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.
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