Abstract
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 omega 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 omega . 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 omega rightarrow infty limit of vanishing electron correlation.
Highlights
The time-independent Schrödinger equation with the Hamiltonian Ĥ = 1 − ∇̂ 21 + ∇̂ 22 + 1 ω2 2 r12 + r22 1 + r12 (1)possesses closed-form solutions for infinitely many values of the confinement strength [1]
The radial components of the natural orbitals (NOs) pertaining to the ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength 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 yield the natural amplitudes
Summary
Possesses closed-form solutions for infinitely many values of the confinement strength [1]. Some recently derived properties of natural orbitals (NOs) and their occupation numbers pertaining to the 1S+ ground state of the two-electron harmonium atom are reported This state has been the subject of numerous studies involving both rigorous mathematical analysis [1, 17,18,19,20,21,22,23,24,25,26,27,28,29,30] and numerical approaches [17, 28], the attention devoted to the corresponding NOs has been limited to investigations of their asymptotic behavior at the → 0 limit [23] and of their collective occupancies at various values of [24,25,26], formulation of accurate approximations to the strongly occupied NOs at. With amelioration of this unsatisfactory state of knowledge as its objective, the present work aims at uncovering universalities in NOs and their occupation numbers that persist throughout the entire range of confinement strengths
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