Pseudoconvexity of the atomic electron density: Lower and upper bounds

The curvature of the spherically averaged ground-state electron density of atoms is the subject of this study. We show that strict pseudoconvexity is the general structural property of the density of atoms which embodies all the structural features that have been empirically attributed to ρ(r). Rigorous upper and lower bounds for the second derivative of ρ(r) at the nucleus and at large distances are obtained. Within the bare Coulomb-field model atom, it is shown that the total electron density of the lowest set of closed shells is convex. Rigorous cusp conditions for the second derivative of the model atom density are derived and shown to be obeyed by model atoms with closed shells. Upper bounds that constrain the curvature of ρ(r) at the nucleus of open-shell model atoms are also developed. The Kohn-Sham formalism is used to obtain a general expression for the second derivative of ρ(r) at the nucleus