Molecular Geometry & Bonding
what the shape decides
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 4 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 Past a fold of 17.90° the pair fold closes the gap at first order and the tilts still do not. The sixteen-electron gap's first-order change along three distortions whose four lifts add to zero, against the fold at which all four ligands are lifted alike. Below 17.90°, where the highest filled level is dz², all three are zero: the blindness. Past it, where the highest filled level is the degenerate dxz, dyz pair, the pair fold closes the gap at −0.0258 per degree whichever way the pairs move, three up and one down at −0.0149, and a tilt still at nothing.

The blindness needed a single top level

Every distortion of a folded sixteen-electron complex whose four ligand lifts add to zero leaves its gap unchanged to first order — while the highest filled level is dz². Past a fold of 17.90° it is the degenerate dxz, dyz pair, and the pair fold, which splits that pair, closes the gap at 0.026 eσ a degree whichever way the pairs move; the tilts, which cannot split it, stay blind. Exactly at the crossing the tilts see too, seventeen times more faintly. And the energy of the filled levels sees none of it at first order: its own turn along the pair fold comes at 13.8°, before the crossing, where nothing in the gap changes.

applied · Electron count The pair fold is the road from a square to a tetrahedron. The flattening angle of the lowest-energy geometry of a sixteen-electron complex, lifting one trans pair of ligands and lowering the other, against the weight given to the filled d levels' energy relative to the ligands' repulsion. With no weight it is the regular tetrahedron at 35.26°; it flattens continuously as the weight rises and reaches the square at 0.0762, beyond which the square is the minimum. The sketches above show the geometry at four weights.

The pair fold was the road to a tetrahedron

Past a fold of 13.8° both of the model's energies fall along the pair fold, and the question left was where a folded sixteen-electron complex goes. It goes nowhere from there, because a folded square is not at rest: both energies rise along the common fold. Applied to the flat square, the same pair fold is the road to a tetrahedron, and on it the minimum of any weighting of the two energies is a flattened tetrahedron that flattens continuously, as a square root, into the square at a weight of 0.0762. The d levels cannot tell the two roads apart.

applied · Electron count

Named alongside it

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

Angular overlapCoordination complexd orbitalsDegeneracyElectron countModel limitIrreducible representations

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