In Frank Jensen Introduction to Computational Chemistry from 2007, I stumbled upon this paragraph about whether to assign meaning to the KS orbitals or not
Another difference is that the unoccupied orbital energies in Hartree–Fock theory are determined in the field of N electrons and therefore correspond to adding an electron, i.e. the electron affinity. The virtual orbitals in density functional theory, on the other hand, are determined in the field of N − 1 electrons and therefore correspond to exciting an electron, i.e. unoccupied orbitals in DFT tend to be significantly lower in energy than the corresponding HF ones, and the highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gaps are therefore much smaller with DFT methods than for HF. This also means that orbital energy differences in DFT are reasonable estimates of excitation energies, in contrast to HF methods where excitation energies involve additional Coulomb and exchange integrals.
I can't seem to understand why the Kohn-Sham equations are solved in the presence of N-1 electrons and not N. From my understanding the Kohn Sham and the Hartree Fock equations are almost identical. Where does this difference come from?