In molecules, it is well known that the Kohn-Sham HOMO-LUMO gap is much below the I -A difference. This is due to the fundamental difference that the Kohn-Sham system has an attractive potential due to the exchange-correlation hole of -1 electron also for the virtual levels, while the Hartree-Fock system lacks this attractive hole potential for the virtual levels. In the same way, the presence of this v hole xc potential lowers the LUMO level (bottom of the conduction band, BCB) in solids strongly. The exchangecorrelation hole in solids is pretty localized -at a given point r, its size is usually well within a unit-cell range around r and therefore its potential is strongly stabilizing. In a delocalized excitation, from an occupied Bloch state to an empty Bloch state, the excited electron does not benefit from this stabilization. Neither does an added electron -the excitation energy to this delocalized state is understandably close to the fundamental gap. So, physically we cannot expect the Kohn-Sham band gap to match approximately the fundamental gap or a delocalized excitation energy. Excitons in a solid (except for Frenkel excitons) typically have a large size, extending over many unit cells. They have excitation energies not much lower than the delocalized excitations, so also for them the attractive Kohn-Sham potential v hole xc does not fit reality. The situation is different in molecules since there the physical hole that the excited electron leaves behind is roughly mimicked by the attractive exchange-correlation hole in the Kohn-Sham potential. Hence the Kohn-Sham virtual-occupied orbital energy differences have the nice property that they do approximate excitation energies in molecules; see contribution (2.4.9).