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

Abstract

. 66, 1005 (1988).The one-electron potential, v 2 G ( r ) / 2 g P ( r ) , appearing in the Schrodinger 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 Cle~nenti and Roetti and McLean and McLean, were performed in order to investigate the concept of shell structure as defined by this potential. v2%$(r)/2*(r) 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 V2p(r). The significant difference is that v 2 V $ ( r ) / 2 f l ( r ) displays all seven shells in the heavier elements of the periodic table, whereas V 2 p ( r ) displays only five shells; i.e. compared with V 2 p ( r ) , v ' G -( r ) / 2 G ( r ) 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 zeroextremum 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-Schrodinger theory of an atom. The odd-numbered zeros in v 2 V $ ( r ) / 2 f l p ( r ) 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. Introduction Density functional theory (1) offers the possibility of reducing the N-electron problem into a more viable one-electron one. However, there are difficulties that are encountered such as N-representability, i.e. to guarantee that a computed density is representative of an ensemble of N fermions (2). In the case of the wavefunction, this N-representability is achieved by making the wavefunction antisymmetric with respect to exchange of coordinates of any two of the N indistinguishable fermions.The Schrodinger equation for the electron density as proposed by Hunter (3) is of interest since this equation is an "exact dynamical model of the motion of a single electron in a many-electron system" (3). Thus, this particular Schrodinger equation is useful for the interpretation of molecular and atomic structure based on the potential,1 {U(r) -E} = v 2 G ( r ) / 2 6 ( r ) , within which any one electron moves.

A benchmark for the non-covalent interaction (NCI) index or… is it really all in the geometry?

“…This means that the topology of s recovers all AIM critical points, but also those related to shells 7 (like the One Electron Potential or OEP 10,11 ) and to very weak interactions (for which ∇ 2 ρ( r) >0). The latter cases are usually related to very weak intramolecular interactions, where the balance between geometric constrains and the weak interaction leads to an absence of critical point.…”

Atomic shells from the electron localizability in momentum space

“…It probably started with Bartell and Brockway [1], who reported peaks in the radial charge density. Different functions have been used since then in order to resolve, at least partially, the shell structure of atoms in position space [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

Occupation numbers for atomic shells in direct space bounded by the maxima of the one‐electron potential

“…The topological properties of the OEP for atoms were extensively analyzed by Sagar et al 13. Using the idea of classically allowed and forbidden regions, where an atomic shell may be defined as successive negative and positive OEP intervals (this definition is somewhat more restrictive than the one adopted in the present study, not allowing for shells solely located in classically forbidden regions), they concluded that the odd‐numbered OEP zeros are best suitable to indicate the atomic shells.…”

“…The picture of atomic shells successively filled up by electrons is not reflected by this simple distribution. Recently, the analysis of some specific properties derived from the electron density revealed, at least partially, the atomic shell structure 4–10, 12–16. However, only the average local electrostatic potential 17 and the electron localization function (ELF) 18, 19, as well as ELF modified by the incorporation of the electrostatic potential 20 were able not only to resolve the atomic shell structure, but also to yield reasonable electron numbers for the shell occupations.…”