Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus:II.Spin‐singlet (excited) and spin‐triplet (lowest) states of helium sequence

Abstract

AbstractAs a continuation of Part I (Int. J. Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He‐like and Li‐like isoelectronic sequences for nuclear chargesZ ≤ 20, a few ultra‐compact wave functions in the form of generalized Hylleraas‐Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4–5 significant digits [s.d.]) of two excited states of He‐like ions: the spin‐singlet (first) excited state 21Sand lowest spin‐triplet 13Sstate. For both states, it provides absolute accuracy for energy ∼10−3 a.u., exact values for cusp parameters and also for six expectation values the relative accuracy ∼10−2. Bressanini‐Reynolds observation about the special form of nodal surface of 21Sstate of helium is confirmed and extended to He‐like ions withZ > 2. Critical chargesZ = ZB, where ultra‐compact trial functions lose their square‐integrability, are estimated:ZB(11S) ≈ ZB(21S) ∼ 0.905 andZB(13S) ∼ 0.902. For both excited states, the Majorana formula—the energy as the second degree polynomial inZ—provides accurately the 4–5 significant digits forZ ≤ 20.