Natural orbitals of the ground state of the two‑electron harmonium atom

The radial components of the natural orbitals (NOs) pertaining to the 1S+ ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength ω at which the respective correlation factors are given by polynomials. Together with the angular momentum l of the NOs, the degrees of these polynomials determine the orders of the differential equations, eigenvalues of which (arising from well-defined boundary conditions) yield the natural amplitudes. In the case of l=0, analysis of these equations uncovers certain properties of the NOs whereas application of a WKB-like approximation produces asymptotic expressions for both the NOs and the corresponding natural amplitudes that hold when the latter are small negative numbers. Extensive numerical calculations reveal that these expressions remain valid for arbitrary values of ω. The approximate s-type NOs, which are remarkably accurate at sufficiently small radial distances and exhibit universal scaling, differ qualitatively from the eigenfunctions of the core Hamiltonian even at the ω→∞ limit of vanishing electron correlation.