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\begin{document}

\title{Ultra-Compact accurate wave functions for He-like and Li-like
iso-electronic sequences and variational calculus. III. Spin-quartet
state of the Lithium sequence}



\author[1]{Daniel Julian Nader}%
\author[2]{Juan Carlos del Valle}%
\author[3]{Juan Carlos Lopez Vieyra}%
\author[3]{Alexander Turbiner}%
\affil[1]{Universidad Nacional Autonoma de Mexico}%
\affil[2]{Universidad Nacional Autonoma de Mexico Instituto de Ciencias Nucleares}%
\affil[3]{UNAM}%


\vspace{-1em}



  
  \date{May 20, 2024}


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\begin{abstract}
As a continuation of Part I, dedicated to the ground state of He-like
and Li-like isoelectronic sequences for nuclear charges \$Z
\textbackslash{}leq 20\$, and Part II, dedicated to two excited states
of He-like sequence, two ultra-compact wave functions in the form of
generalized Guevara-Harris-Turbiner functions are constructed for
Li-like sequence. They describe accurately the domain of applicability
of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (2-3
significant digits (s.d.)) of the spin-quartet state \$1\^{}40\^{}+\$ of
Li-like ions (in static approximation with point-like, infinitely heavy
nuclei). Variational parameters are fitted in \$Z\$ by 2nd degree
polynomials. The most accurate ultra-compact function leads to the
absolute accuracy \$\textbackslash{}sim
10\^{}\{-3\}\$\textbackslash{},a.u. for energy, and
\$\textbackslash{}sim 10\^{}\{-4\}\$ for the normalized electron-nuclear
cusp parameter for \$Z \textbackslash{}leq 20\$. Critical charge
\$Z=Z\_B\$, where the ultra-compact trial function for the
\$1\^{}40\^{}+\$ state looses its square-integrability, is estimated,
\$Z\_B(1\^{}4\textbackslash{},0\^{}+) \textbackslash{}sim 1.26 - 1.30\$.
As a complement to Part I, square integrability for the compact
functions constructed for the \{\textbackslash{}it ground, spin-doublet
state\} \$1\^{}2\textbackslash{},0\^{}+\$ of the Li-like sequence is
discussed. The critical charge, for which these functions stop to be
normalizable, is estimated as \$Z\_B( 1\^{}2\textbackslash{},0\^{}+) =
1.62 - 1.65\$. It implies that at \$Z=2\$ - the negative helium ion
He\$\^{}-\$ - both states \$1\^{}2\textbackslash{},0\^{}+\$ and
\$1\^{}4\textbackslash{},0\^{}+\$ exist as states embedded to continuum.%
\end{abstract}%



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\textbf{Hosted file}

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