One-to-one Correspondence of the Normalization and Coulomb Hole Sum Rules for Approximate Wave Functions.$^{1}$

For approximate wave functions, we prove the theorem that there is a one-to-one correspondence between the constraints of normalization, and of the Fermi-Coulomb and Coulomb hole sum rules. This correspondence is surprising because normalization depends on the probability of finding an electron at some position, whereas the Fermi-Coulomb/Coulomb hole sum rules depend on the probability of two electrons staying apart due to Pauli-Coulomb/Coulomb correlations. We demonstrate the theorem by example using wave function functionals${}^{2}$. The significance of the theorem for DFT lies in the fact that the extensively employed LYP correlation energy functional${}^{3}$ is based on a wave function (that of Colle-Salvetti${}^4$) which satisfies the Coulomb hole sum rule only approximately, and that wave function is therefore not normalized.\newline \newline 1 Supported by RF CUNY \\ 2 X.-Y. Pan\emph{ et al}, Phys. Rev. Lett. \textbf{93}, 130401 (2004) \\ 3 C. Lee \emph{et al}, Phys. Rev. B \textbf{37}, 785 (1988) \\ 4 R. Colle and O.Salvetti, Theor. Chim. Acta \textbf{37}, 329 (1975).