Molecular Geometry & Bonding
what the shape decides
Concept

Equatorial — where it appears

A site round the waist of an arrangement, as against one along its principal axis. In a trigonal bipyramid the two kinds have different bond lengths and different demands, which is why five-coordinate molecules are not all alike.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Five sites are not five of a kind. The minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey.

Five sites are not alike

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.

shape · VSEPR Eight points: the cube loses. The cube and the minimised arrangement of eight points on a sphere, with the repulsion energy of each computed. The minimum is a square antiprism — the cube twisted by forty-five degrees on one face — and the margin is about one part in three hundred.

The shapes above six coordination

Eight points on a sphere do not arrange themselves in a cube. They twist one face by forty-five degrees, and above six the arrangements stop being the ones anybody would name and start being the ones a minimisation finds.

shape · VSEPR Which site a long bond takes, and it is not the one the rule says. The energy of the axial placement minus the energy of the equatorial one, against the odd bond's length. Above the line the equatorial placement wins and below it the axial one does. The textbook sentence is that a bulkier group goes equatorial where there is more room; given a longer bond and nothing else, the repulsion model puts it AXIAL, at every length, by a margin that grows. A site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites left behind — and three equatorial with one axial is the better set of four.

The sites are not the same size

Every arrangement in this collection puts its sites on one sphere, which is an assumption about bond lengths made silently. Give the repulsion model a bond length and it predicts that the long bond goes axial — the opposite of the rule the model is always cited for.

shape · VSEPR Where the orphan pair actually sits. A σ-only Hückel model of each molecule, built from its own coordinates: the central atom's four valence orbitals, one σ orbital on each ligand, and the coupling between them the direction cosine of that ligand. Lone pairs need no special handling — they come out of the diagonalisation as the central orbitals no ligand combination transforms like. The charge is measured rather than assigned, and PF₅ comes out with two kinds of fluorine at 3 at -0.13 and 2 at -0.3, the more charged pair being the axial one that carries the orphan.

The count is the population

The census counted how many ligand combinations have no partner on the central atom and called the count n + L − 4. A σ-only model built from each molecule's own coordinates says what that count is worth: with no electronegativity difference anywhere, the mean charge on a ligand is minus the orphan count divided by the ligand count, exactly, in all ten cases. And the prediction the census made — that the charge grows with the orphan count — is refused by the divisor.

beyond · Hypervalency The even sharer is the one the repulsion likes least. trigonal bipyramid: 2 kinds of ligand, spread 0.1137, repulsion 6.4747; square pyramid: 2 kinds of ligand, spread 0.1658, repulsion 6.4844; pentagonal planar: 1 kind of ligand, spread 0.0000, repulsion 6.8819. The planar arrangement gives all five ligands exactly the same charge and costs 6.3 per cent more in repulsion than the bipyramid, which is the arrangement chemistry actually adopts — so the two models disagree about which arrangement is preferred, and about how much charge is moved.

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

beyond · Hypervalency A long bond goes axial at five and equatorial at seven. The energy of the odd site placed axially minus placed equatorially, against the odd bond's length relative to the others, for the trigonal bipyramid and the pentagonal bipyramid. Above zero the equatorial site is preferred. At five a longer bond goes axial and a shorter one equatorial; at seven, from 0.7 to 1.3, every sign is reversed — a shorter bond axial, a longer one equatorial. An open marker is a placement that is not a minimum: at seven the site each bond avoids is one it would slide out of.

The long bond goes to the crowded site

Given one bond longer than the others, the repulsion model puts it axial in a trigonal bipyramid — against the rule it is usually cited for. The reason is a crowding count, and at seven sites the count reverses: the pentagonal bipyramid's crowded site is equatorial, so a long bond goes equatorial and a short one axial. PF₅'s long bonds are axial and IF₇'s are equatorial. And at seven the site a bond avoids is not even a minimum.

shape · VSEPR

Named alongside it

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

AxialRepulsionTrigonal bipyramidCoordination numberMinimisationModel limitVSEPRBond lengthHypervalencyLone pairNon-bonding orbitalsPartial charge

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