Atomic shell structure in Hartree—Fock theory

Average local electrostatic potential and the core‐valency separation in atoms

“…Smith and co-workers [17] identified the extrema in the double logarithmic derivative function, d In p / d In r, with the shell structure. In the outer region of the radial distances, such extrema have been found to coincide with the inflection points of the log/log plots in [14].…”

“…Unfortunately, the radial density distribution function D ( r ) = 47ru%(u) does not express the outermost shell as a distinct maximum for several atoms beyond Ar. Using the outer inflection points in D ( r ) derived from the numerical Hartree-Fock (HF) density [ 131, Sen. et al [14] succeeded in recovering the shell structure for atoms beyond Ar. Angulo et al [15] recently confirmed the findings of Sen et al [14] for atoms using the analytical HF wave functions [16].…”

“…Using the outer inflection points in D ( r ) derived from the numerical Hartree-Fock (HF) density [ 131, Sen. et al [14] succeeded in recovering the shell structure for atoms beyond Ar. Angulo et al [15] recently confirmed the findings of Sen et al [14] for atoms using the analytical HF wave functions [16]. Smith and co-workers [17] identified the extrema in the double logarithmic derivative function, d In p / d In r, with the shell structure.…”

Nonconvexity of the atomic charge density and shell structure

“…, the structure of these functions is not enough to detect all the shells occupied in all of the atomic systems. Some recent results in this laboratory, based on a suggestion of Sen, Slamet, and Sahni [20], indicate that functionals connected with the second derivative of the radial density D(r) exhibit the correct number of shells for all ground-state atoms [25]. Some of these structural features have allowed the derivation of numerous relationships among the density and its first derivative at the nucleus, p(0) and p'(0), respectively, as well as some radial expectation values (r )-: fr p(r)dr [9,16], increasing greatly the interest in the study of the monotonicity characteristics of the one-particle density p(r }.…”

“…Other attempts to study the atomic shell structure by means of functions related to the charge density p(r) have been made [e.g., the radial density 4m.r p(r) [15,20], the logarithmic derivative p'( r) /p( r ) [21], the Laplacian p"(r)+(2lr)p'(r) [22,23] and other related functionals [24]]. However, in all these studies, there is always at least one shell missing for some atoms, i.e.…”