Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus: III. Spin‐quartet state (1s2s3s) of the lithium sequence

As a continuation of Part I, dedicated to the ground state of He‐like and Li‐like isoelectronic sequences for nuclear charges Z≤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 non‐relativistic Quantum Mechanics of Coulomb Charges (QMCC) for energies (2–3 significant digits [s.d.]) of the spin‐quartet state 140+ 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 ∼10−3 a.u. for energy, and ∼10−4 for the normalized electron‐nuclear cusp parameter for Z≤20. Critical charge Z=ZB, where the ultra‐compact trial function for the 140+ state loses its square‐integrability, is estimated, ZB140+∼1.26−1.30. As a complement to Part I, square integrability for the compact functions constructed for the ground, spin‐doublet state 120+ of the Li‐like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as ZB120+=1.62−1.65. It implies that at Z=2—the negative helium ion He−—both states, 120+ and 140+, exist as states embedded in the continuum. 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 14 0+ of Li‐like ions (in static approximation with point‐like, infinitely heavy nuclei). Variational parameters are fitted in Z (the nuclear charge) by 2nd degree polynomials. The most accurate ultra‐compact function leads to the absolute accuracy ~10−3 a.u. for energy, and ~10−4 a.u. for the normalized electron‐nuclear cusp parameter for Z ≤ 20. Critical charge Z = ZB, where the ultra‐compact trial function for the14 0+ state loses its square‐integrability, is estimated, ZB (14 0+) ~ 1.26–1.30.