Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. I. Ground state

TL;DR

This paper develops ultra-compact, highly accurate wave functions for the ground states of He-like and Li-like ions, achieving significant precision with simple parameter fits and providing practical variational functions.

Contribution

It introduces new ultra-compact wave functions for He-like and Li-like ions that attain high accuracy and are fitted with simple smooth functions of nuclear charge Z.

Findings

Achieved 4-5 significant digit accuracy for ground state energies.

Provided explicit approximate wave functions with ~10^{-3} a.u. energy accuracy.

Demonstrated the wave functions' effectiveness across a range of Z values.

Abstract

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 $Z=3$.