  We assessed three schemes for propagating a variable-width (thawed) Gaussian
wave packet moving under the influence of Morse or double-well potentials with
parameters that are chemically representative. The most rigorous scheme is
based on the time-dependent variational principle (TDVP); it leads to realistic
behaviour of the center and width of a wave packet in all investigated regimes.
Two other approximate schemes, Heller's and the extended semiclassical ones,
demonstrate various aberrations. Heller's scheme does not properly account for
various zero-point energy-related effects, is unable to predict tunneling, and
more importantly, exhibits completely nonphysical unbound width oscillations.
The extended semiclassical scheme, which was developed to address some of the
shortcomings of the Heller counterpart, demonstrates another unphysical
behaviour: self-trapping of a trajectory in both Morse and double-well
potentials. We conclude that only the TDVP-based scheme is suitable for
problem-free dynamical simulations. This, however, raises the question of how
to utilize it efficiently in high-dimensional systems.
