Molecular Geometry & Bonding
what the shape decides
Concept

Barycentre — where it appears

The mean energy of a set of orbitals, which a field redistributes levels about and cannot move. Splittings are therefore differences about a fixed centre, and a stabilisation of one set is paid for exactly by the other.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A d shell in an octahedral field. The five d energies in octahedral coordination, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

The splitting is a symmetry statement

Put six ligands round a metal and the five d orbitals stop being degenerate. Which of them stay together, and how many sets there are, follows from the point group alone — before any account of what the ligands are made of, and before any number is computed.

applied · Ligand field The same d shell in four fields. The five d energies in octahedral, tetrahedral, cubic, square planar coordinations, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

Two models, one ratio

A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.

applied · Ligand field 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.

The double hump and what removes it

The hydration enthalpies of the first transition series do not lie on a line — they rise, dip at manganese, rise and dip again at zinc. Subtract the ligand field stabilisation computed from the same model that describes their spectra and what is left is a line, with the one fitted parameter landing inside the range a spectrum measures.

applied · Ligand field How anisotropic a structure has to be. The fourth moment of a layered band, divided by the square of its second so that the width drops out, against the coupling between layers. The curve is the closed form that follows from adding cumulants — 3 − 3(2 + λ⁴)/(2(2 + λ²)²) — with nothing fitted; the points are eigenvalues. At λ = 1 it is the three-dimensional value 2.5 and at λ = 0 the two-dimensional 2.25, and it is halfway between them at λ = 0.47 — so the question of when a band is effectively two-dimensional has a number rather than an opinion.

Two bands, and the shape of each

A band's shape has a closed form when the structure it belongs to factorises. Put a second band beside it and the form survives exactly while the two do not talk, and fails as the square of the coupling once they do. Make one direction weaker instead and the form survives everywhere — 3 − 3(2 + λ⁴)/(2(2 + λ²)²), agreeing with eigenvalues to nine decimal places — so how anisotropic a structure has to be before its band is two-dimensional is a number: λ = 0.46518.

solids · Bands in a solid

Named alongside it

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

d orbitalsLigand fieldSplittingAngular overlapCrystal fieldDegeneracyApproximationBands in a solidBand gapBand widthCharacter tableClosed form

All concepts