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We review some salient points in the derivation of density functional theory (DFT) and of the local density approximation (LDA) of it. We then articulate an understanding of DFT and LDA that seems to be ignored in the literature. We note the well-established failures of many DFT and LDA calculations to reproduce the measured energy gaps of finite systems and band gaps of semiconductors and insulators. We then illustrate significant differences between the results from self consistent calculations using single trial basis sets and those from computations following the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). Unlike the former, the latter calculations verifiably attain the absolute minima of the occupied energies, as required by DFT. These minima are one of the reasons for the agreement between their results and corresponding, experimental ones for the band gap and a host of other properties. Further, we note predictions of DFT BZW-EF calculations that have been confirmed by experiment. Our subsequent description of the BZW-EF method ends with the application of the Rayleigh theorem in the selection, among the several calculations the method requires, of the one whose results have a full, physics content ascribed to DFT. This application of the Rayleigh theorem adds to or completes DFT, in practice, to preserve the physical content of unoccupied, low energy levels. Discussions, including implications of the method, and a short conclusion follow the description of the method. The successive augmentation of the basis set in the BZW-EF method, needed for the application of the Rayleigh theorem, is also necessary in the search for the absolute minima of the occupied energies, in practice

Understanding density functional theory (DFT) and completing it in practice