Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus: I. Ground state

Several ultra‐compact accurate wave functions in the form of generalized Hylleraas‐Kinoshita functions and Guevara‐Harris‐Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), for the ground state energies (4‐5 significant digits [s.d.]) of He‐like and Li‐like iso‐electronic sequences in the static approximation with point‐like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in Z by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He‐like two‐electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy ∼10−3 a.u. and the same relative accuracies ∼10−2 to 10−3 for both the cusp parameters and the six expectation values, is found. For the Li‐like three‐electron sequence the most accurate ultra‐compact function taken as the variational trial function provides absolute accuracy for energy ∼10−3 a.u., 2 to 3 s.d. for the electron‐nuclear cusp parameter for Z ≤ 20 and 3 s.d. for the two expectation values for Z = 3. Ultra‐compact accurate wave functions in the form of generalized Hylleraas‐Kinoshita functions and Guevara‐Harris‐Turbiner functions, which describe the domain of applicability of Quantum Mechanics of Coulomb Charges for the ground state energies of Helium‐like (formula and picture shown) and Li‐like iso‐electronic sequences for Z ≤ 20 are constructed. Both cusp parameters are accurately described (see figure for the case of He‐like).