Molecular Geometry & Bonding
what the shape decides
Concept

Susceptibility — where it appears

The magnetisation a substance takes up per unit of applied field. It is what a magnetic measurement returns at each temperature, and every moment quoted in the literature is extracted from it by fitting a model.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

A moment that is not an integer's worth of anything. The effective magnetic moment against temperature for an iron(II) complex whose two spin states lie close together, with the moments belonging to whole numbers of unpaired electrons drawn across. The curve spends its time between them and settles on neither.

A moment between two integers

A magnetic moment is celebrated as one of the few chemical measurements that returns an integer: count the unpaired electrons, feed the count into a formula, and nine first-row ions come out right. That works when one state lies far below the others. Sit a complex at its own crossover and the same measurement returns 0.30 at 80 K and 3.61 at 400 K — a quantity that counts nothing and is a temperature in disguise.

applied · Magnetism The same sample, fitted over four temperature ranges. A pair coupled at -50 cm⁻¹, its susceptibility computed exactly, fitted to a Curie–Weiss law over four ranges. The moment and the Weiss temperature the fit reports both depend on which range was used, and the quality of the fit does not warn about it.

The moment a fit invents

One coupled pair of spins, its susceptibility computed exactly, fitted to a Curie–Weiss law over four temperature ranges. The moments reported are 2.471, 2.535, 2.566 and 3.590 Bohr magnetons, and the Weiss temperatures −51, −78, −78 and −292 K — from one sample, measured perfectly, with three of the four fits agreeing with their own data to better than a part in three hundred.

applied · Magnetism 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 The coupling a fit reports, and the coupling the sample has. Exact susceptibilities of Heisenberg chains of two, four, six and eight spins, every one of them coupled at -50 cm⁻¹, each fitted with the two-spin expression over 80–600 K. The two-spin sample returns its own coupling exactly; every longer chain returns one too large, by more the longer it is, up to 20.9 per cent. Every one of those fits has an R² above 0.99, so nothing in the fit reports that anything is wrong.

The model is what is fitted

Fit a pair of coupled spins with the two-spin expression and it hands back the coupling exactly, from any temperature range. Fit a chain of eight with the same expression and it hands back −66.7 where the sample has −50, with a residual of 0.998 and a g factor of 1.973 — three numbers of which only the last says anything is wrong, and it is the one nobody looks at.

applied · Magnetism How much of a curve each extra parameter has left to work with. The singular values of the design matrix for a susceptibility curve, for two, three and four parameters fitted to the same data, on a logarithmic scale. With four they run 12.411, 2.026, 0.149, 0.025 — a span of 500 — so one per cent data fix the first two to under 3 per cent and the last to 37. Each value is what is left of the measurement after the directions above it have taken their share, so a short bar is not a hard parameter but an absent one.

How many parameters a curve is worth

A susceptibility curve routinely carries four fitted parameters and the question of whether it can support them is never asked. It has an arithmetic answer: the four directions the fit sees span a factor of five hundred, so one per cent data fix the first two to under three per cent and the last to thirty-seven — and forty points reaching two kelvin are worth more than sixteen thousand starting at twenty.

applied · Magnetism

Named alongside it

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

BoltzmannMagnetic momentSpin stateTemperatureClosed formConventionExchange couplingLeast-squaresModel limitUnpaired electronsDegeneracyExact diagonalisation

All concepts