An examination of the shell structure of atoms and ions as revealed by the one-electron potential,

Abstract

The one-electron potential, [Formula: see text], appearing in the Schrödinger equation for the charge density is calculated for the neutral atoms from hydrogen to uranium, and the singly positive ions, from helium to barium and lutetium to radium. These computations, utilizing the nonrelativistic SCF wavefunctions of Clementi and Roetti and McLean and McLean, were performed in order to investigate the concept of shell structure as defined by this potential. [Formula: see text] exhibits a number of zeros and extrema corresponding to classically allowed and forbidden regions, with its topology being very similar in nature to that of [Formula: see text]. The significant difference is that [Formula: see text] displays all seven shells in the heavier elements of the periodic table, whereas [Formula: see text] displays only five shells; i.e. compared with [Formula: see text], [Formula: see text] has additional zeros and extrema at large distances corresponding to the two outer shells. The positions of these additional zeros and extrema, when plotted against atomic number Z, exhibit large deviations from the Bohr model of the hydrogen atom. The outermost zero-extremum is not displayed in the transition elements; i.e. where d-orbital filling is present. The model of classically allowed and forbidden regions is shown to provide an alternative definition of shell structure to that of the simple Bohr–Schrödinger theory of an atom. The odd-numbered zeros in [Formula: see text] provide a topological feature capable of representing all shells. Furthermore, these zeros appear at precisely the atomic number as the corresponding shell predicted by the aufbau principle.