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تسجيل مستخدم جديدUltra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. I. Ground state
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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 $Z=3$.
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As a continuation of Part I (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z leq 20$, a few ultra-compact wave functions in the form of generaliz
ed 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 $2^1 S$ and for lowest spin-triplet $1^3 S$ state. For both states it provides absolute accuracy for energy $sim 10^{-3}$,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $sim 10^{-2}$. Bressanini-Reynolds observation about the special form of nodal surface of $2^1 S$ state for Helium is confirmed and extended to ions with $Z > 2$. Critical charges $Z=Z_B$, where ultra-compact trial functions loose their square-integrability, are estimated: $Z_B(1^1 S)approx Z_B(2^1 S)sim 0.905$ and $Z_B(1^3 S)sim 0.902$. For both states the Majorana formula - the energy as the second degree polynomial in $Z$ - provides accurately the 4-5 significant digits for $Z leq 20$.
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