Molecular Geometry & Bonding
what the shape decides
Concept

Magnetism — where it appears

The response of a substance to a magnetic field, which for most molecules comes from unpaired electron spins. What is measured is a susceptibility at each temperature, and a moment is extracted from it by fitting a model.

Named by 4 essays across 3 fields — 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 A coupling that is second order in the hopping. The singlet–triplet splitting of a two-site Hubbard model, and the same quantity multiplied by U. The product settles on −4t² — -4 at U = 64 — which is what makes the coupling a second-order effect rather than a term somebody put in.

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

applied · Magnetism 60 electrons in 60 levels. The density of states of a ring of 60, drawn with the energy up the page, and the 60 electrons filled in from the bottom. Where the filling stops is what decides whether the system has cheap excitations.

A half-filled band is not always a metal

Every band picture rests on an approximation that a whole class of materials refuses — each electron moving in an average field, never seeing another one individually. Where the repulsion is strong enough, a half-filled band describes an insulator, and no amount of care with the band fixes it.

wrong · Metal Two kinds of answer to a flux, and only one of them is a curve. The π binding of three rings against the magnetic flux through them, in units of beta and of the flux quantum, measured from each ring's own value at zero flux. Beta is negative, so a binding that FALLS is an energy that rises: benzene's does, which is what a diamagnetic ring current is. Cyclobutadiene's rises in both directions from a corner — its energy has no second derivative at zero field at all, and the two one-sided slopes differ by 12.57.

Two rules that share no arithmetic

Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.

symmetry · Aromaticity

Named alongside it

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

DegeneracyElectron correlationMagnetic momentUnpaired electronsAntiaromaticityApproximationAromaticityBands in a solidBand gapClosed formConjugationCoordination complex

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