Pseudoconvexity of the atomic electron density: A numerical study

The curvature, '' (r), of the atomic electron density (r) is studied using results from a bare-Coulomb-field (BCF) model, Hartree-Fock (HF), and configuration-interaction (CI) calculations. A region of nonconvexity in (r), previously reported by Angulo, Dehesa, and G\'alvez Phys. Rev. A 42, 641 (1990) for light atoms in a Hartree-Fock framework, is investigated for all atoms up to Z=92 and is found not to be an artifact of the basis set or the HF model. Numerical results for the BCF model show that the total electron density of an arbitrary number of closed shells is convex. However, for the same model with electrons filling orbitals according to Stoner's restriction we find that nonconvexity of the density is a periodic property appearing around closed-shell ground-state hydrogenic configurations. Cusp conditions, reported earlier by Esquivel et al. Phys. Rev. A 47, 936 (1993) for the second derivative of the BCF density are verified for model atoms with s and p subshells. Using wave functions of near-HF accuracy we have found a region of nonconvexity in (r) for atoms with Z=3--6, 16--32, and 45--92. Highly correlated densities of CI and Hylleraas-type quality for atoms of Li and Be isoelectronic sequences show that the nonconvex region of (r) is largely unaffected by the inclusion of electron correlation. These results, coupled with those from the BCF model, lead us to suggest that it is the bare Coulomb field of the nucleus that is mainly responsible for the appearance of nonconvex regions in atoms. Furthermore, the degree of nonconvexity is shown to decrease as Z increases along the isoelectronic series. The contributions of different spin densities to the nonconvex electron densities is also studied. Finally, the behavior of the curvature of the electron density far from the nucleus is investigated. The ratio '' (r) / (r) is found to approach an asymptotic value from above or below, according to the magnitude of the ionization potential.