Molecular Geometry & Bonding
what the shape decides
Figure

Two humps and a dip, which is not what a trend looks like

The measured enthalpy of hydration of the first transition series, in kilojoules per mole, against a straight line fitted through it. The measurements do not fall on the line: they rise and dip at manganese, rise and dip again at zinc. Both dips are at configurations with no ligand field stabilisation — d⁵ high spin and d¹⁰ — and the line alone accounts for only 73 per cent of the variation.
Two humps and a dip, which is not what a trend looks like. The measured enthalpy of hydration of the first transition series, in kilojoules per mole, against a straight line fitted through it. The measurements do not fall on the line: they rise and dip at manganese, rise and dip again at zinc. Both dips are at configurations with no ligand field stabilisation — d⁵ high spin and d¹⁰ — and the line alone accounts for only 73 per cent of the variation.

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.

Three 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.

In the essays

The double hump and what removes it

The measured hydration enthalpies of the first transition series against a straight line fitted through them. The line accounts for 73 per cent of the variation, and what it misses is not noise: the residuals rise and fall twice in a pattern that repeats itself across the row.

The same enthalpies with the computed stabilisation subtracted. The double hump is gone; what is left rises smoothly across the series, which is what the shrinking-ion argument says it should do. The one fitted scale factor is the splitting, and the next section is about where else that number can be measured.

The correction itself, which is the one quantity in the argument that is neither measured nor fitted. Each ion’s stabilisation is computed from the angular overlap model in units of the octahedral splitting, and it is drawn for both spin states: high spin gives the two humps with zeros at d⁰, d⁵ and d¹⁰, and low spin gives a single rise with one zero. The measured enthalpies follow the first, which is the evidence that these ions are high spin.

A moment between two integers

The quantity the gap is a difference of, measured where it can be: the ligand field stabilisation read out of the hydration enthalpies of a whole transition series. Which spin state is occupied changes that number, so a compound near its crossover contributes an average here too.

The integral that cannot count electrons

Five measured splittings against three models: one overlap, two overlaps, and two overlaps with each ligand’s π character supplying the sign. The dashed line is agreement.

The two integrals, ligand by ligand. The sideways overlap is between a half and nearly all of the head-on one — chloride’s diffuse 3p makes its π overlap 97 per cent of its σ overlap — so this is not a small correction being added to a large term.

The stabilisation the integrals are supposed to reproduce, computed for both spin states. The sign of the π parameter is what decides whether an interaction pulls the lower set down or pushes it up — the same two orbitals overlapping either way — and the shape of this curve is what a wrong sign would ruin.

Every figure · Every orbital, by what it encloses · All essays