The count is the population

Four is all s and p can match reduced the ligand σ set of ten molecules in each one’s own point group and matched it against the central atom’s four valence orbitals. What came out was an identity rather than a rule: with n ligands and L lone pairs, the number of ligand combinations with no partner is n + L − 4, and the electron count around the centre exceeds eight by exactly twice that. Hypervalency is n + L > 4 and nothing else.

It ended by naming what a census cannot do. What it does not do is compute the one number that would make it quantitative: how much of the orphan pair’s density actually sits on the ligands, which is a population rather than a count.

And it made a prediction. The ligand charge in a hypervalent molecule grows with the orphan count, so sulfur hexafluoride’s fluorines should carry more than phosphorus pentafluoride’s axial ones, which carry more than its equatorial ones.

The population is computed here. Two thirds of the prediction is right and the general form of it is wrong, and being wrong in a specific way is what turns it into a law.

A σ model built from coordinates #

The model is the smallest one that can answer the question, and every element of it comes from the molecule rather than from a table.

The basis is the central atom’s four valence orbitals — s, and the three p — and one σ orbital on each ligand, pointing at the centre. The coupling between a p orbital and a ligand is the component of that ligand’s direction along the p axis, which is the angular part of an overlap and is the same construction the angular-overlap model uses for a d shell. The geometry supplies every off-diagonal element; nothing is fitted.

Lone pairs need no special treatment, and that is worth saying because the census had to handle them by hand. A lone pair is a central orbital that no ligand combination transforms like, so it comes out of the diagonalisation as a non-bonding level and is filled by the electron count. Xenon difluoride’s three lone pairs are its s and its two perpendicular p orbitals, and the model finds them rather than being told.

The identity #

Set the ligand’s Coulomb integral equal to the centre’s — no electronegativity difference anywhere — and the answer is exact.

$$\bar{q}_{\text{ligand}} = -\frac{\text{orphan pairs}}{\text{ligands}}$$

Xenon difluoride: one orphan, two fluorines, −0.5 each. Sulfur hexafluoride: two orphans, six fluorines, −0.3333. Chlorine trifluoride: one over three. Phosphorus pentafluoride: one over five. Xenon tetrafluoride: two over four. And the five molecules with no orphan pair — carbon dioxide, water, ammonia, boron trifluoride, methane — put exactly nothing on their ligands, to twelve decimal places.

That last group is the control and it is what makes the rest a result. A model in which every ligand acquired charge would be measuring its own arithmetic; a model in which four-coordinate species put nothing on their ligands and hypervalent ones put a definite amount is measuring the orphan pair.

The mechanism is visible in the three-centre case and generalises. The three-centre four-electron arrangement has a non-bonding orbital with exactly no amplitude on the middle atom, so the pair in it sits entirely on the two ends: half an electron each, and a whole positive charge on the centre. Every hypervalent molecule here is that arrangement repeated as many times as it has orphan pairs, and the pair spreads over however many ligands the symmetry lets it reach.

The prediction, refused by its own divisor #

The census predicted an ordering and got two thirds of it.

Phosphorus pentafluoride’s axial fluorines do carry more than its equatorial ones. The model finds two sets without being told there are two — three at −0.1333 and two at −0.3000 — and the two are the axial pair, which is the orphan’s own. Sulfur hexafluoride’s fluorines at −0.3333 do carry more than phosphorus pentafluoride’s axial ones at −0.3000.

What fails is the general form. Xenon difluoride has one orphan pair and charges its fluorines by 0.5 — half again as much as sulfur hexafluoride’s two orphan pairs manage. Ordering the ten by orphan count does not order them by ligand charge, and the reason is the divisor: an orphan pair is two electrons however many ligands they have to spread over.

So the census’s prediction was a statement about the numerator with the denominator left out, and the three cases it named happen to be three where the denominators cooperate. That is exactly the kind of near-miss worth recording, because a prediction that is right about the cases it names and wrong as a rule is the hardest kind to catch. A ranking is not a difference is the same failure in a different subject: an ordering that holds for the examples in front of somebody, taken for a law.

What this says about which molecules exist #

The identity has a consequence for chemistry that the census could only gesture at.

Each ligand of a hypervalent molecule has to carry a definite amount of negative charge, set by the arithmetic above and not negotiable. A ligand unwilling to carry it makes the arrangement expensive, which is why the hypervalent species are the fluorides, the oxides and the chlorides — and why SH₆ and PH₅ are not compounds. That argument has been in this collection since hypervalency is about the ligands, stated qualitatively; the identity puts a number on the demand. Six bonds and four orbitals is where the symmetry half of it was settled.

And the number is largest for the smallest hypervalent molecules. Xenon difluoride demands half an electron per fluorine; sulfur hexafluoride demands a third. A reader would expect the demand to grow with the coordination number, and it falls.

What the electronegativity adds #

Switching the ligand’s Coulomb integral to a realistic value adds a second, ordinary contribution on top of the identity.

Sulfur hexafluoride goes from −0.3333 to −0.4667 per fluorine; methane goes from exactly zero to −0.2425 per hydrogen. So a real molecule’s ligand charge is the sum of two things — what the orphan pair puts there, which is the identity, and what the electronegativity difference puts there, which every molecule has — and they are separable because one of them survives the other being switched off.

That separation is the reason the identity was computed at zero electronegativity rather than at a realistic one. At h = 1 the ten molecules give a smooth-looking spread of ligand charges with no exact relation in it at all, and the relation would have been invisible.

Why the divisor is the interesting half #

It is worth asking why an orphan pair spreads over every ligand rather than over the two it belongs to, because the answer is the reason the identity has a denominator at all.

In xenon difluoride the orphan is a single combination of two fluorine σ orbitals, and there are only two fluorines for it to be on. In sulfur hexafluoride there are two orphan combinations — the pair transforming as eg in the octahedral group — and the pair has amplitude on all six fluorines, because that is what an eg combination of six σ orbitals looks like. Nothing chooses two of the six.

So the denominator is a symmetry statement rather than an averaging convention. The orphan combinations are irreducible representations of the whole ligand set, and an irreducible representation of a set of six equivalent objects has amplitude on all six. The count is a count of combinations and the population is spread over the ligands those combinations are built from, and the two are related by the size of the set.

That also says when the identity should be expected to fail: whenever the ligands are not equivalent. Chlorine trifluoride’s three fluorines split one and two, phosphorus pentafluoride’s five split three and two, and in both cases the mean is still the identity while no individual ligand carries it. A molecule with five inequivalent ligands would have five different charges averaging to the identity, and quoting the mean as though it were a ligand’s charge would be quoting something no atom has.

The charges sum, which is the check #

The arithmetic check is that the central atom’s charge and the ligands’ sum to the molecule’s, exactly, in every case. They do, to a nanounit.

That is not decoration. A population analysis of this kind assigns each electron’s density to the orbital it is in and each orbital’s density to the sites its coefficients are on, and any slip in the bookkeeping shows up as a total that is not zero. It is the same check the three-centre charges were validated with, extended from three sites to ten.

The molecular-orbital level scheme of an octahedral set is the object being filled: two of its ligand combinations transform as nothing on the central atom, so they sit at the ligand energy and hold their pairs there. That is the same identity in the case where the central atom has d orbitals available and still cannot use them all.

Where the orphan pair actually is #

A number is more convincing with a picture behind it, and the picture is the same one the three-centre argument has always had.

Take xenon difluoride. Three orbitals — a xenon p and two fluorine σ — combine into a bonding one, a non-bonding one and an antibonding one. Four electrons fill the first two. The non-bonding orbital is the antisymmetric combination of the two fluorines with exactly no amplitude on the xenon, because there is nothing of that symmetry on the central atom for it to mix with; so its two electrons are on the fluorines and nowhere else, half an electron each, and the xenon carries a whole positive charge.

Nothing about that is approximate. The zero on the central atom is a symmetry statement, and it is why the identity comes out at twelve decimal places rather than at three.

The same arrangement extended gives five centres holding three pairs and seven holding four, and the chain calculation gives the charges directly. The ends carry what the count demands, at every length, and the identity in this essay is the general form of that observation.

Sulfur hexafluoride’s six fluorines are one set by symmetry and carry −0.3333 apiece: two orphan pairs, four electrons, six ligands. That is the identity in its simplest form, and the reason a molecule that looks like it needs d orbitals does not need them to balance its charge.

Everything the model knows about which ligand combinations can find a partner comes from the group generated from the molecule’s own operations, and from nothing else. No parameter enters, which is why the identity is exact rather than approximate.

What this cannot say #

σ only. There is no π bonding anywhere in this model, and a fluorine ligand has lone pairs of π symmetry that donate some charge back. Real fluorine charges in SF₆ are smaller in magnitude than −0.47 for exactly that reason, and the identity is about the σ framework rather than about the molecule.

Populations are Mulliken-like. The density is assigned by squared coefficients in a non-orthogonal-free basis, which is the crudest partition available. A weight that depends on how it is weighed is the general caution, and the identity survives it only because at h = 0 every orbital is either entirely on the ligands or symmetric between them.

The lone-pair counts are the Lewis structures’. The census took them from the received structures and they are inherited here. The model does not decide how many lone pairs xenon has; it decides where the electrons go once it has been told.

No repulsion between the ligands. The charges above are all of one sign and all on the outside of the molecule, which is a situation in which they repel one another — and nothing in a one-electron model charges for that. The sites are not the same size is where this collection prices that repulsion, and the two calculations have never been put together.

And the geometries are the accepted ones. Nothing here optimises a structure, so the split between axial and equatorial in phosphorus pentafluoride is a consequence of the geometry rather than a prediction of it.

Whether the ligands can be repaid, and the symmetry answer #

The obvious objection to the identity is that it counts only σ. A fluorine has lone pairs of π symmetry, they sit close to the central atom, and if they donate back then the charge the σ arrangement forces onto the ligands is partly returned — which would make the identity an upper bound rather than an answer.

The reduction settles it without any calculation, and the answer is that for a main-group centre the repayment channel is almost entirely closed.

Take the octahedral case. Six ligands with two π functions apiece give twelve combinations, and in the octahedral group they span $t_{1g} \oplus t_{1u} \oplus t_{2g} \oplus t_{2u}$. Now ask what the central atom has to offer them. Its s transforms as $a_{1g}$, which appears nowhere in that list. Its three p orbitals transform as $t_{1u}$, which does appear — and which is already spent, in full, on the σ set.

So of the four π species the ligands present, three have no partner of any kind on a centre with only s and p, and the fourth competes with the σ bonding for the same three orbitals. There is no fifth channel. The donation the objection appeals to is not weak; it is forbidden.

Two consequences follow, and the second is an irony worth stating plainly.

The identity stands as an answer rather than a bound. The charge the σ arrangement puts on the ligands stays there, because symmetry supplies no route back.

And the orbitals that would supply one are the d orbitals. A $t_{2g}$ set and an $e_g$ set are exactly what the leftover π species need, and $t_{2g}$ is among them. So the d orbitals the whole hypervalency argument removes from the σ account turn out to be the only acceptors the π donation could use — which is not a rehabilitation of the $d^2sp^3$ story, since that story was about σ bonding and about holding twelve electrons, but it does locate the one place where a small d contribution is doing something a symmetry argument cannot dismiss.

The size of that contribution is a question about energies and this model has none. What the reduction establishes is narrower and is the part the identity needed: whatever the d orbitals do, the s and p framework has no way to return the charge, so the demand the census counts is a demand the ligands actually meet.

The argument also runs the other way and explains a preference the census does not contain. If π donation cannot repay the σ demand, then a ligand’s π lone pairs are simply parked — they cost the arrangement nothing and buy it nothing — and the only thing that matters about a ligand is how willingly it accepts σ charge. That is why the hypervalent compounds are fluorides and oxides rather than, say, amides or alkyls: the selection is entirely on the σ side, and a ligand’s π properties, which decide so much in transition-metal chemistry, decide nothing here.

Which is a real difference between the two kinds of centre rather than a detail of the bookkeeping. A transition metal has five d orbitals available to receive π density and its whole ligand chemistry is organised around them; a main-group centre has none, so its ligand chemistry is organised around one variable instead of two.

What was checked #

Charge is conserved: the central atom’s and the ligands’ sum to the molecule’s, to a nanounit, in all ten cases.

With no electronegativity difference the mean ligand charge is exactly minus the orphan count over the ligand count, to twelve decimal places, for every molecule — the identity the census asked for.

A molecule with no orphan pair puts exactly nothing on its ligands, which is the tripwire.

Phosphorus pentafluoride has two kinds of fluorine, found rather than assumed, and the axial pair carries the more.

The raw orphan count does not order the ligand charges, checked as a requirement that a reversal exist — so the census’s own prediction is refused by a computation rather than by a remark.

And an electronegative ligand takes more than the count gives it, which separates the two contributions.

Still open: π donation, and how the ligands share #

The obvious open question is the π donation the model has no room for. A fluorine’s lone pairs of π symmetry can donate into whatever is left on the central atom, and how much they donate is computable in the same framework with one more orbital per ligand. What that would say is how much of the σ demand is repaid — and whether the repayment is larger for the molecules that demand most, which would make the identity above an upper bound rather than an answer.

The nearer question is about the geometry. The identity says a hypervalent molecule’s ligands must share a fixed amount of charge, and it says nothing about how they share it; the axial and equatorial split in phosphorus pentafluoride comes out of the coordinates. So the arrangement that shares it most evenly is a computable thing, and whether real hypervalent geometries are the even-sharing ones — or whether they are set by the repulsion argument computed separately — is a comparison between two models that have never been put on one axis.