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 non-idempotent one. The problem is absent for the original Kohn-Sham integro-differential equations or if a *strictly* complete basis set is assumed.
I. Mayer, I. Pápai, I. Bakó and Á. Nagy: J. Chem. Theory Comput., 13, 3961-3963 (2017) http://dx.doi.org/10.1021/acs.jctc.7b00562