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Partial Wigner Transforms and the Quantum--Classical Liouville Equation

Please always quote using this URN: urn:nbn:de:0297-zib-3983
  • In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The {\em quantum-classical Liouville equation} (QCL) describes most atoms of the molecular system under consideration by means of classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the classical part'' of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a first order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of {\em non-adiabatic quantum effects,} especially near so-called {\em avoided crossings} where most quantum-classical models fail.

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Author:Christof Schütte
Document Type:ZIB-Report
Tag:QCMD; Wigner transform; asymptotic expansion; nonadiabatic effects; quantum-classical Liouville equation; surface hopping
MSC-Classification:81-XX QUANTUM THEORY / 81Qxx General mathematical topics and methods in quantum theory / 81Q15 Perturbation theories for operators and differential equations
81-XX QUANTUM THEORY / 81Qxx General mathematical topics and methods in quantum theory / 81Q20 Semiclassical techniques, including WKB and Maslov methods
Date of first Publication:1999/04/12
Series (Serial Number):ZIB-Report (SC-99-10)
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