Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
J Chem Phys. 2019 Nov 14;151(18):184107. doi: 10.1063/1.5123669.
Although both the natural orbitals (NOs) {ψ(r→)} and their occupation numbers {ν} pertaining to the ground state of two electrons confined by a central potential are completely determined by the spatial component Ψ(r→,r→) of the underlying wavefunction through a homogeneous Fredholm equation of the second kind in which Ψ(r→,r→) 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, 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→,r→). 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.
尽管两个电子被中心势限制在基态时的自然轨道（NOs）{ψ(r→)}及其占据数{ν}完全由基础波函数的空间分量 Ψ(r→,r→) 通过一个同伦第二类 Fredholm 方程决定，其中 Ψ(r→,r→) 作为核函数，对于具有单一正值自然振幅的物种（对应于强烈占据的 NO），这些数量几乎完全取决于“在顶部”的波函数 Ψ(r→,r→)。对于此类物种，发现占据数具有 n 的大 n 渐近性，从 Ψ(r→,r→) 中的电子-电子合并尖端推断出一个仅涉及 Ψ(r→,r→) 的通用表达式，该表达式可以非常准确地再现弱占据的 NO。这些理论预测通过氦原子、核电荷从 3 到 5 的等电子阳离子以及无限强限制下的双电子谐振子原子的基准质量 NO 之间的比较得到了完全证实。
