Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
J Chem Phys. 2019 Feb 21;150(7):074111. doi: 10.1063/1.5065791.
Employment of exact numerical quadratures in the evaluation of matrix elements involving highly accurate wavefunctions of helium (and its isoelectronic congeners) generated with the help of the regularized Krylov sequences of Nakatsuji results in an efficient algorithm for the calculation of natural orbitals and the corresponding natural amplitudes {λ}. The results of such calculations are presented for the total of 600 natural orbitals pertaining to the ground state of the helium atom. The benchmark-quality values of {λ} computed for 1 ≤ n ≤ 100 and 0 ≤ l ≤ 5 reveal gross inaccuracies in the previously published data. In particular, the dependence of λ on n is found to follow very closely a simple power-scaling law λ≈-A (n+B) with A that, contrary to previous claims, varies only weakly with l. Even more importantly, the numerical trends observed in the present calculations strongly suggest that in the case of the ground state of the helium atom, the only positive-valued natural amplitude is that pertaining to the strongly occupied orbital, i.e., λ. The relevance of this finding to the existence of unoccupied natural orbitals pertaining to the ground state wavefunction of the H molecule is discussed.
在评估涉及氦（及其等电子同系物）高度精确波函数的矩阵元素时，使用正则化 Nakatsuji 序列的精确数值积分，可以得到一种计算自然轨道和相应自然幅度 {λ} 的有效算法。为氦原子基态的总共 600 个自然轨道计算了这样的结果。对于 1 ≤ n ≤ 100 和 0 ≤ l ≤ 5，计算出的 {λ} 的基准质量值显示了先前发表的数据中的严重不准确。特别是，发现 λ 对 n 的依赖性非常接近简单的幂律 λ≈-A (n+B)，与之前的说法相反，A 仅随 l 弱变化。更重要的是，在本计算中观察到的数值趋势强烈表明，对于氦原子的基态，唯一的正值自然幅度是属于强烈占据轨道的幅度，即 λ。讨论了这一发现对 H 分子基态波函数的未占据自然轨道存在的相关性。
