Molecular Geometry & Bonding
what the shape decides
Concept

Spin-only — where it appears

The formula √(n(n+2)) giving a magnetic moment from a count of unpaired electrons. It works when the orbital angular momentum is quenched and a single state is populated, and fails predictably when either condition breaks.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The spin-only count against nine measured moments. Each ion's magnetic moment computed from the number of unpaired electrons alone, √(n(n+2)) Bohr magnetons, beside the measured value. The two agree to a hundredth for the first five and the measurement exceeds the count by up to 0.93 for Co²⁺ — always in the same direction, which is what an omission looks like rather than noise.

A moment counts electrons, not orbitals

A magnetic moment is one of the few chemical measurements that returns an integer. Feed the count of unpaired electrons into √(n(n+2)) and nine first-row ions come back within a hundredth for five of them — and the five that miss all miss the same way, which is what a missing term looks like.

applied · Magnetism What the angular momentum operator connects. The five real d functions, with a line between each pair the z component of orbital angular momentum connects and the size of the connection on it. Written in this basis the operator is i times an antisymmetric matrix, so its expectation in any real function is exactly zero — that is the quenching, and it holds before any field is applied. The three t₂g functions are connected among themselves, so as a SET they carry eigenvalues 1, 0, -1; the two eg functions are each connected only to something outside the pair, so as a set they carry nothing.

An orbital carries no angular momentum

The d orbitals every chemist draws carry exactly no orbital angular momentum, and the proof is one line about a matrix being antisymmetric. A set of three of them carries a whole unit, which is why the spin-only formula works for most ions and fails for cobalt by nearly a Bohr magneton.

applied · Magnetism The total is the same in both columns; nothing else is. Ten complexes counted by both conventions. The neutral method gives the metal its group number and every ligand what it brings as a neutral fragment; the ionic method assigns an oxidation state and gives every anionic ligand a pair. The two totals agree in every row. The oxidation state and the d count do not agree wherever the oxidation state is not zero, and the moments the two d counts predict differ by as much as 2.83 Bohr magnetons.

The same count, two oxidation states

Count a complex by the neutral method and by the ionic one and the total is the same integer every time — eighteen for ferrocene, sixteen for tetrachloroplatinate, twenty for hexaaquanickel. The oxidation state and the d count are not: hexaaquairon is d⁶ on one convention and d⁸ on the other, and the two predict spin-only moments of 4.90 and 2.83 against a measured 5.40.

applied · Electron count

Named alongside it

The objects these essays reach for when they reach for this one.

d orbitalsLigand fieldMagnetic momentDegeneracyModel limitUnpaired electronsAngular momentumConventionCoordination complexEighteen-electron ruleElectron countExpectation value

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