Universalities among natural orbitals and occupation numbers pertaining to ground states of two electrons in central...

Although both the natural orbitals (NOs) {ψ_nlm(r→)} and their occupation numbers {ν_nl} pertaining to the ground state of two electrons confined by a central potential are completely determined by the spatial component Ψ(r→₁,r→₂) of the underlying wavefunction through a homogeneous Fredholm equation of the second kind in which Ψ(r→₁,r→₂) plays the role of the kernel, for the species with a single positive-valued natural amplitude that corresponds to the strongly occupied NO ("the normal sign pattern"), these quantities turn out to depend almost entirely on the "on-top" wavefunction Ψ(r→,r→). For such species, for which the occupation numbers are found to have the large-n asymptotics of n^-8, a universal expression involving only Ψ(r→,r→) that reproduces the weakly occupied NOs with remarkable accuracy is inferred from the electron-electron coalescence cusp in Ψ(r→₁,r→₂). These theoretical predictions are fully confirmed by comparisons among the benchmark-quality NOs computed for the helium atom, the isoelectronic cations with the nuclear charges ranging from 3 to 5, and the two-electron harmonium atom at the limit of an infinitely strong confinement.