The total is the same in both columns; nothing else is

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The same count, two oxidation states

Ten complexes counted both ways. The two totals agree in every row. The oxidation state, the d count and the moments those d counts imply do not agree wherever the oxidation state is not zero, which is six of the ten.

The integer nobody measured, beside the charge nobody agrees on, across every complex here. The total counts orbitals and the oxidation state counts a convention, and the census is the two columns side by side — one of them the same number in every row of a pair and the other different.

For the four complexes here with measured moments, the spin-only moment each convention’s d count predicts, against the measurement. The ionic count is nearer in every case, and by more than the spin-only formula’s own error: the group-number count is out by 1.73, 2.57 and 3.20 Bohr magnetons in the three where the two differ.

An integer nobody measured

The metal’s charge against how much of each shared pair the ligand is given. Half each is Mulliken’s rule; the whole to the ligand is the assumption an oxidation state makes; and the answer runs over 2.06 electrons between them. The oxidation state itself is zero, which is off the end of that range. The electron count is the same integer at every point on the same axis.

Six octahedral complexes, each with its oxidation state as a square and its computed charge from two population analyses beside it. The gap runs to two electrons and it is the same size for every complex in the model, because in this model it is a property of the bonding rather than of the metal.

What one extra interaction does to the two quantities. Adding a π channel to the same octahedral d⁶ complex moves the metal’s charge by a whole electron across the range of sharing conventions, and leaves the electron count exactly where it was. The count is eighteen with the π channel and eighteen without it, which is the property the rule depends on and the charge does not have.