Partial-wave decomposition of the ground-state wavefunction of the two-electron harmonium atom

An exact formula for the collective occupancy of natural orbitals with an angular momentum l is derived for the ground state of the two-electron harmonium atom. For confinement strengths ωω that correspond to polynomial correlation factors as well as at the weak (ω→∞ω→∞) and strong (ω→0ω→0) correlations limits, it reduces to closed-form expressions. At the former limit, a similar result obtains for the partial-wave contributions to the ground-state energy. Slow convergence of the collective occupancies to their leading large-l asymptotics provided by Hill’s formula is uncovered. As the rate of convergence decreases strongly with ωω, a complete breakdown of Hill’s formula ensues upon the confinement strength becoming infinitesimally small. The relevance of these findings to the performance of the extrapolation schemes for the estimation of the complete-basis-set limits of quantum-mechanical observables is discussed.