Few-electron atomic ions in non-relativistic QED: the Ground state energy

Abstract

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for $Z\le 20$ the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with $Z\le 20$ the finite nuclear mass effects do not change 4-5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for $Z\le 12$, and not less than 12 s.d. for $Z\le 50$ for helium-like as well as for $Z\le 20$ for lithium-like ions by a compact meromorphic function in $\lambda =\sqrt{Z-Z_B}$ ($Z_B$ is the 2nd critical charge, see {TLO:2016}), $P_9(\lambda )/Q_5(\lambda )$. It is found that the Majorana formula - a second degree polynomial in $Z$ with two free parameters - and a fourth degree polynomial in $\lambda$ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for $Z\le 20$ in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that $\gtrsim 99.9\%$ of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for $Z\le 20$, respectively. It may imply that these trial functions are, in fact, {\it exact} wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in $Z$ for $Z\le 20$.

32 pages, 4(6) figures, 6 tables, 35 references, final version to be published at Annals of Physics

References in corpus (5)

• High-precision measurement of the atomic mass of the electron
• Relativistic, QED, and finite nuclear mass corrections for low-lying states of Li and Be${}^{+}$
• Testing fundamental interactions on the helium atom
• {H${}_2^{+}$, HeH and H${}_2$}: approximating potential curves, calculating rovibrational states
• Three-body problem in 3D space: ground state, (quasi)-exact-solvability

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• Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. III. Spin-quartet state of the Lithium sequence
• From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics
• HCl, DCl and TCl diatomic molecules in their ground state: predicting Born-Oppenheimer rovibrational spectra
• Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions
• Ultra-Compact accurate wave functions for He-like iso-electronic sequences and variational calculus. IV. Spin-singlet states $(1sns)$ $n{}^{1}S$ family of the Helium sequence