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), or, equivalently, the Non-Relativistic QED (NRQED), 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 sim 10^-3\,a.u. and the same relative accuracies sim 10^-2-10^-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 sim 10^-3\,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for Z leq 20 and 3 s.d. for the two expectation values for Z=3.