  We develop and test an ab initio time-dependent Basis Function (tBF) method
to solve non-perturbative and time-dependent problems in quantum mechanics. For
our test problem, we apply this method to the Coulomb excitation of the
deuteron by an impinging heavy ion. In the tBF method applied to deuterium, we
employ wave functions for its bound and excited states to calculate its
transition probabilities and the r.m.s. radius during the scattering process.
For comparison, corresponding results based on first-order perturbation theory
are also provided. For the Coulomb excitation process in a weak and
time-varying Coulomb field, where higher-order effects are negligible, we
obtain good agreement of the results based on these two methods. The tBF method
is then applied to the Coulomb excitation process with stronger external field.
The higher-order effects, such as those appearing in the reorientation of the
polarization of the deuteron system, are analyzed.
