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

TL;DR

This paper develops ultra-compact wave functions for the spin-quartet state of Li-like ions, achieving high accuracy in energy calculations and estimating critical nuclear charges where these functions lose normalizability.

Contribution

It introduces new generalized Guevara-Harris-Turbiner wave functions for Li-like sequences, extending previous work and providing precise energy estimates and critical charge values.

Findings

Achieves ~10^{-3} a.u. energy accuracy for Z ≤ 20

Estimates critical charge Z_B ~ 1.26-1.30 for the spin-quartet state

Determines Z_B ~ 1.62-1.65 for the ground state

Abstract

As a continuation of Part I, dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, 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 $\sim 10^{-3}$\,a.u. for energy, and $\sim 10^{-4}$ for the normalized electron-nuclear cusp parameter for $Z \leq 20$. Critical charge $Z=Z_B$, where the ultra-compact trial function for the $1^40^+$ state looses its square-integrability, is estimated, $Z_B(1^4\,0^+) \sim 1.26 - 1.30$. As a complement to Part I, square integrability for the compact functions constructed for the {\it ground, spin-doublet state} $1^2\,0^+$ of the Li-like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as $Z_B( 1^2\,0^+) = 1.62 - 1.65$. It implies that at $Z=2$ - the negative helium ion He${}^-$ - both states $1^2\,0^+$ and $1^4\,0^+$ exist as states embedded to continuum.