The attractive properties of the exact Kohn-Sham orbitals and orbital energies have been expounded in some contributions; see contributions (2.4.9), (2.2.4), (2.2.13), and (2.2.11). A salient feature of the exact Kohn-Sham model is that the LUMO is not at -A (given that the HOMO is at -I) but much lower: the HOMO-LUMO gap is approximately equal to the first excitation energy. It should be made clear that contribution (2.2.15) does not contradict these properties of the exact Kohn-Sham model. It refers to a different family of Kohn-Sham models, usually called the generalized Kohn-Sham models. These generalized models make it possible to include, for instance, part of the exchange operator (a nonlocal potential) of the Hartree-Fock model and adjust the local part of the potential so that the density remains exact and adjust the exchange-correlation functional so that the energy also remains exact. In such a scheme, the orbital energies are different from those generated by the exact local Kohn-Sham potential. In such a generalized Kohn-Sham model, one may strive to obtain that the HOMO is again at -I and that the LUMO is now at -A, as is also done in the Koopmans-compliant functionals. The LUMO then becomes more diffuse and one loses the simple representation of excitations in TDDFT with just one or a few orbital transitions (2.2.17 and) on the physical meaning of the HOMO and LUMO in DFT, it is important to separate the two types of one-electron Kohn-Sham Hamiltonians. The first one is from the ground-state calculation with a given DFA E DFA xc , which yields the density, orbitals and orbital energies of the noninteracting reference system, as developed in the original Kohn-Sham paper. This is called the direct approach. The second one is from an inverse calculation, generating the local potential v σ s (r) that reproduces a given ground-state electron density, which can be the exact density or an accurate density from high-level calculations. We called this the inverse Kohn-Sham or inverse OEP approach , the potential so obtained is also called the "exact Kohn-Sham" potential by Baerends in contribution (2.2.16).