Four is all that s and p can match

The argument began by removing d orbitals from the account of sulfur hexafluoride and finding that nothing was lost: the molecule can be described with the central atom’s s and p alone, plus a set of ligand combinations that are not bonded to anything.

Everything since has worked outwards from that — three-centre bonding computed, the ligands’ role, the family of odd chains that generalises the three-centre bond.

What follows is the census. Ten molecules, each reduced in its own point group from its own coordinates, and one question asked of each: how many ligand combinations can find a partner among the four orbitals the central atom has?

The ceiling #

That ceiling is not an assumption. For each molecule the operations are generated from the coordinates, sorted into conjugacy classes, and used to reduce the representation the ligand σ functions span; the central atom’s p orbitals span whatever the translations do, and its s orbital is totally symmetric in any point group, because every operation fixes the centre.

Then the two lists are matched species by species, and the total dimension matched is counted. The result is at most four everywhere — and would not have been if some geometry had produced a σ set with five components among the s and p species.

Where the ceiling bites, and where it does not #

Reading the census row by row is the essay, because the ten molecules divide into four different situations.

Methane is the exact fit. Four ligands, σ spanning A₁ ⊕ T₂, and the central s and p spanning A₁ and T₂ — four matched, nothing left over, and eight electrons in four bonds. The octet is what happens when the ceiling is reached exactly.

Boron trifluoride is short. Three ligands span A₁′ ⊕ E′, and the central atom has an A₂″ p orbital pointing where there is no ligand. Three matched, one orbital spare — which is boron’s empty p, and is why boron trifluoride is a Lewis acid. Six electrons around the centre.

Water and ammonia reach the ceiling with lone pairs. Two ligands and two lone pairs; three and one. Four orbitals used, eight electrons, nothing left over, and the octet again.

And from ten electrons upwards something is left over. That is the hypervalent set, and it splits into two kinds.

Two ways to be hypervalent #

The census separates two mechanisms that the single word conceals, and the separation is the finding nobody expected.

Sulfur hexafluoride and phosphorus pentafluoride are hypervalent by symmetry. Six fluorines span A₁g ⊕ Eg ⊕ T₁u in an octahedron; the sulfur’s s is A₁g and its p is T₁u, so four are matched and the Eg pair has no central orbital of its symmetry at all. Not one that is occupied — one that does not exist among s and p. The same happens for the second A₁′ of the trigonal bipyramid.

Xenon difluoride and chlorine trifluoride are hypervalent by occupancy. Xenon difluoride’s two fluorines span exactly what carbon dioxide’s two oxygens span, and carbon dioxide is not hypervalent. The difference is that xenon has three lone pairs, so three of its four orbitals are already full and only one is available. Two combinations, one orbital, one left over.

Xenon tetrafluoride is both at once. Its four fluorines span A₁g ⊕ B₁g ⊕ Eu; the B₁g has no partner among s and p, and one of the three that do is taken by a lone pair. One orphan of each kind, and twelve electrons around a centre that holds eight.

The identity #

Once the ceiling is established the rest is arithmetic, and it is worth writing out because it makes hypervalency a definition rather than a phenomenon.

With $n$ ligands and $L$ lone pairs, the central atom has four orbitals, $L$ of them holding lone pairs, so at most $4 - L$ are available to match ligand combinations. The number left with no partner is therefore

$$\text{orphans} = \max(0,\; n + L - 4)$$

and the electron count around the centre, $2n + 2L$, exceeds eight by exactly twice that.

So hypervalency is $n + L > 4$, it is not about ligand count alone, and the excess electrons are not on the central atom: they are in ligand combinations that have no central partner, which is to say they are on the ligands.

That last sentence is the one the whole argument has been building towards, and it is checked in the census: the electrons on the centre never exceed eight in any row.

Where the four comes from #

The number four is doing all the work, and it is worth saying what it is and what it is not.

It is the number of valence orbitals a main-group atom has: one s and three p. It is not a statement about how many bonds an atom can make, and it is not a statement about how many electrons fit around it — those are the conclusions, and they follow from the four together with the ceiling.

Nothing in the symmetry analysis produces the four. It comes from the shell structure, which is a fact about the radial part of an atom’s orbitals and about where the next shell sits in energy. What the symmetry analysis produces is the matching, and the matching is what turns a count of orbitals into a count of bonds.

That division of labour is why the argument transfers so cleanly to transition metals, where the count is nine rather than four, and the eighteen-electron rule takes the octet’s place with no change in the reasoning — a point the eighteen-electron count makes from the other side of the periodic table.

Why the ligands are negative #

The picture the census produces has a measurable consequence, and it is why hypervalency turns out to be about the ligands.

A ligand combination with no central partner is a non-bonding orbital localised on the ligands, and the pair in it belongs to them. So a hypervalent molecule has more electron density on its ligands than a count of bonds suggests, and the effect grows with the number of orphan pairs.

That is why hypervalent molecules are found with the most electronegative ligands — fluorine, oxygen, chlorine — and essentially never with hydrogen or carbon. The orphan pairs have to be held somewhere, and an electronegative ligand can hold them.

The prediction is testable in the reverse direction too, and it holds: SH₆ does not exist, SF₆ does; PH₅ does not, PF₅ does. A count of ligands is the same in each pair and the electronegativity is not, which is a discrimination the older account — where the central atom expands its own valence shell — has no way to make. Electronegativity is a difficult quantity and it is being asked here only for an ordering, which is the use it can bear.

The same conclusion arrives from the other direction in odd chains of centres holding one pair more than they have bonds: the ends carry the charge the arrangement demands, and the count works without any orbital the central atom does not have.

The smallest case is three centres with four electrons and two levels occupied — one bonding and one non-bonding with a node at the centre, so the second pair sits entirely on the two ends. That is where the charge the hypervalent count requires actually goes.

What d orbitals would do #

The whole census is taken with four valence orbitals. Allowing d orbitals changes the ceiling from four to nine and the result would be that nothing is ever left over — SF₆’s Eg would find the metal’s own Eg, and the molecule would be six ordinary two-centre bonds.

That is the old account, and the reason for rejecting it is not that the arithmetic fails but that the d orbitals are too high in energy to participate meaningfully, which is a quantitative claim put in numbers without d orbitals.

What the census adds is that the alternative account is not a workaround. It produces an exact identity, it predicts which molecules are hypervalent from a count of ligands and lone pairs, and it puts the excess electrons where the measurements find them.

Where the model stops #

A lone-pair count is not a computed quantity. It comes from the Lewis structure, and for every molecule here it is unambiguous. For a molecule where it is not — where the number of lone pairs on the centre is itself in dispute — the census returns whatever it was given.

Symmetry says which combinations can mix, not how much they do. Two orbitals of the same species interact by an amount that depends on their energies and overlap, and a match in the census can be a weak interaction in the molecule.

The matching is by species and by count, not by energy order. Two components of the same species are matched to two central orbitals of that species without asking which is lower, and a bonding combination is not simply the lower of two when overlap is kept.

Nothing here is an energy. The census counts orbitals and electrons; whether the resulting molecule is stable is a different question and needs the quantitative arguments about orbital energies.

And the linear molecules are handled in a working group. An infinite-order axis has infinitely many operations and the reduction formula divides by the order, so carbon dioxide and xenon difluoride are reduced in D₂ₕ and the species names translated — which loses the distinction between two of them and does not affect any count here.

A descent from Oh to D₄ₕ splits the eg pair and the t₁u set, so the number of species available to match against changes with the symmetry. That is worth stating because it means the shortfall this essay is about is not a fixed number: it is a comparison between two reductions, and both depend on the group.

The census as a rule of thumb #

Stated as advice rather than as arithmetic, the census says three things a chemist can use without computing anything.

Count ligands and lone pairs together. Their sum, against four, decides whether a molecule is hypervalent. Two ligands and three lone pairs is as hypervalent as six ligands and none.

Expect the excess on the ligands. Every pair beyond the fourth is in a ligand-only combination, so a hypervalent molecule’s central atom is more positive than a count of bonds suggests and its ligands more negative.

And expect the geometry to follow the ligand set rather than the central orbitals. Four matched combinations cannot dictate the shape of six ligands, so what sets an octahedron’s angles is the repulsion between the ligands themselves — which is exactly what the repulsion model of shape computes by minimising repulsion on a sphere and never mentions orbitals in.

None of the three requires a character table. All three are consequences of one, and the third is the reason a repulsion model works as well as it does on precisely the molecules where an orbital account is hardest.

What is quoted, and what is computed #

Quoted: the lone-pair counts, which are the Lewis structures’; the structures themselves, which are measured; and the claim that d orbitals are too high to participate, which is computed separately and taken as settled here.

Computed: every point group, from the coordinates; every set of operations, by closure; every conjugacy class, by conjugation; every reduction, by the orthogonality relations; and every count in the census.

What was checked #

At most four ligand combinations transform as one of the central atom’s valence orbitals, in every geometry. This is the computed half and it is not arithmetic: a geometry whose σ set had five components among those species would break it.

The orphan count is n + L − 4 wherever that is positive, and what is left of the four is 4 − n − L wherever that is, and a centre never has both a spare orbital and an orphaned pair.

The electrons on the centre never exceed eight, and the count in excess of the octet is exactly twice the orphan count.

Every orphan is orphaned for one of two reasons, and the two are counted separately: no orbital of that symmetry, or an orbital of that symmetry already holding a lone pair.

And the refusal: a tetrahedron leaves nothing over and an octahedron leaves two. If both came out the same the census would be counting something other than what it claims.

What the word hypervalent should mean #

The term was coined for molecules whose central atom appears to exceed an octet, and it has been argued about ever since — some authors abandoning it, others keeping it while denying the orbital expansion it was named for.

The census suggests a definition that is neither: hypervalent means $n + L > 4$, which is a statement about a Lewis structure and can be read off one in a second. It carries no claim about orbitals, it applies to xenon difluoride and sulfur hexafluoride alike, and it comes with a prediction — the number of ligand-localised pairs — that is checkable.

What it gives up is the implication that something unusual is happening at the centre. Nothing is: the centre has four orbitals and uses at most four, in these molecules exactly as in methane. What is unusual is that some of the electrons in the Lewis structure are not at the centre at all, and the Lewis structure has no way of drawing that.

What the count does not explain, and what does #

The definition has a gap in it that is worth naming, because it is the gap where the d-orbital story used to live and something has to fill it.

Nothing in $n + L > 4$ mentions the period. The reduction, the orphan count and the two-electrons-per-orphan arithmetic run identically for a second-row centre and a third-row one: nitrogen has four valence orbitals exactly as phosphorus does, and the counting says a five-coordinate nitrogen would have one orphan combination and one ligand-localised pair, exactly as phosphorus pentafluoride does.

Nitrogen pentafluoride does not exist. Neither does an oxygen tetrafluoride, nor any stable five-coordinate carbon.

The old account had an answer to that — the second row has no accessible d orbitals — and the answer was wrong for the reason the argument began with. What replaces it is not an orbital argument at all. A second-row atom is too small. Five or six ligands round a nitrogen would sit far closer to one another than round a phosphorus, and the ligand–ligand repulsion is what refuses. The constraint is geometric, it has nothing to do with the central atom’s orbitals, and it is a different kind of statement from anything in the census.

The evidence that it is the right replacement is that the arrangement appears at carbon whenever the geometry is affordable, and it appears as exactly the structure the counting predicts.

The most-studied transition state in organic chemistry is a hypervalent carbon. A nucleophile approaching a carbon from behind a leaving group produces a five-coordinate arrangement with three ordinary bonds in a plane and a long, weak, three-centre four-electron system through the axis — the entering group, the carbon, and the departing one, sharing one bonding pair. Its two axial bonds are long and its three equatorial ones are normal, which is the sulfur tetrafluoride pattern in a carbon compound.

That structure is not stable and it is not exotic; it is the mechanism of a reaction class, and it is transient for a reason the counting argument makes precise. Carbon can adopt the arrangement and cannot hold it, because five groups round an atom that small is affordable only while two of them are half-detached.

So the account of hypervalency divides into two parts that were previously confused. Whether a hypervalent arrangement is electronically available is the counting question, and the answer is that it always is, for any main-group centre, with no orbital expansion required. Whether a particular molecule can hold it is a question about size and repulsion, and it is what excludes the second row and what makes the third row’s fluorides ordinary reagents.

Still open: how much of the orphan pair sits on the ligands #

The census counts. 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.

That is computable with models already in hand — the three-centre model gives it in closed form, and the chain generalisation gives it for five and seven centres — and putting a number on it would connect the symmetry argument to the measured charges. It would also make a prediction the census cannot: that 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.