The square that wastes an orbital

Comparing two shape models puts two accounts of a five-coordinate molecule’s shape side by side, and on the way turns up something about the count they use:

It was treated throughout as a property of the molecule — ligands plus lone pairs minus four — and the planar arrangement shows it is a property of the arrangement, because a central orbital pointing nowhere useful is a central orbital that does not count.

That is a suspicion rather than a measurement, and the measurement is available. The orphan count is not a formula in the first place: it is what comes out of reducing a molecule’s ligand σ set in its own point group and asking how many of the combinations find a central orbital of the same species with nothing already in it. Running that on a different arrangement of the same atoms costs a different set of coordinates and nothing else.

Twenty-six arrangements of the same ten molecules can be worked in a tabulated group. The formula is right for twenty-three of them.

The three that break it #

Three failures, one cause. In every other arrangement here with four or more ligands, four of the ligand σ combinations transform as one of the centre’s four valence orbitals. In these three, only three do.

Why a flat arrangement is short by one #

A main-group centre has four valence orbitals: an s, which is totally symmetric in any group, and three p orbitals, which transform the way the three directions in space do.

Put every ligand in one plane and the third direction has nothing in it. Every ligand σ function lies in the plane, so no combination of them changes sign across it — and the p orbital perpendicular to the plane transforms in a way that nothing in the σ set does. It is not that the orbital is high in energy or badly oriented. There is no combination for it to pair with, as a matter of species, and that is a statement about the arrangement rather than about the atoms in it.

It is worth seeing the species rather than the count, because the species is where the argument is airtight. Six ligands in a plane give a σ set spanning a₁g ⊕ e₁u ⊕ e₂g ⊕ b₁u in D₆ₕ. The centre’s s is a₁g and its three p orbitals are e₁u — the two in the plane — and a₂u, the one perpendicular. There is no a₂u anywhere in the σ set, and there is an e₂g and a b₁u in it that the centre has nothing to offer. Three combinations match, three do not, and one central orbital is left holding nothing.

The same molecule as an octahedron gives a₁g ⊕ eg ⊕ t₁u. The centre’s s is a₁g and its p orbitals are t₁u, so four match; the eg pair is what is left over, and it is left over for the reason at the heart of hypervalency — there is no central orbital of that species at all, which is a shortage of orbitals rather than a shortage of directions.

Two different failures, and the census separates them. A combination can go unmatched because the centre has no orbital of its species — that is hypervalency, and it happens in every arrangement — or because the orbital of its species is pointing somewhere the ligands are not, which is what a flat arrangement adds. The first is a fact about four orbitals; the second is a fact about three directions.

So a flat arrangement offers three usable orbitals and every other arrangement here offers four, and the corrected count is

$$\text{orphans} = n - \min(m,\; 4 - L)$$

with $m$ the number of ligand combinations that match by species — three for a flat arrangement and four otherwise. The usual formula is this one with $m$ taken to be four always, which is right for every arrangement a molecule of this kind actually adopts.

The molecule that is square planar and does not break it #

Xenon tetrafluoride is the case that turns the finding from an oddity into a piece of chemistry, and it is the control the census supplies for free.

It is square planar. Its ligand set matches three of the centre’s orbitals, exactly as methane’s would in the same arrangement. And the formula is right for it.

The difference is the two lone pairs. They sit in the orbital pointing out of the plane — the orbital a square arrangement cannot use — so the arrangement wastes nothing. Methane in a square would waste it, because methane has nothing to put there.

There is a second reading available and the census rules it out. It might have been that a square arrangement of four ligands is simply a poor description — that the reduction is right and the lone pairs have been put in the wrong place. Xenon tetrafluoride settles that: the same square, the same three matched combinations, and the formula holds. What changes between the two molecules is not the reduction and not the arrangement but what is already sitting in the orbital the arrangement cannot reach.

Which is a reason rather than a coincidence. An arrangement that leaves a central orbital with nothing to do while leaving a ligand pair with no partner is an arrangement that has thrown away a bonding interaction, so it is not the arrangement the molecule takes. The formula is right for every real molecule because the counterexamples are structures nothing adopts — and that is the same shape of argument as the count that is safe because the population moves instead: a rule can be reliable and still be true for a reason other than the one given.

The coincidence that was impossible and is not #

The usual census claims, molecule by molecule, that a centre cannot have both a spare orbital and an unmatched ligand pair. It is a natural thing to believe — an orbital with nothing to do and a pair with nowhere to go ought to find each other — and it is true of every real molecule in the census.

It is false in exactly the three flat arrangements.

The reason is worth stating carefully because it is what the whole census is about: the matching is by species, not by counting. A leftover orbital and a leftover combination pair up only if they transform the same way, and in a flat arrangement they provably do not — one changes sign across the plane and the other cannot. Two quantities can both be positive because the thing that would cancel them is forbidden.

That failure is invisible from inside a census of real molecules, and not because anything in it is wrong. Every molecule in it is one whose arrangement uses everything it has.

How far the count moves #

Six of the ten formulas give one orphan count whatever arrangement they are put in. Four give two.

Which four they are is not random. Methane, phosphorus pentafluoride and sulfur hexafluoride are the three with four or more ligands and no lone pairs, and they are the three whose flat arrangement wastes something; carbon dioxide’s two arrangements agree because two ligands cannot leave anything over however they are placed. A molecule’s count is arrangement-dependent exactly when it has enough ligands to fill a plane and nothing already occupying the perpendicular orbital — which is a condition on the composition after all, though not the one the formula states.

What does not move is the bound. No arrangement of $n$ ligands leaves fewer than $n - 4$ pairs over, because a main-group centre has four orbitals and no shape can give it more. So the half of the argument that matters survives untouched: hypervalency is not about d orbitals and it is not about the shape either — a molecule with more than four ligands has unmatched pairs in every arrangement whatever. What the shape decides is only how many more than the minimum.

What this does to the three-centre picture #

The orphan count is not an abstraction: each unmatched pair is a three-centre four-electron bond, a filled combination of two ligands with a non-bonding partner and no central orbital in it. The count is therefore a count of objects, and an arrangement changing the count changes how many of those objects there are.

That has a consequence worth stating. In the three flat arrangements, one of the extra orphaned pairs is orphaned for a reason that has nothing to do with the ligands being too many. A flat pentagon of five fluorines around phosphorus needs two three-centre systems where a trigonal bipyramid needs one — and the second is not paying for a fifth ligand, it is paying for the shape. A structure can be hypervalent by more than its formula requires, and the excess is a property of the arrangement.

It also says which way the energy runs, without computing one. Each extra three-centre system puts a pair into a non-bonding combination rather than a bonding one, so a flat arrangement is short of a bonding interaction that a three-dimensional one has — quite apart from whatever the ligand–ligand repulsion is doing. The two effects push the same way, which is probably why no molecule in the census has to choose between them.

What is quoted, and what is computed #

Nothing is quoted. There is no measurement here at all: the input is a set of directions, and the output is a set of integers.

Each arrangement is built as a centre at the origin and ligands on the unit sphere — bond lengths are all one, because nothing in a symmetry reduction can see a bond length, and saying so is what makes a computed orphan count a statement about shape rather than about a structure. The point group is then found the way this collection always finds one: by applying every candidate operation and keeping those that permute the atoms, rather than by looking up what a five-coordinate molecule is supposed to be.

The lone pairs are the one input that is not computed. They are assigned to central orbitals in the usual order, which is a Lewis structure and not a calculation — and it is where the two counts could have differed for a reason that had nothing to do with symmetry.

Four of the thirty arrangements are not in the table. Three are linear, and a linear molecule has an infinite group that is handled through a finite stand-in — which is legitimate for a molecule that is linear and misleading for a comparison between arrangements, since the stand-in would be answering about a different group from the ones the other rows are in. The fourth is a pentagonal pyramid, whose group C₅ᵥ has no table in this census.

What this cannot say #

Nothing here says which arrangement a molecule takes. The census is symmetry and occupancy only; there is no energy in it, and the observation that the counterexamples are unadopted structures is an interpretation of the pattern rather than a result. What would test it is a repulsion energy for each arrangement beside its orphan count, and a repulsion model could supply one.

The correction is stated for flat arrangements and demonstrated on three of them. Any arrangement whose ligands span fewer than three independent directions has the same problem, and the flat ones are the only such case here. A linear arrangement of four ligands would be short by two, and nothing in the census reaches it.

The lone-pair assignment is an input and it does the work in one place. Xenon tetrafluoride’s formula is right because its two lone pairs are put in the orbital a square wastes, and nothing here computes that they go there — it is the Lewis structure. Putting them elsewhere would give a different count and no check here would notice. What can be said is that the assignment is the same one used at every other arrangement of the same molecule, so the comparison between arrangements does not depend on it even though the absolute count does.

And the count is about σ only. Every π interaction is left out, as it is in every σ-only count — which matters most for exactly the flat arrangements, since a ligand with a π function perpendicular to the plane could pair with the orbital a σ set cannot reach. That would not repair the arithmetic; it would change what is being counted.

What the census requires #

The formula holds at every real arrangement, molecule by molecule rather than on average — which is the standard result, re-derived on a different route, and it had to survive or the census would be measuring an error.

And some arrangement breaks it. Stated as an existence rather than as a proportion, because the finding is that the count can be broken at all: a check demanding three failures would fail the day a better character table admitted the pentagonal pyramid.

No arrangement matches more than four, which is the bound the whole argument rests on and the one thing here that cannot depend on shape.

A tetrahedron and a square of the same four ligands leave different numbers over. That is the refusal: if the two agreed there would be nothing to find, and the claim names both numbers rather than their difference.

The same wasted orbital, in a different part of the periodic table #

The mechanism found here is not peculiar to main-group hypervalency. It is one geometric fact with two well-known consequences, and putting them beside each other is the strongest evidence that the reading is right.

A planar arrangement of ligands has a horizontal mirror plane, and every σ combination the ligands can form is symmetric under it. The central atom’s p orbital perpendicular to the plane is antisymmetric under the same mirror. It therefore has no partner among the ligand σ set, at any planar geometry, whatever the ligands are and however many of them there are.

That single orbital is what goes missing from a main-group centre’s four, leaving three and breaking the census formula.

It is also, exactly, the orbital that makes a square-planar transition-metal complex count sixteen electrons rather than eighteen. A metal brings nine valence orbitals; in a square plane the perpendicular p is unusable for the same reason; eight orbitals remain; sixteen electrons fill them.

One mirror plane, one orphaned p orbital, two counting rules in two different fields — and neither is usually stated as a consequence of the other.

The unification is worth having because it says what to expect elsewhere. Any centre in a planar ligand environment loses one p orbital from its usable set, so its count drops by two electrons relative to the three-dimensional case. That is a prediction about arrangements rather than about elements, and it is the same statement this essay makes about a square of four ligands round a main-group atom.

Still open: the energy, and six coordination #

The obvious open question is the energy the census deliberately does not have. Every arrangement here has a computable repulsion energy and a computed orphan count, and putting the two on one axis would test the reading above directly — that the arrangements breaking the formula are the ones nothing adopts. If the flat arrangements are also the expensive ones, the two accounts agree and the formula’s reliability has an explanation; if some cheap arrangement breaks it, the explanation is wrong and the formula is merely lucky.

The nearer question is six coordination, which matters for a different reason too. An octahedron and a trigonal prism both leave two pairs over — the census says so above — so the orphan count cannot distinguish them, exactly as the charge-sharing criterion could not. Two independent counts that are both silent on the same comparison is a stronger statement than either being silent alone, and it says that whatever decides between an octahedron and a prism is not a count of anything.