The Laplacian of the intracule and extracule densities and their relationship to the shell structure of atoms

Abstract

The shell structure of atoms and the Laplacian of the charge density J. Chem. Phys. 88, 4375 (1988); 10.1063/1.454711The Laplacian of the charge density and its relationship to the shell structure of atoms and ions Numerical evidence of the monotonicity of both the intracule density I(r) and the extracule density E(R) in atoms are presented. However, it is found that their associated Laplacian fields V1E(R) and V 2 I(r) show a rich structure. Roots and extrema of these functions have been compared with the corresponding values of the Laplacian of the charge density V 2 p(r). Radii corresponding to minima in V 2 I(r) are found to match closely their corresponding values in V 2 p(r). Linear relationships between the inverses of the roots ofV 2 E(R) and V 2 I(r) and the nuclear charge Z are found. Calculations demonstrate that atomic shell structure is well resolved by the odd numbered zeros of V 2 E(R) and by the radiicorresponding to minima in V 2 I(r).

The relevance of the Laplacian of intracule and extracule density distributions for analyzing electron–electron interactions in molecules

“…1͑c͒ and 1͑d͒ is that the topology of the molecular ٌ 2 I(r) and ٌ 2 E(R) maps reveals the shell structure of atoms in the molecule, as previously noted by Sarasola et al 10 for isolated atoms. In particular, the two shells of carbon atoms can be clearly identified.…”

“…9,10 In this aspect, the recent description of a more efficient algorithm to compute I(r) and E(R) ͑Ref. 14͒ provides a way to scrutinize in more detail the topology of electron-pair densities in small molecules and to extend this kind of analyses to larger molecules.…”

“…So far, I(r) and E(R) calculations have been mainly performed on atomic systems [8][9][10] and small molecules.…”

The mapping of the local contributions of Fermi and Coulomb correlation into intracule and extracule density distributions

“…1,32,33 Moreover, atomic shell structure has also been discussed from the point of view of the Laplacians of the intracule and extracule densities. 34,35 Even though similar shell structures were found for ٌ 2 (r), ٌ 2 I(r), and ٌ 2 E(R), the interpretation is different in each case. 35 Table II collects the HF and CI electron-pair shell populations for the Ne and Ar atoms.…”

Second-order quantum similarity measures from intracule and extracule densities

“…Calculation of the respective Laplacians r 2 qr, r 2 sr, and r 2 iR revealed the typical pairs of spherical regions of alternating density concentration (negative values) and depletion (positive values) which re¯ect the shell structure of atoms [14,26]. Thus, for example, the shell structures revealed by r 2 qr and r 2 sr for the Neon atom can be visually compared in Fig.…”