SC 99-11 Burkhard Schmidt, Petra Zdanska: Solution of the Time-Dependent Schroedinger Equation for
Highly Symmetric Potentials
Abstract: A method to solve the time-dependent Schrödinger
equation for a
highly symmetric potential energy surface is developed.
The angular
dependence of the quantum-mechanical wavepacket to be
propagated is
expanded in spherical harmonics where the number of
close-coupled
equations for the corresponding radial functions can be
efficiently
reduced by symmetry adaption of the rotational basis.
Various
techniques to generate symmetry adapted spherical
harmonics (SASHs)
for the point groups of highest symmetry (octahedral,
icosahedral) are
discussed. Numerical instabilities occur for high angular
momentum
states when applying the standard technique based on
transformations
of spherical harmonics. Two methods to circumvent
numerical
instabilities occuring for the standard technique are
suggested. The first is a recursive algorithm to generate
high
order SASHs from lower order ones. The second is a
numerical scheme
based on Gaussian quadratures which yields exact and
stable results
for a modest number of quadrature points. The technique of
symmetry
adapted wavepackets is applied to the photodissociation
quantum
dynamics of diatomic hydrogen-containing molecules
embedded in rare
gas clusters. Upon excitation of the molecule to a
repulsive state,
the hydrogenic wavepacket diffuses into the symmetric cage
of the rare
gas cluster.
Keywords: Schroedinger Equation,
quantum mechanics,
symmetry,
group theory
PACS: 03.65.F, 31.15.H