  We revisit recent developments in the theoretical foundations of
time-dependent density functional theory (TDDFT). TDDFT is then applied to the
calculation of total cross sections for ionization processes in the
antiproton-Helium collision system. The Kohn-Sham potential is approximated as
the sum of the Hartree-exchange potential and a correlation potential that was
proposed in the context of laser-induced ionization. Furthermore, some
approaches to the problem of calculating the ionization probabilities from the
density are discussed. Small projectile energies below 5keV are considered as
well as those in the range from 5 to 1000 keV. Results are compared with former
calculations and with experimental data. We find that the correlation potential
yields no obvious improvement of the results over the exchange-only
approximation where the correlation potential is neglected. Furthermore, we
find the problem of calculating the desired observables crucial, introducing
errors of at least the same order of magnitude as the correlation potential.
For the case of small energies we find that trajectory effects play an
important role: the ionization cross sections are enlarged significantly if
curved instead of straight-line trajectories are used for the projectile
motion.
