Molecular Geometry & Bonding
what the shape decides
Concept

Crystal field — where it appears

A model that treats ligands as point charges and computes what their electrostatic potential does to a metal's d orbitals. It gets every degeneracy right, because those follow from symmetry, and gets the ordering of ligands by strength wrong.

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 What orders the spectrochemical series, and what does not. Nine ligands' measured octahedral splittings, plotted against charge and against the π parameter that says whether the ligand donates or accepts π density. Charge puts 3 of 20 mixed-charge pairs in the order it predicts; the π parameter puts 35 of 35 in order.

The spectrochemical series is not electrostatics

Ligands can be put in order by how hard they split a d shell, and the order is highly reproducible. It is not the order of charge — three of twenty mixed-charge pairs come out the way a point-charge model predicts, which is worse than tossing a coin — and the two ligands at the strong end are electrically neutral.

wrong · Ligand field The sixteen-electron gap, from a plane to a tetrahedron. The gap above eight d electrons as four ligands are folded out of a square plane towards a tetrahedron, at three π strengths. It is 2eσ exactly at the plane whatever the π strength is, and exactly zero at the tetrahedron whatever it is — the upper three levels there are the degenerate t₂ set. In between the three curves separate, and the separation is what the π channel is doing.

The gap that only a tetrahedron closes

The sixteen-electron gap is 2eσ exactly, and a square-planar π set cannot touch it because d(z²) has no partner there. Fold the ligands out of the plane and the gap survives almost intact for fifteen degrees, closes to nothing only at the tetrahedron — and loses its exactness at the very first degree.

applied · Electron count

Named alongside it

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

d orbitalsLigand fieldAngular overlapDegeneracySplittingBarycentrePi acceptorPi-donorBack-donationCharacter tableClosed formCoordination

All concepts