Companions to paper 1, Atoms in four spatial dimensions with a 1/r interaction: a self-consistent periodic table. With the force between charges pinned at 1/r in four dimensions there's no shell theorem: a cloud of charge round an atom doesn't act as if it sat at the centre. First, why that law: try the others. Then what it does. Every toy runs on the paper's own code and baked table.
One electron, one nucleus, four dimensions. Under a pull −1/rp its levels group into shells only when p = 1: 2s sits exactly with 2p, and 3s with 3p and 3d. That grouping is what makes shells of 2, 8, 18, 32 and the table's rows. Move p off 1 and the shells come apart.
each shell's levels against its own s level, in percent of its binding: s, p, d; at p = 1 they all sit on the dashed line
Gauss's law would give −α/r² instead, and that has no length in it at all. Below α = ½ nothing binds. Above it the electron falls to the centre: its lowest level is set by how close to the nucleus the calculation is allowed to look, and moves without limit as that point moves in.
the lowest level |ε| (hartree) against the grid's inner end r₀: a settled atom would be a flat line; 1/r₀² dashed
Pick a 4D element. The page solves it twice, in its ground configuration: once as this universe has it, and once with the shell theorem forced on (the clouds' pull computed as if each sat at the centre, the way ours would). The outer electron is held much harder without the shell theorem: the inner electrons screen it only partly.
the charge the outer electron sees inside radius r, Z_eff(r) = −r V(r): 4D, shell theorem forced on; shaded, where the outer electron is
the noble gases' first ionization (eV): 4D as solved, with the shell theorem forced on (from the bake)
In our world a neutral atom is invisible from outside: its nucleus and its cloud cancel exactly, by the shell theorem. In four dimensions the cloud's pull falls short of the nucleus's push, by a little that dies away as the cube of the distance. Walk an electron away from the atom: what it feels approaches −S/8s³, with S = Σ⟨r²⟩ over the atom's electrons.
left: |U(s)| (hartree, log scales), from the cloud against S/8s³; right: their ratio, going to 1
The whole table · fields in four dimensions (step 3 has a toy shell) · the paper's data · the paper's outline · ORBITAL, 4D hydrogen's orbitals to slice and turn