102

Cited by 14 publications

(13 citation statements)

References 14 publications

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“…It is worthwhile to mention that the HK propagator is the lowest order term in a series expression for the full quantum mechanical propagator [71,72]. Considering higher order terms in this series, the description of (deep) barrier tunneling is within the reach of semiclassical IVR methods [73,74]. The numerical effort to determine the correction terms is quite formidable, however.…”

Section: A the Herman-kluk Propagatormentioning

confidence: 99%

Investigating quantum transport with an initial value representation of the semiclassical propagator

2009

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Quantized systems whose underlying classical dynamics possess an elaborate mixture of regular and chaotic motion can exhibit rather subtle long-time quantum transport phenomena. In a short wavelength regime where semiclassical theories are most relevant, such transport phenomena, being quintessentially interference based, are difficult to understand with the system's specific long-time classical dynamics. Fortunately, semiclassical methods applied to wave packet propagation can provide a natural approach to understanding the connections, even though they are known to break down progressively as time increases. This is due to the fact that some long-time transport properties can be deduced from intermediate-time behavior. Thus, these methods need only retain validity and be carried out on much shorter time scales than the transport phenomena themselves in order to be valuable. The initial value representation of the semiclassical propagator of Herman and Kluk [Chem. Phys. 91, 27 (1984)] is heavily used in a number of molecular and atomic physics contexts, and is of interest here. It is known to be increasingly challenging to implement as the underlying classical chaos strengthens, and we ask whether it is possible to implement it well enough to extract the kind of intermediate-time information that reflects wave packet localization at long times. Using a system of two coupled quartic oscillators, we focus on the localizing effects of transport barriers formed by stable and unstable manifolds in the chaotic sea and show that these effects can be captured with the Herman-Kluk propagator.

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Section: A the Herman-kluk Propagatormentioning

confidence: 99%

Investigating quantum transport with an initial value representation of the semiclassical propagator

2009

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“…͑2͒ to fluctuate irregularly. Similar behavior for the g t ͑n͒ , observed under analogous circumstances for a double-well system, 20 was taken as a symptom of divergence of the asymptotic series ͚ n ប n g t ͑n͒ , and was treated accordingly. However, in the present case, this divergence does not seem relevant.…”

Section: ͑0͒mentioning

confidence: 94%

“…We note that previous applications of the HOHK method have been restricted to systems with one degree of freedom. 8,20 The calculations presented here are the first applications of HOHK treatment to multidimensional systems.…”

Section: Tunneling In Two-dimensional Systems Using a Higher-order Hementioning

confidence: 99%

Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

Hochman

Kay

2009

The Journal of Chemical Physics

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A principal weakness of the Herman–Kluk (HK) semiclassical approximation is its failure to provide a reliably accurate description of tunneling between different classically allowed regions. It was previously shown that semiclassical corrections significantly improve the HK treatment of tunneling for the particular case of the one-dimensional Eckart system. Calculations presented here demonstrate that the lowest-order correction also substantially improves the HK description of tunneling across barriers in two-dimensional systems. Numerical convergence issues either do not arise or are easily overcome, so that the calculations require only a moderate number of ordinary, real, classical trajectories.

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“…Fifteen years later, a couple of papers in the chemical literature critically questioned the validity of the Herman-Kluk approximation, and it was Kay (2006) who developed the elaborate integration by parts performed in Theorem 5.3 to justify a semiclassical expansion in powers of the semiclassical parameter, which has the Herman-Kluk propagator as its leading-order term. The first correction term of this expansion was numerically explored in Hochman and Kay (2006). The first mathematically rigorous error estimates are due to Swart and Rousse (2009).…”

Section: Notesmentioning

confidence: 99%

Computing quantum dynamics in the semiclassical regime

Lasser

Lubich

2020

Acta Numerica

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The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

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Semiclassical corrections to the Herman-Kluk propagator

Cited by 14 publications

References 14 publications

Investigating quantum transport with an initial value representation of the semiclassical propagator

Investigating quantum transport with an initial value representation of the semiclassical propagator

Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

Computing quantum dynamics in the semiclassical regime

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