Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table

TL;DR

This paper analyzes the Schrödinger equation for atoms with 1 to 10 electrons in the large nuclear charge limit, explicitly determining energy levels and eigenstates that align well with experimental data and classical periodic table models.

Contribution

It provides explicit asymptotic solutions for atomic ground states and energy levels in the large nuclear charge limit, confirming and refining classical semi-empirical models.

Findings

Asymptotic energies match experimental data for ions.

Ground state quantum numbers agree with experiments.

Predictions differ from Hund's rules in some cases, aligning better with experiments.

Abstract

This paper is concerned with the Schr\"odinger equation for atoms and ions with N=1 to 10 electrons. In the asymptotic limit of large nuclear charge $Z$, we determine explicitly the low-lying energy levels and eigenstates. The asymptotic energies and wavefunctions are in good quantitative agreement with experimental data for positive ions, and in excellent qualitative agreement even for neutral atoms ($Z=N$). In particular, the predicted ground state spin and angular momentum quantum numbers ($^1S$ for He, Be, Ne, $^2S$ for H and Li, $^4S$ for N, $^2P$ for B and F, and $^3P$ for C and O) agree with experiment in every case. The asymptotic Schr\"odinger ground states agree, up to small corrections, with the semi-empirical hydrogen orbital configurations developed by Bohr, Hund and Slater to explain the periodic table. In rare cases where our results deviate from this picture, such as the ordering of the lowest ${}^1D^o$ and ${}^3S^o$ states of the Carbon isoelectronic sequence, experiment confirms our, not Hund's, predictions.