Non-Nuclear maxima and the universality of Bright Wilson's justification of the first Hohenberg Kohn theorem revisited

Highlights: • The radial derivative of spherically-averaged density is shown, analytically, to vanish everywhere except at atomic nuclei. • The Kato condition yields zero-charge for non-nuclear attractors/maxima (NNMs/NNAs). • The Bright Wilson justification of the 1st Hohenberg-Kohn Theorem applies to densities with NNMs/NNAs. • Kato's condition cannot be used to determine the atomic number in excited states. The Kato cusp condition involves a spherically-averaged derivative of the electron density with respect to distance from a given nucleus weighted by the electron density at that nucleus. This "Kato ratio" is shown analytically to vanish at all points of the electron density (and of the underlying external potential) that do not exhibit a singularity including non-nuclear maxima/attractors (NNM/NNAs). Using prototypical test cases, the analytical result is confirmed numerically. This result, while quite simply demonstrated and possibly tacitly known, could not be located in the literature to the best of the authors' efforts. Since the Kato ratio is zero at a NNM/NNA, this saves the DFT plausibility argument of E. Bright Wilson. The consideration of these ideas in excited electronic states is discussed.