Band Gap Studies in Density Functional Theory*

We examine the exact properties of Density Functional Theory (DFT) in one dimension and compare it with the numerically exact answer provided by Density Matrix Renormalization Group. Using the exact answers, we can compare exact Density Functional Theory quantities against commonly used approximations. Approximations tend to underestimate the band gap of the material. We compare the exact DFT quantities with the approximations to explore the band gap problem and provide numerical proofs of principle.