Bijectivity of the Normalization and Fermi-Coulomb Hole Sum Rules for Approximate Wave Functions

We prove the bijectivity of the constraints of normalization and of the Fermi-Coulomb hole charge sum rule at each electron position for approximate wave functions. This bijectivity is surprising in light of the fact that normalization depends upon the probability of finding an electron at some position, whereas the Fermi-Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion. We further demonstrate the bijectivity of these sum rules by example.