SC 99-10 Christof Schuette.: Partial Wigner transforms and the quantum--classical
Liouville equation
Abstract: In molecular dynamics applications there is a growing
interest in mixed quantum-classical models. The article is concerned
with the so-called quantum-classical Liouville equation (QCL). This model
describes most atoms of the molecular system by the means of classical phase space
density but an important, small portion of the system by the means of a wavefunction.
The QCL results from the full quantum dynamical (QD) description via 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.
In most quantum-classical simulations the Born-Oppenheimer
model (BO) is used. In this model, the wavefunction is
adiabatically coupled to the classical motion which leads to serious
approximation deficiencies with respect to non-adiabatic effects in
the fully quantum dynamical description of the system. In contrast
to the BO model, the QCL model does include non-adiabatic processes,
e.g., transitions between the energy levels of the quantum system. It
is demonstrated that the QCL yields good
approximations of such non-adiabatic effects in full quantum
dynamics.
Keywords: QCMD,
quantum-classical Liouville equation,
surface hopping,
Wigner transform,
asymptotic expansion,
nonadiabatic effects
MSC: 81Q15, 81Q20