Molecular Geometry & Bonding
what the shape decides
Concept

Trigonal bipyramid — where it appears

The five-coordinate arrangement with three equatorial sites and two axial ones. Its two kinds of site are inequivalent, which is why five-coordinate molecules have a substituent preference and readily exchange between the two.

Named by 8 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 Angles at exponents 1, 2, 3, 6, 12. The distinct angles of the minimised arrangement of 4, 5, 6, 7 points, under a repulsion going as one over r to the power 1, 2, 3, 6, 12. Where the arrangement is the maximally symmetric one the angles do not move at all; where it is not, both the angles and how many of them there are depend on the law assumed.

Which angles are symmetry and which are the model

VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.

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 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 Five ligands: which shape the σ energy wants, by one number. The fourth-order cost of the trigonal bipyramid (flat at one sixth) and of the best square pyramid, against q — the ratio of the net-direction coefficient to the frame coefficient, which with equal overlaps is r(1 + r) for p at r times s's height. The two cross at q = 0.862. The bipyramid is a minimum only above q = 2/3 and the pyramid only below 0.955, so between them both are minima. The model's own heights give q = 3.75, and any main-group centre, with p above s, has q above 2.

Five ligands, and no ideal to aim at

A σ-only bond energy prefers, at the order where geometry first enters, an arrangement that points nowhere and treats every direction alike — a spherical 2-design. Four ligands have one and so does every count from six. Five does not, and the energy has to trade one virtue against the other. One number decides the trade, and across it the trigonal bipyramid and the square pyramid exchange the roles of minimum and transition state on Berry's pseudorotation path.

beyond · Hypervalency A ligand repulsion makes a main-group square pyramid only if it is steeper than the fifteenth power. At a σ parameter q = 2, where the bipyramid is the only σ minimum, the contact weight at which the square pyramid takes over, as a share of the bipyramid's σ cost, against the power s of a 1/rˢ contact between the ligands. For every power below 15.05 — Coulomb, 1/r⁶, 1/r¹² — no weight does it. Above it every power does, at 1123 at s = 16, 233 at s = 20, 101 at s = 30.

Only a steep contact makes five square

A σ-only bond energy gives a main-group centre with five ligands one minimum, the trigonal bipyramid, and yet pentaphenylantimony is a square pyramid. The obvious repair is a repulsion between the ligands. A repulsion does not help by being there: five points repelling as any power of their distance below about the fifteenth prefer the bipyramid themselves, so Coulomb and every soft law only widen the bipyramid's region, at any weight. Above the fifteenth power they prefer the square pyramid, and then every main-group centre switches at a weight that rises with its σ parameter. An exponential contact between ligand atoms 2.2 Å out does it only below a range of 0.215 Å — which is the fifteenth power again, measured at the ligands' separation.

beyond · Hypervalency The exact σ energy makes the contact's price fall with the coupling. The contact share at which a 1/r²⁰ contact makes the square pyramid win, against the coupling β, with the σ energy exact (lines) and to fourth order (dashes, which do not depend on β in this normalisation). Weak coupling reproduces the fourth-order shares; by β = 1 the share at q = 2 is 9.3 against 233, 25 times smaller, and at β = 1.5 the four q run 0.77, 3.1, 6.4, 15.

Priced against the whole bond

A steep contact between ligands makes a main-group square pyramid only past the fifteenth power, and its price was quoted as a hundred to two thousand times the bipyramid's fourth-order σ cost. With the σ energy exact the threshold and the pyramid do not move, and the price falls by up to two hundred times as the coupling grows — but measured against the whole σ bond energy it settles instead at a fraction set by the contact's steepness alone: a seventh of the bond under the thirtieth power, a third under the twentieth, more than the bond itself under the sixteenth, and the same for every main-group centre.

beyond · Hypervalency

Named alongside it

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

Coordination numberModel limitRepulsionAxialEquatorialMinimisationVSEPRHypervalencyLocal minimumBond angleBond lengthMolecular orbital

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