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Notes on DFT.. Pierre Hohenberg and Walter Kohn introduced density functional theory in 1964. A year later, Kohn and Sham provided a practical, local density approximation (LDA) of the theory. Density functional theory (DFT) is based on two theorems.


 * The first theorem states that the external potential acting of an enclosed system of electrons is a functional of the electron density - except for an additive constant.
 * The second theorem asserts that the energy content of the Hamiltonian of the system reaches its minimum if and only if the charge density is that of the ground state. In other words, the energy content of the Hamiltonian reaches its absolute minimum, i.e., the ground state, when the charge density is that of the ground state.

While the first theorem has been totally understood in the literature, there remain some questions relative to the second one. Indeed, current practice of calculations using DFT potentials has resulted in a recalcitrant disagreement between the calculated band gaps of semiconductors and insulators and the corresponding, experimental values. While this disagreement is ascribed to DFT, it appears that DFT is not at fault, rather, the computational methods or approaches seem to bear the blame.

When a calculation is performed with a basis set comprising N functions, after the iterations, the energy content of the Hamiltonian reaches a minimum. One should resist the temptation of claiming that minimum energy to be the correct answer. The reason for that stems from the fact that the same calculation performed with (N+1) functions generally leads to eigenvalues that are lower than (or equal to) corresponding ones obtained with N functions. In short, the minimum energy resulting from a calculation is relative to the basis set employed. The sum of the occupied eigenvalues is the energy content of the Hamiltonian. So, to find he absolute, lowest energy content of the Hamiltonian, it is necessary to perform successive calculations where, except for the first one, the basis set of a calculation is that of the preceding one plus one orbital - up to the point where three successive calculations produce the same occupied energies, i.e., the ground state. The first of these three calculations (with the smallest of the three basis sets) is the one providing the DFT solution. While the two other calculations do not change the occupied energies, they always lower some unoccupied energies from their corresponding values obtained with the first calculation. This lowering of unoccupied energies while the occupied ones do not change is an artifact of the Rayleigh theorem for eigenvalues. And this lowering is plausible explanation of the wide spread underestimation of band gaps by DFT calculations employing a single basis set that is generally taken to be quite large - in order to ensure completeness. It appears that such large basis sets are over-complete for description of the ground state, i.e., the absolute minimum of the energy content of the Hamiltonian. The referenced underestimation of band gaps is extensively discussed in review articles on DFT.