For an N-electron system, a Kohn-Sham calculation with an exchange-correlation functional that is an explicit and continuous functional of the electron density leads directly to E v (N -1) and E v (N + 1), the ground-state energies of the corresponding (N -1) and (N + 1) electron systems. Similar connections follow for a generalized Kohn-Sham calculation with an exchangecorrelation functional that is an explicit and continuous functional of the noninteracting reference density matrix. This is true because of the following: (1) it has been proved that the HOMO/LUMO energy is the chemical potential for electron removal/addition, (See Table ) (2) the PPLB condition shows that the chemical potential of the N-electron system is -I and -A. Thus the band gap can be predicted from the HOMO-LUMO gap, in either Kohn-Sham calculations with an explicit functional of electron density or generalized Kohn-Sham calculations with an explicit functional of the noninteracting reference density matrix. This connection is independent of the functional approximation. However, the accuracy of the prediction depends on the quality of the functional used. For functionals with minimal delocalization error, the prediction is comparable to, or better than, that of GW approaches. Similarly to the access to the ground state information of the corresponding (N -1) and (N + 1) electron systems, it has been argued recently that ε (N), the orbital energies of orbitals above LUMO and below HOMO also approximate the corresponding quasiparticle energies ω +/-(N) as follows: