Concept

Ligand repulsion — where it appears

The energy two ligands on one centre cost each other by being close, taken as a function of the angle between them. It decides a shape only through the arrangement that minimises its sum, so a change in the law it is written with can move a geometry without changing any count of electrons.

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

Four arrangements, and only the first has a symmetry to exchange its pairs. A square-planar centre with both trans pairs folded ten degrees out of the plane, drawn at four ratios of the two pairs' bond lengths. At a ratio of one, exchanging the two pairs maps the arrangement onto itself — so the gap and the ligand–ligand repulsion are both even functions of the antisymmetric fold, and both have a vanishing first derivative along it. At every other ratio that exchange is not a symmetry, and nothing requires either derivative to vanish.

One was a symmetry and one was not

A square-planar complex has a direction the sixteen-electron gap does not move along, and the repulsion falls along it — so the distortion looked free in both senses at once. Both results came from four ligands at the same bond length, and the two behave completely differently when that is relaxed. Two per cent of a bond length gives the gap a first-order slope; the repulsion keeps falling out to a fifty per cent mismatch, and its curvature never changes sign. The blindness was a symmetry; the fall is a property of the arrangement.

applied · Electron count

The gap's gradient over four separate lifts has four equal parts. At each fold angle where all four ligands are lifted alike, the rate at which the sixteen-electron gap changes as each ligand alone is lifted a little further, drawn as four bars. They are equal to eleven figures or better: the gradient lies on the one direction in which all four move together, so any distortion whose lifts add to zero leaves the gap unchanged to first order.

Four lifts and one that matters

The direction a sixteen-electron gap cannot see was found in a plane of two folds, where it was the antisymmetric fold of the two trans pairs. That plane hid a choice: the ligands were moved in pairs and they were all alike, and nothing separated which of those the blindness needed. Let each of the four ligands lift by its own angle and the answer is plain. The gap's gradient has four equal parts, so every distortion whose lifts add to zero is blind — three directions, not one — and the pairing never mattered. What sorts the three is the second order, and there the two models agree: the pair fold is rewarded by both and the tilt of a pair is penalised by both.

applied · Electron count

Named alongside it

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

Angular overlap Coordination complex d orbitals Degeneracy Electron count Irreducible representations Model limit

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