The possibilities to use Kohn-Sham density functional theory to
make accurate predictions of core photoelectron binding energies and
chemical shifts are explored through a series of calculations on
compounds of different size and type. The recent proposal
of an ``unrestricted generalized transition state'' (UGTS) method,
for this purpose is put to test versus
full separate state optimizations of the ground and core hole states,
that is a ``$\Delta$Kohn-Sham'' ($\Delta$KS) method. It is found that
while internal parametrizations in terms of grid and basis set
expansions can be well controlled there is still
a notable dependency of the absolute binding energies on the
choice of functional for both the UGTS and $\Delta$KS methods. As
for pure ab initio $\Delta$SCF the former method must still be viewed
as an approximation of the latter. Keeping these dependences in mind,
Kohn-Sham calculations seem to provide a promising tool for
predicting binding energies and chemical shifts of an accuracy that
approaches that of experiment.