Near-Origin Structure of the Hooke's Atoms

Abstract

The Hooke's atoms with two or more than two electrons give rise to an interesting quantum mechanical model with valuable practical applications. In this work, we study the electronic properties near the origin of the harmonic potential. It is seen that the spherically averaged density, ρ̄, exhibits an interesting character — it has only even order terms in its small r expansion. The spherical average of the Hartree potential, v̄H, and the spherical average of the Kohn—Sham exchange-correlation potential, v̄xc, are also shown to have the same property — all odd order terms in their expansions vanish. Furthermore, the analysis and results extend also to the case of two-dimensional models. While only models interacting via. the Coulomb potential are primarily considered in the article, the results also extend to models interacting via. other potentials (viz. Van der Waals potential).