The spin-only count against nine measured moments

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.

Five 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.

A moment counts electrons, not orbitals

The spin-only count against nine measured moments. Each pair of dots is one ion: the count from √(n(n+2)) and the measurement. Five agree to a hundredth. Four do not, and all four miss in the same direction — the measurement is larger, by up to 0.93 Bohr magnetons for cobalt(II).

Which ions are allowed an orbital contribution and which actually show one. The permission is a symmetry statement — whether the angular momentum operator connects two occupied orbitals of the same energy — and the two columns do not agree, which is the whole finding: permission is necessary and not sufficient.

The same nine ions, ordered by the size of the discrepancy rather than by d count. Cobalt(II), iron(II), nickel(II) and copper(II) are the four with a real excess, and they are consecutive — the late first row, where the ground states are the ones the orbital contribution can reach.

The pairing energy decides the moment

The measured moments the whole competition reports through. Iron(II) is highlighted: high spin it measures 5.40, low spin it is diamagnetic, and the same ion does both depending on what is round it — which is the competition of this essay, read off an instrument.

An orbital carries no angular momentum

The operator drawn as a graph: which of the five real d functions it connects to which, and by how much. Every connection is off-diagonal, which is what makes the matrix antisymmetric and every diagonal expectation zero.

Each ion, its configuration, whether the group permits an orbital contribution, the spin-only value, the measurement and the difference. The two largest excesses are both permitted; every forbidden ion is within four tenths of a Bohr magneton of spin-only.

The spin-only formula against the nine measurements. Every departure is in the same direction — the measurement is always larger — which is what identifies the missing term as an addition rather than as scatter, and is the observation this essay sets out to account for.

The g-value is the orbital coming back

The angular momentum operator restricted to each subspace of an octahedral field. The t₂g set carries a unit of momentum and the eg set carries none, which is the quenching stated as a computation rather than as a rule — and it is the same operator the g-shift needs, in its other components. The set that carries none is the one a d–d transition is weak because of, for a related reason: both are statements about what a symmetry forbids a matrix element to be.

Three ions, their computed g-values and their measured ones. λ is quoted; the matrix elements and the energy denominators are not. The last column is what is left over.

The nine ions sorted by how badly the spin-only count misses, which is the ordering the rest of this essay has to explain. The ions at the top of that list are the ones whose g-value departs furthest from the free-electron figure, and the two facts are the same fact: an orbital contribution that shifts the resonance line is an orbital contribution that moves the moment.

A moment between two integers

The nine ions the count works for, with the measured moment against the spin-only value. Every one of them has a ground state far below anything else, which is the condition the count needs — and the condition is met often enough that the count looks like a law.