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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

J. C. Lopez Vieyra, A. V. Turbiner

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 $(1s\,ns)$ 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 $Z_B^{(1)}=Z_B^{(2)}\,=\,0.904$, $Z_B^{(3)}=Z_B^{(4)}\,=\,0.928$, $Z_B^{(5)}\ =\ 0.939$.

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

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 type excited states of the He-like sequence is studied with an emphasis on the : states, for nuclear charges . Particular attention is given to finding of critical charges at which the ultra-compact wave functions lose their square-integrability. For each 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 of He-like ions (in the static approximation with point-like, infinitely heavy nuclei) for and any \,. All energies are well described by second degree polynomials in (the Majorana formula). Critical charges , where the ultra-compact trial function for the states loses its square-integrability, are estimated: for all studied states increases slowly with ; it seems they lie in the interval , in particular, with , , .
Paper Structure (11 sections, 65 equations, 5 figures, 12 tables)

This paper contains 11 sections, 65 equations, 5 figures, 12 tables.

Table of Contents

  1. Spin-singlet $3{}^1S$ state
  2. Generalities and results
  3. Majorana formula for $3^1S$ state
  4. Square-integrability of the $3{}^1S$ excited state
  5. Spin-singlet $4{}^1S$ state
  6. Generalities and results
  7. Majorana formula for $4^1S$ state
  8. Square-integrability of the $4{}^1S$ excited state
  9. Spin-singlet $5{}^1S$ state
  10. Spin-singlet $n{}^1S$ state
  11. Majorana Formulae

Figures (5)

  • Figure 1: Parameters $A_1^{3S}, A_2^{3S}$ for different $Z$ and their interpolations (\ref{['A1-3S']})-(\ref{['A2-3S']})
  • Figure 2: Parameters $A_1^{4S}, A_2^{4S}$vs.$Z$, the lines represent the interpolations (\ref{['A1-4S']})-(\ref{['A2-4S']})
  • Figure 3: Parameters $A_1^{5S}, A_2^{5S}$ for different $Z$ and their interpolations (\ref{['A1-5S']})-(\ref{['A2-5S']})
  • Figure 4: Coefficient $M_0(n)$ in the Majorana formula (\ref{['Majon']}) vs.$n$
  • Figure 5: Coefficient $M_1(n)$ in the Majorana formula (\ref{['Majon']}) vs.$n$