Hartree-Fock (HF) theory makes the prediction that for neutral atoms the chemical potential (μ) is equal to minus the ionization potential (I). This has led us to inquire whether this intimate relation is sensitive to electron correlation. We present here therefore some discussion of the predictions for neutral atoms and atomic ions, and some homonuclear diatomic molecules. An account of fairly recent progress in obtaining the HF ionization potentials for the isoelectronic series of He, Be, Ne, Mg, and Ar-like atomic ions is first considered. The inline image expansion for total non-relativistic energy of atomic ions evokes that inline image is not very sensitive to the introduction of electron correlation. The connection between μ and I for neutral atoms via the Pauli potential (VP) is then examined. We focus on the relation of VP to more recent advances in density functional theory (DFT) plus low-order density matrix theory. In this context, the example of nonrelativistic Be-like atomic ions is treated. Afterward, we introduce the bosonized equation for the density amplitude inline image, which emphasizes the major role that plays inline image in DFT. For spherical atomic densities, the bosonized potential argument strongly suggests also that inline image remains valid in the presence of electron correlation. Finally, numerical estimates of μ and I from natural orbital functional (NOF) theory are presented for neutral atoms ranging from H to Kr. The predicted vertical I by means of the extended Koopmans' theorem are in good agreement with the corresponding experimental data. However, the NOF theory of μ lowers the experimental values considerably as we approach to noble gas atoms though oscillatory behavior is in evidence.