The Herman-Kluk approximation: Derivation and semiclassical corrections

Kenneth G. Kay

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The Herman-Kluk (HK) approximation for the propagator is derived semiclassically for a multidimensional system as an asymptotic solution of the Schrödinger equation. The propagator is obtained in the form of an expansion in h, in which the lowest-order term is the HK formula. Thus, the result extends the HK approximation to higher orders in h. Examination of the various terms shows that the expansion is a uniform asymptotic series and establishes the HK formula as a uniform semiclassical approximation. Successive terms in the series should allow one to improve the accuracy of the HK approximation for small h in a systematic and purely semiclassical manner, analogous to a higher-order WKB treatment of time-independent wave functions.

Original languageEnglish
Pages (from-to)3-12
Number of pages10
JournalChemical Physics
Issue number1-2
StatePublished - 6 Mar 2006

Bibliographical note

Funding Information:
This work was supported by the Israel Science Foundation (Grant No. 85/03).


  • Herman-Kluk theory
  • Semiclassical approximations
  • Time-dependent propagator


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