Below you find the ground state energies produced by realQM with electrons filling a spherical shell structure homogenised to spherical symmetry. realQM then reduces to a wave equation in one radial variable in terms of a wave function defined on a sequence of intervals/spherical shells with a certain total charge density /number of electrons in each interval/shell. The ground state is computed iteratively by energy minimisation over each shell and and update of the free boundary separating shells to reach continuity. See below for details.

The basic shell structure is given by the sequence 2, 8, 18, 32,…, as the number of electrons in fully filled shells of increasing radius, that is with electrons i shell . Here the factor 2 reflects the structure of Helium with two electrons filling two half-spherical shells meeting a separating plane as free boundary with a Bernoulli free boundary, with subsequent regular subdivision into subregions. As an example, the electron shell structure for Neon then shows to be 2+4+4 with the second full shell subdivided into two subshells with 4 electrons in each.

In general computed energies agree with observed energies up to about three digits. Here you can inspect sample results with the shell structure indicated.

- Helium
- Helium1+
- Lithium
- Beryllium
- Boron
- Boron1+
- Carbon
- Carbon1+
- Nitrogen
- Oxygen
- Oxygen1+
- Oxygen1-
- Fluorine
- Fluorine1+
- Neon
- Neon1+
- Sodium
- Sodium1+
- Magnesium
- Aluminium
- Chlorine
- Sulfur
- Phosphorus
- Silicon
- Argon
- Xenon
- Krypton
- Radon

For an atom/ion with kernel charge and electrons realQM takes the following form in spherical coordinates with the electrons filling an expanding sequence of spherical shells and homogenize the electron charge distribution in each shell into spherical symmetry of the same total charge. Find the wave function as a function of distance to the atom kernel of the form

supported by a partition of the interval into intervals with in and outside, satisfying

- (1)

and the normalization condition with and the number of electrons in shell $S_j$:

- , (2)

which minimises the total energy

- ,

where

- ,

with

- , (3)

where for is a reduction factor due to lack of self-repulsion, and else.

Recall that the potential generated by a spherically symmetric charge distribution of total charge at kernel distance , is given by the formula which motivates (1).

The boundary condition at reflects balance of -terms in Schrödinger’s equation in spherical coordinates.

The ground state is computed iteratively by minimization of over satsifying (1) and (2) by a gradient method with vanishing gradient condition

- ,

for , where the are Lagrange multipliers for the charge conservation (2) combined with update of the free boundary to reach continuity of .