  Exact exchange (EXX) Kohn-Sham calculations within an all-electron
full-potential method are performed on a range of semiconductors and insulators
(Ge, GaAs, CdS, Si, ZnS, C, BN, Ne, Ar, Kr and Xe). We find that the band-gaps
are not as close to experiment as those obtained from previous pseudopotential
EXX calculations. Full-potential band-gaps are also not significantly better
for $sp$ semiconductors than for insulators, as had been found for
pseudopotentials. The locations of $d$-band states, determined using the
full-potential EXX method, are in excellent agreement with experiment,
irrespective of whether these states are core, semi-core or valence. We
conclude that the inclusion of the core-valence interaction is necessary for
accurate determination of EXX Kohn-Sham band structures, indicating a possible
deficiency in pseudopotential calculations.
