Ultra-Compact accurate wave functions for He-like iso-electronic sequences and variational calculus. IV. Spin-singlet states $(1sns)$ $n{}^{1}S$ family of the Helium sequence

Abstract

As a continuation of Parts I \cite{Part-1:2020}, II \cite{Part-2:2021}, III \cite{Part-3:2022}, where ultra-compact wave functions were constructed for a few low-lying states of He-like and Li-like sequences, the family of spin-singlet $(1sns)$ type excited states $n{}^{1}S$ of the He-like sequence is studied with an emphasis on the $n=3,4,5$: $3{}^{1}S,4{}^{1}S,5{}^{1}S$ states, for nuclear charges $Z\le 20$. Particular attention is given to finding of critical charges $Z=Z_B$ at which the ultra-compact wave functions lose their square-integrability. For each ${}^{1}S$ state an ultra-compact, seven-parametric trial function is constructed, which describes the domain of applicability of the non-relativistic Quantum Mechanics of Coulomb Charges (QMCC) for the total energies (4-5 significant digits (s.d.)) and reproduces 3 decimal digits (d.d.) of the spin-singlet states $n{}^{1}S$ of He-like ions (in the static approximation with point-like, infinitely heavy nuclei) for $n=1,2,3,\dots$ and any $Z\le 20$\,. All energies are well described by second degree polynomials in $Z$ (the Majorana formula). Critical charges $Z=Z_B^{(n)}$, where the ultra-compact trial function for the $n^{1}S,n=1,2,3,\dots$ states loses its square-integrability, are estimated: for all studied states $Z_B^{(n)}$ increases slowly with $n$; it seems they lie in the interval $Z_B(n^{1}S)\sim 0.90-0.95$, in particular, with $Z_B^{(1)}=Z_B^{(2)}=0.904$, $Z_B^{(3)}=Z_B^{(4)}=0.928$, $Z_B^{(5)}=0.939$.

31 pages, 5 figures, 12 Tables; dedicated to the memory of Frank E Harris; extended, typos fixed, accepted for publication at Advances in Quantum Chemistry