Conceptual Problem with Calculating Electron Densities in Finite Basis Density Functional Theory

It is discussed that finite basis Density Functional Theory (DFT) calculations using a single Kohn-Sham determinant cannot reproduce, in a strict mathematical sense, the exact electron density corresponding to the same finite basis. This is because the DFT density derives from an idempotent first order density matrix, while the exact (full CI) density can only be obtained from a nonidempotent one. The problem is absent for the original Kohn-Sham integro-differential equations or if a strictly complete basis set is assumed.