Only a steep contact makes five square

Five ligands, and no ideal to aim at found that a σ-only bond energy has to trade two things at five-coordination, because five points on a sphere cannot be both balanced and isotropic. At fourth order in the coupling the trade is set by one number, q, built from the ratio of the ligand’s overlaps with the central s and p orbitals and the ratio of those orbitals’ heights; below q = 0.862 the square pyramid is the lower of the two shapes and above it the trigonal bipyramid. Main-group centres, whose p orbitals sit well above their s, have q above 2, and for them the bipyramid is the only σ minimum.

That essay named what it had left out. Pentaphenylantimony, Sb(C₆H₅)₅, is a square pyramid in its crystal, and the pentaphenyl compounds of phosphorus and arsenic are bipyramids; a σ-only model with q above 2 cannot produce the first. The obvious omission is the ligands themselves: what a missing orbital costs was a model of bonds, and five phenyl rings do not only bond to a centre, they repel each other. The lead was to add a repulsion between the ligands with one weight and map the regimes in q and the weight.

A repulsion does not help by being there. Whether it can make a square pyramid at all is decided by its steepness, and the threshold is a number that has nothing to do with chemistry.

The σ energy and a contact #

The σ energy is the essay before’s, to fourth order: q times the squared length of the ligands’ summed direction, plus the frame excess — how far the second-moment tensor is from treating the three directions of space equally — both in units of the frame coefficient. The bipyramid pays nothing for balance and one sixth for anisotropy; the square pyramid pays for both, and its apex angle is chosen to minimise what it pays.

To that is added a contact: a law of the distance between each pair of ligand positions on the unit sphere, summed over the ten pairs, with a weight. The laws are powers of the distance, 1/rˢ for s from 1 to 30, and an exponential of range ρ between ligand atoms 2.2 Å from the centre — antimony–carbon in the pentaphenyl compound. The weight is reported as a share: the contact energy of the bipyramid divided by the bipyramid’s σ cost of one sixth, the one normalisation that does not depend on the arbitrary units of either term.

The refusal is the σ energy alone. At zero weight the square pyramid’s margin over the bipyramid must be the essay before’s branch difference exactly, at q = 0.5, at its switch of 0.862 and at 2 — and it is, to a part in a billion.

Five points have a switch of their own #

Before any σ energy is involved, the contact has a preference of its own, and it is the whole story.

Five points repelling as a Coulomb law have the bipyramid as their least-energy arrangement; that is the Thomson problem at five, and the square pyramid lies 0.14 per cent above it. Make the law steeper and the gap first widens, then closes. At s = 15.05 the two are equal, and for every steeper law the square pyramid is lower. This is a known result in the mathematics of point energies on a sphere — the five-point Riesz problem changes its answer between the fifteenth and sixteenth powers — and here it is found directly, by bisection on s, between 15.050 and 15.051.

The reason is geometric. A steep law is dominated by the closest pairs. The bipyramid’s closest pairs are its six axial–equatorial pairs at 90°; the square pyramid, with its basal ligands pushed slightly below the equator, has four apex–base pairs a little over 90° and four base–base pairs at a little under, and for a steep enough law spreading the close contacts evenly over eight pairs beats concentrating them in six. For a soft law the far pairs still count, and the bipyramid’s three equatorial pairs at 120° and axial pair at 180° win it the sum.

A soft repulsion widens the bipyramid’s region #

So a Coulomb repulsion between the ligands, added to the σ energy, prefers the same shape the σ energy prefers at main-group q, and cannot turn a bipyramid into a pyramid at any weight. It does the opposite at small q.

With no repulsion the shapes exchange at q = 0.862. As a Coulomb repulsion grows to five times the bipyramid’s σ cost the exchange moves down to 0.816: the bipyramid takes territory from the square pyramid, never the reverse. The same is true for 1/r⁶ and 1/r¹²; at σ parameters from 1 to 5, no share up to a million gives a square pyramid under any power below 15.05.

That rules out the explanation the lead most naturally suggests. Five bulky ligands do repel, and a repulsion that grows as the ligands crowd does push them apart — but “apart” means the bipyramid for any law that is not extremely steep, because the bipyramid is how five mutually repelling points arrange themselves.

Why the preference grows so fast past fifteen #

The threshold is sharp, and what happens just past it explains the size of every share below. At the fifteenth power the two shapes’ contact energies are equal; at the sixteenth the square pyramid is lower by 0.41 per cent of the bipyramid’s contact energy, at the twentieth by 2.3 per cent, and at the thirtieth by 6.7 per cent. The preference grows roughly as the power’s excess over the threshold, and much faster than linearly once the closest pairs are all that count.

That is the arithmetic of a steep law dominated by its shortest distances. The bipyramid has six pairs at exactly 90°. The best square pyramid under a steep law alone has four base–base pairs at about 89° — each about one per cent closer than a 90° pair — and four apex–base pairs at 96° to 98°, far enough that under the sixteenth power each counts for about a third of a 90° pair. So the pyramid trades six contacts at 90° for four slightly closer ones and four much weaker ones. Steepening the law makes the four closer contacts dearer and the four weaker ones cheaper still, and past the fifteenth power the second effect wins; the apex swings down towards the equator as it does, from 97.8° at the sixteenth power to 95.6° at the thirtieth, making the trade more lopsided. The threshold is high because the distances being traded differ by only a per cent: a law has to be very steep before a one-per-cent difference in distance outweighs having two fewer close pairs.

A steep contact switches every main-group centre #

Above the fifteenth power the contact prefers the square pyramid, and it then competes with the σ energy’s preference for the bipyramid.

At q = 2 the switch comes at a share of 1,120 for 1/r¹⁶, 230 for 1/r²⁰ and 100 for 1/r³⁰. Just above the threshold power the contact’s preference for the pyramid is tiny — the two shapes differ by a few parts in ten thousand of the contact energy — so an enormous share is needed to overcome the σ term; further above it the preference grows and the share falls.

Every steep law switches every σ parameter tried, from 1 to 5, and the share rises with q under each: a larger q charges more for the net direction the square pyramid carries, so the contact has to be stronger to pay it. Under 1/r²⁰ the share is 170 at q = 1 and 430 at q = 5. A search from ten seeded random starts at points on both sides of a switch confirms that the true minimum is always one of the two shapes, never a third.

The shares are large — a contact energy a hundred to a thousand times the σ energy’s fourth-order geometric term. That is not as extravagant as it sounds: the fourth-order term is the small part of the σ energy that depends on geometry at all, and a steric contact between crowded ligands is not a small energy. But it does mean the square pyramid arrives only when the contact is both steep and large, and the model cannot say from inside itself whether pentaphenylantimony’s is.

An exponential needs the same steepness #

A repulsion between closed-shell ligand atoms is usually written as an exponential, not a power, and an exponential has no single steepness: its logarithmic slope at a distance r is r/ρ. The power-law result predicts that an exponential should switch the shape when that local slope, taken at the separation that matters, passes 15.

It does. With the ligand atoms 2.2 Å from the centre, an exponential contact switches the shape at q = 2 for every range up to 0.215 Å, at shares from 89 at 0.10 Å to 860 at 0.20, and never at 0.25 Å or more. At the threshold the exponential’s local steepness at the ninety-degree separation of $\sqrt{2} \times 2.2$ = 3.11 Å is 14.5 — the power-law switch of 15.05 to within four per cent, the difference being that an exponential’s steepness is not the same at every pair distance.

So the chemical version of the requirement is a statement about range: a contact between ligand atoms must fall off over less than about a fifteenth of their separation. Born and Mayer’s 0.345 Å, the conventional range for closed-shell ions, is well outside it. A range of 0.2 Å at 3.1 Å is the kind of steepness a contact between hard, bulky groups has near their van der Waals separation, where the repulsion rises very sharply — which is where five phenyl rings round one antimony atom would be.

The pyramid a contact makes #

The square pyramid the σ energy makes at small q and the one a steep contact makes at main-group q are not the same shape.

The σ energy’s pyramid at q = 0.5 has its basal ligands 110° from the apex — well below the equator, spreading the base to reduce anisotropy. The steep contact’s pyramids have them at 97.9°, 97.3° and 96.2°, close to right angles, spreading the eight close contacts evenly. The two mechanisms make recognisably different pyramids, and a measured apex–base angle near 98° against one near 110° is the kind of datum that could tell which one made a given molecule.

How the claims can fail #

Each statement is checked where the figures are drawn. At zero weight the margin must be the σ-only branch difference at three values of q — the refusal. Five points’ own switch must lie between the fifteenth and the 15.1th power. Under 1/r, 1/r⁶ and 1/r¹² no share up to a million may give a square pyramid at any q from 1 up, and a Coulomb repulsion must move the σ-only switch to lower q at every step. Under 1/r¹⁶, 1/r²⁰ and 1/r³⁰ every q must switch, at a share rising with q and, at q = 2, falling with the power. An exponential contact at 2.2 Å must switch at a range of 0.1 Å and at no range of 0.25 Å or more, and its critical range must put $\sqrt{2}\,L/\rho$ within five per cent of the power-law switch. And at three points a seeded search must find no minimum lower than the better of the two shapes.

Where the model stops #

Fourth order in the coupling. The σ energy is the essay before’s fourth-order expansion, which the exact energy confines further: the exact σ energy restricts where a square pyramid can win even with no contact. The switch shares here are against the fourth-order term, and against the exact energy they would move.

Equal bonds. Every ligand sits at one distance on one sphere. A bipyramid’s axial bonds are longer than its equatorial ones in real molecules, and the sites of a five-coordinate molecule are not the same size; a contact between ligands at different distances has different closest pairs.

Points, not rings. A phenyl ring is a flat object with an orientation, and its contacts with its neighbours depend on how the rings are turned. The model’s contact is between points on the sphere, which is the limit of a ligand whose bulk is round.

And one crystal. Pentaphenylantimony is a square pyramid in its unsolvated crystal and a bipyramid in a cyclohexane solvate, which says the energy between the two is small enough for packing to decide. The model’s switch between two shapes is a statement about an isolated molecule, and a real one this close to its switch is a molecule whose shape its surroundings choose.

The same term decides a different question elsewhere #

A contact between ligand atoms has now decided the shape of a molecule twice before in these essays, in opposite ways. In iodine heptafluoride the range of an exponential contact decided the sign of its effect on a flat ring: shorter than 0.346 Å it puckered the ring, longer it held it flatter. In xenon hexafluoride every contact range tried moved the lone pair from an axis into a face, and the range set only the price. Here the range decides again, and the threshold — 0.215 Å at a 2.2 Å bond — sits below the heptafluoride’s.

What the three have in common is that the contact acts on a choice the bonds alone have already nearly made, and its effect is decided by which pairs of ligands are closest in each of the competing arrangements. When the arrangements differ in their closest pairs only by a small shortening, as five-coordination’s two shapes do, only a very steep law can tell them apart. When they differ by a whole ring of close neighbours, as xenon hexafluoride’s pyramid and octahedron do, any law can.

A number from point energies decides a question about bonds #

The σ energy chooses between two five-coordinate shapes by trading balance against isotropy, and the essay before found where it trades. Adding a repulsion between the ligands seemed to add a second chemical consideration to the same trade. It turns out to add a mathematical one: five points repelling each other have their own answer to which shape is best, and the answer changes at the fifteenth power, for reasons that belong to the geometry of five points on a sphere and to nothing about antimony or phenyl rings.

That makes the chemical question sharper rather than vaguer. “Bulky ligands favour the square pyramid” is true only of ligands whose repulsion is steep — rising more than fifteen-fold in its logarithm across its own separation — and false of every softer repulsion, which favours the bipyramid more strongly than the bonds do. VSEPR’s own account of five-coordination never distinguishes the two, because it treats domains as repelling without saying how steeply — and the bond-counting side of the story never sees the ligands’ contact with each other at all.

Still open: the exact energy, and the π rows #

The obvious open question is the exact σ energy. The switch shares here are against the fourth-order term alone, and the essay before found that the exact energy confines the square pyramid to a narrow window of ratio and coupling even with no contact. Whether the exact energy makes the steep contact’s job easier or harder — whether the higher-order terms favour the 98° pyramid or the bipyramid — is the same bisection with the exact energy in place of the expansion, and it decides whether the shares of a hundred to a thousand are the right order at all.

The nearer question is the lead’s second half, left untouched: π donation. Two π functions on each ligand add rows the σ-only energy does not have, and π donation from phenyl rings into an empty central orbital is one of the explanations offered for pentaphenylantimony’s shape. Whether π rows move a main-group centre’s effective q towards the window below 0.862 — where the square pyramid needs no contact at all — would say whether the steep contact is necessary or only one of two sufficient causes.