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On the Singular Limit of the Quantum-Classical Molecular Dynamics Model

Please always quote using this URN: urn:nbn:de:0297-zib-2761
  • \noindent In molecular dynamics applications there is a growing interest in so-called {\em mixed quantum-classical} models. These models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of quantum mechanics. A particularly extensively used model, the QCMD model, consists of a {\em singularly perturbed}\/ Schrödinger equation nonlinearly coupled to a classical Newtonian equation of motion. This paper studies the singular limit of the QCMD model for finite dimensional Hilbert spaces. The main result states that this limit is given by the time-dependent Born-Oppenheimer model of quantum theory---provided the Hamiltonian under consideration has a smooth spectral decomposition. This result is strongly related to the {\em quantum adiabatic theorem}. The proof uses the method of {\em weak convergence} by directly discussing the density matrix instead of the wave functions. This technique avoids the discussion of highly oscillatory phases. On the other hand, the limit of the QCMD model is of a different nature if the spectral decomposition of the Hamiltonian happens not to be smooth. We will present a generic example for which the limit set is not a unique trajectory of a limit dynamical system but rather a {\em funnel} consisting of infinitely many trajectories.

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Author:Folkmar A. Bornemann, Christof Schütte
Document Type:ZIB-Report
Date of first Publication:1997/02/14
Series (Serial Number):ZIB-Report (SC-97-07)
Published in:Appeared in: SIAM J. Appl. Math. 59 (1999) 1208-1224
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