The Physical Content of Eigenvalues from Density Functional Theory (DFT)

The density functional theory (DFT) of Hohenberg and Kohn rests on the energy functional E$_{\mathrm{v}}$[n] assuming its minimum for the \textit{correct density} n(\textbf{r}), with the admissible functions restricted by the condition$N\left[ n \right]=\int {n\left( r \right)} dr=N$, where N is the number of particles in the system under study. We show that, for such a system, there is an infinite number of basis sets (of localized orbitals) for which N is fixed while the density is not necessarily the correct one. Consequently, the eigenvalues obtained with self consistent DFT calculations using a single basis set do not necessarily have any particular physical content. The physical content is ensured only by the search and utilization of \textit{the optimal basis set} that yields \textit{the minima of the occupied energies} and physically meaningful values of low laying unoccupied energies. Further, by virtue of the Rayleigh theorem, there exist many basis sets larger than the optimal one [and that contain it] for which some unoccupied energies are lowered on account of a mathematical artifact. We illustrate these points in the cases of ZnO, TiO$_{2}$, and SrTiO$_{3}$. The calculated band gaps and other properties of these materials are in excellent agreement with experiment. Work funded by in part by the National Science Foundation, through LASiGMA [NSF AwardEPS-1003897, No. NSF (2010-15)-RII-SUBR, and No. HRD-1002541], LONI [Award No. 2-10915], and the Louisiana Space Consortium (LaSPACE).