Uniform electron gases. III. Low-density gases on three-dimensional spheres

Harmonium atoms at weak confinements: The formation of the Wigner molecules

The formation of the Wigner molecules in three-dimensional assemblies of equicharged particles upon weakening of harmonic confinements with spherical symmetry is investigated using five electronic states of the four-electron harmonium atom as a representative example. Numerical results of accurate explicitly-correlated calculations, carried out for a wide range of confinement strengths ω, reveal nonmonotonic convergence of the total energies and one-electron densities to those predicted by the formalism asymptotically exact at the ω→0 limit. The convergence rates are strongly state- and property-dependent, which implies the nonexistence of a single measure quantifying the extent of the formation of the Wigner molecule in a particular state of the harmonium atom. In light of these observations, it is concluded that, like in two dimensions, the emergence of the Wigner molecules in Coulombic systems confined by spherically symmetric harmonic potentials is a complex and gradual process that takes place over a range of confinement strengths spanning several orders of magnitude.

Natural occupation numbers in two-electron quantum rings

Natural orbitals (NOs) are central constituents for evaluating correlation energies through efficient approximations. Here, we report the closed-form expression of the NOs of two-electron quantum rings, which are prototypical finite-extension systems and new starting points for the development of exchange-correlation functionals in density functional theory. We also show that the natural occupation numbers for these two-electron paradigms are in general non-vanishing and follow the same power law decay as atomic and molecular two-electron systems.