Ultra-Compact accurate wave functions for He-like iso-electronic sequences and variational calculus. IV. Spin-singlet states $(1s\,ns)$ $n\,{}^1 S$ family of the Helium sequence
| Autor: | Vieyra, J. C. Lopez, Turbiner, A. V. |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Advances in Quantum Chemistry 89 (2023) Chapter 5, 305-339 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/bs.aiq.2023.11.001 |
| Popis: | 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 $(1s\,ns)$ 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 $Z \leq 20$. 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 $n=1,2,3,\ldots$ and any $Z \leq 20$\,. 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 $n^1 S, n=1,2,3,\ldots$ 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 $Z_B^{(1)}=Z_B^{(2)}\,=\,0.904$, $Z_B^{(3)}=Z_B^{(4)}\,=\,0.928$, $Z_B^{(5)}\ =\ 0.939$. Comment: 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 |
| Databáze: | arXiv |
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