Derivative discontinuity, band gap and LUMO orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schluter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of band gaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for band gap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital in describing electron removal. (2) Every term in the total energy functional contribute to the energy gap because of the discontinuity of the derivative of density/density matrix with respect to the number of electron, at intergers. (3) Consistent with the Perdew-Levy-Sham-Schluter conclusion that the exact Kohn- Sham energy gap differ from fundamental band gap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be a continuous explicit functional of the electron density, either local or nonloal. The last result is further strengthened, when we consider Mott insulator. There, the exact exchange-correlation functional needs to have an explicitly discontinuous dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electron