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

As 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 charges Z ≤ 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 21S and lowest spin‐triplet 13S state. 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 21S state of helium is confirmed and extended to He‐like ions with Z > 2. Critical charges Z = ZB, where ultra‐compact trial functions lose their square‐integrability, are estimated: ZB(11S) ≈ ZB(21S) ∼ 0.905 and ZB(13S) ∼ 0.902. For both excited states, the Majorana formula—the energy as the second degree polynomial in Z—provides accurately the 4–5 significant digits for Z ≤ 20. Ultra‐compact accurate wave functions in the form of generalized Hylleraas‐Kinoshita functions which describe the domain of applicability of quantum mechanics of Coulomb charges for spin‐singlet (first) excited state 21S and for lowest spin‐triplet 13S state of He‐like ions are constructed. For both states, we get an absolute accuracy for energy 〜10−3 a.u., exact values for cusp parameters and also for six expectation values a relative accuracy 〜10−2. Bressanini‐Reynolds observation about the special form of nodal surface of 21S state for helium is confirmed and extended to ions with Z > 2. Critical charges Z = ZB, where ultra‐compact trial functions lose their square‐integrability, are estimated: ZB(11S) ≈ ZB(21S) 〜 0.905 and ZB(13S) 〜 0.902.