Highest occupied orbital Kohn-Sham theory

It has been discovered that excitations from the highest occupied orbital are in agreement with differences of Kohn-Sham eigenvalues. See CJ Umrigar, A Savin, X Gonze. In JF Dobson, G Vignale, MP Das, eds. Electronic Density Functional Theory Recent Progress and New Directions. New York Plenum Press, 1998, pp 167-176. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

In Kohn-Sham density functional theory, the ionization potential is the negative of the eigenvalue of the highest occupied Kohn-Sham orbital. 86-88 The IP = —sH0M0 relation holds, however, only for the exact exchange-correlation potential. Numerical confirmations for this relation exist for model systems such as the... 

So the highest occupied Kohn-Sham orbital has a fractional occupation number to. The fact that is uniquely defined by follows directly from the Hohenberg-Kohn theorem applied to the non-interacting system. The proof of the Hohenberg-Kohn theorem for systems with noninteger number of electrons proceeds along the same lines as for systems with integer particle number (alternatively it can be obtained from the zero temperature limit of temperature dependent density functional theory [82]). We further split up v, as... 

The highest occupied Kohn-Sham orbital energy is equal to the exact first ionization energy. This is a property that is very desirable in qualitative MO theory in general and is often simply assumed in such theories. 

---