The Kohn-Sham gap, the fundamental gap and the optical gap: the physical meaning of occupied and virtual Kohn-Sham orbital energies

A number of consequences of the presence of the exchange-correlation hole potential in the Kohn-Sham potential are elucidated. One consequence is that the HOMO-LUMO orbital energy difference in the KS-DFT model (the KS gap) is not "underestimated" or even "wrong", but that it is physically expected to be an approximation to the excitation energy if electrons and holes are close, and numerically proves to be so rather accurately. It is physically not an approximation to the difference between ionization energy and electron affinity I - A (fundamental gap or chemical hardness) and also numerically differs considerably from this quantity. The KS virtual orbitals do not possess the notorious diffuseness of the Hartree-Fock virtual orbitals, they often describe excited states much more closely as simple orbital transitions. The Hartree-Fock model does yield an approximation to I - A as the HOMO-LUMO orbital energy difference (in Koopmans' frozen orbital approximation), if the anion is bound, which is often not the case. We stress the spurious nature of HF LUMOs if the orbital energy is positive. One may prefer Hartree-Fock, or mix Hartree-Fock and (approximate) KS operators to obtain a HOMO-LUMO gap as a Koopmans' approximation to I - A (in cases where A exists). That is a different one-electron model, which exists in its own right. But it is not an "improvement" of the KS model, it necessarily deteriorates the (approximate) excitation energy property of the KS gap in molecules, and deteriorates the good shape of the KS virtual orbitals.