Core ionization in systems with several equivalent atoms gives rise to a set of near-degenerate core-hole states each associated with the removal of an electron from one of the delocalized orbitals. The energy splitting between the core-hole states is the intrinsic feature of core ionized systems and should not be neglected. The conventional equivalent core model (ECM) predicts strictly degenerate core-hole states because the core-hole is thought of as completely localized on one of the equivalent centers. This failure as well as several others inherent to the ECM are successfully removed in the framework of the corrected ECM presented in this paper. Two approaches are available in accordance with the two representations of the core-hole, either delocalized or localized. The Z+1 approximation is an excellent starting point in the localized representation. It must be modified, however, when the delocalized picture is used. To this end we introduce a (Z+Q) system where Q is the magnitude of point charges added to the nuclear charges of equivalent atoms which share the loss of an electron upon core ionization. Systematic corrections improving the conventional ECM are obtained by establishing a connection between the ECM and core-hole Hamiltonians. The core-hole Hamiltonians, being ideally suited for description of core ionization, are derived both in the delocalized and localized representations. Numerical results are presented for N-2. (C) 2003 American Institute of Physics.