Density Functional Theory (DFT) does not predict the correct band gap (E$_{g})$ of the materials. Which exchange and correlation functional predicts the E$_{g}$ value exactly?
What about the HOMO-LUMO gap? Can we use HOMO-LUMO gap (for non-periodic structures) instead of the gap observed in the density of states and band structures?
Note: Density Functional Theory is, in principle, an exact theory. However, DFT needs an important ingredient known as "exchange and correlation (XC) functional". Since the exact XC functional is still unknown, it has to be approximated. It is often said that the failure to predict properties accurately is because of the use of approximate XC functional.