Hypervalency without d orbitals

Sulfur hexafluoride has six bonds to sulfur and twelve electrons round it. The octet rule says eight. The standard resolution, in a great many textbooks, is that sulfur uses its 3d orbitals to expand its valence shell into a d²sp³ hybrid set.

That explanation is wrong, and it has been known to be wrong since the 1970s.

Why the d-orbital account fails #

Three reasons, of increasing severity.

Energy. Sulfur’s 3d orbitals lie roughly 10 electron volts above its 3p. Orbitals interact in proportion to overlap divided by energy difference, as the overlap essay sets out, and a gap that size makes the mixing negligible.

Size. Free-atom 3d orbitals are far more diffuse than the 3s and 3p, so they are in the wrong place to overlap effectively with a fluorine 2p at bonding distance.

Computation. When the d participation is actually calculated — by population analysis on a good wavefunction — it comes out at a few per cent. That is not nothing, but it is a polarisation function improving the description of the s and p bonding rather than a full set of bonding orbitals.

The d-orbital story was a reasonable guess in the 1930s, when the alternative was no explanation at all. It stopped being reasonable once calculations could be done.

What is actually happening #

The modern account is three-centre four-electron bonding, and it is simpler than the thing it replaces.

Take three atoms in a row: F–S–F, using one sulfur p orbital and one p orbital on each fluorine pointing along the axis. Three atomic orbitals give three molecular orbitals: one bonding, one non-bonding, one antibonding.

Put four electrons in. Two go into the bonding orbital, two into the non-bonding one, and the antibonding orbital stays empty. The result is a net bonding interaction spread over three centres, using one sulfur orbital rather than two.

Three such systems, using sulfur’s three p orbitals along the three axes, give six bonds in an octahedron. The sulfur has used its 3s and three 3p orbitals and nothing else.

Where the electrons are #

The account has a consequence that distinguishes it sharply from the d-orbital one, and the consequence is observed.

The non-bonding orbital of a three-centre system has a node at the central atom — its amplitude is on the two outer atoms only. So the four electrons are not shared equally: substantial negative charge sits on the terminal atoms and positive charge on the centre.

That predicts hypervalent molecules should be strongly ionic in character, with a markedly positive central atom. Calculations agree: the sulfur in SF₆ carries a large positive charge, and describing the molecule as six covalent bonds understates how polar it is.

It also explains a rule that the d-orbital account cannot: hypervalency happens almost exclusively with electronegative terminal atoms. SF₆ exists and SH₆ does not, because the three-centre arrangement requires the outer atoms to accept charge. Fluorine, oxygen and chlorine will; hydrogen and carbon will not.

Why the geometries come out right #

The arrangement of a hypervalent molecule follows from the number of three-centre systems and lone pairs, and the repulsion minimisation gives the same shapes without any bonding model at all.

That agreement is worth noticing rather than treating as confirmation. Both accounts predict an octahedron for six regions, and neither is evidence for the other — the geometry is over-determined by the counting, and it does not discriminate between the two bonding pictures.

What does discriminate is the electronegativity requirement, the charge distribution, and the computed d participation. All three favour the three-centre account.

What the six fluorines are allowed to span #

There is a fourth discriminator, and it needs no calculation at all — only the shape.

Sulfur hexafluoride is octahedral, so its group is Oh: forty-eight operations, generated from the coordinates by multiplying the found operations together until the set closes, and sorted into ten conjugacy classes. Ask what the six fluorine sigma orbitals span under those operations, and the answer is fixed before any bonding question is raised.

The six fluorine orbitals reduce to a₁g ⊕ eg ⊕ t₁u, with every multiplicity coming out a whole number — which is the check. Three species for six combinations, and the sulfur’s s and p supply only two of the three.

They span a₁g ⊕ eg ⊕ t₁u. That is one totally symmetric combination, a doubly degenerate pair, and a triply degenerate set: six orbitals, three symmetry species.

Now compare what sulfur has to offer. Its 3s orbital is a₁g. Its three 3p orbitals are t₁u. Between them they match four of the six ligand combinations, and the eg pair has no partner among sulfur’s s and p orbitals.

That is precisely the observation the d-orbital account was invented to answer. Sulfur’s 3d orbitals include an eg pair, so bringing them in supplies the missing partner and gives six bonds. The argument is elegant and its premise is that every ligand combination must find a partner.

The premise is wrong, and the eg pair is the whole of the difference between the two accounts. Left without a partner it becomes non-bonding, and the two electrons occupying it sit entirely on the fluorines — which is the node at the central atom described above, arrived at from the symmetry rather than from the three-centre picture, and which is why the sulfur ends up strongly positive.

The Oh table the reduction sums over has forty-eight operations in ten classes, generated from the molecule’s own coordinates. The eg row is the one this essay turns on: a two-dimensional species that the central atom’s s and p cannot supply and its d can, which is where the whole dispute lives.

So the two accounts differ over one representation, and they make different predictions about it. On the d-orbital account eg is bonding and its electrons are shared. On the three-centre account it is non-bonding and its electrons are on the fluorines. The computed charge distribution decides, and it decides for the second.

The axial preference, explained #

A case where the three-centre picture predicts something the d-orbital one does not.

In a trigonal bipyramidal molecule like PF₃Cl₂, the more electronegative substituents go axial. That is a solid empirical rule.

Three-centre bonding explains it directly. The two axial positions form one three-centre four-electron system, in which the terminal atoms carry substantial negative charge; the three equatorial bonds are ordinary two-centre bonds. So the axial sites are the electron-rich ones, and the more electronegative substituents prefer them.

The d-orbital account has no mechanism for the preference, because in it all five bonds are equivalent hybrids.

Why the wrong version survives #

Worth asking, because the correction is fifty years old and the textbooks have not all followed.

It is simple. “Expand the octet using d orbitals” is one sentence and requires no new mathematics.

It preserves the octet rule’s framework. Everything is still two-centre two-electron bonds; only the count changes.

Nothing visible goes wrong. The geometry comes out right either way, so a student using the wrong account gets the right shapes and never encounters a contradiction.

That last point is the important one and it recurs throughout chemistry. An explanation that produces correct predictions in the cases a student meets is very difficult to dislodge, because the evidence that would dislodge it is not in the syllabus.

What “hypervalent” even means #

A terminological note, since the word carries an assumption.

“Hypervalent” presupposes that exceeding an octet is anomalous. On the three-centre account nothing is exceeded: sulfur uses four orbitals and forms bonds that are spread over more than two centres, which no rule forbids.

Some authors prefer “hypercoordinate”, which describes the observation — more than four neighbours — without embedding an explanation. That is better practice, and the older word is used here because it is what a reader will meet elsewhere.

The molecules the argument covers #

Worth listing, because “hypervalent” covers a family with a common structure.

Six-coordinate: SF₆, PF₆⁻, SiF₆²⁻. Three three-centre systems along three axes, using the central atom’s three p orbitals.

Five-coordinate: PF₅, PCl₅. One three-centre system along the axis, two ordinary bonds in the equatorial plane, plus one more — which is exactly why the axial and equatorial sites differ in electron distribution as well as in geometry.

Three-coordinate with a lone pair: the trihalide ions like I₃⁻. A single three-centre four-electron system and nothing else, which is the simplest possible example and the one worth learning first.

The common thread is that the central atom is from period three or below, and the terminal atoms are electronegative. Both conditions follow from the mechanism: the central atom needs accessible p orbitals and enough size to accommodate the neighbours, and the terminal atoms have to accept the charge that the non-bonding orbital places on them.

Where the model stops #

Two limits.

Three-centre bonding is itself a description. It is a choice of basis, and a delocalised molecular orbital treatment of SF₆ gives the same wavefunction described differently. The claim being made is not that three-centre bonds are real objects; it is that d orbitals do not contribute significantly, which is a statement about a computed quantity.

The symmetry half is computed; the charges are not. That the six fluorine orbitals span a₁g ⊕ eg ⊕ t₁u is produced here, from the coordinates, and it is exact. That the eg pair is non-bonding rather than stabilised by sulfur’s 3d orbitals is not — it requires knowing how much the 3d orbitals actually contribute, which is a calculation not performed here. The d-participation numbers and the atomic charges quoted come from the literature.

That split is worth stating precisely rather than glossing, because it is where the essay’s argument is load-bearing. Symmetry says that sulfur’s s and p orbitals cannot reach the eg combination and that the 3d orbitals can. It cannot say whether they do. What symmetry contributes is the identification of the single quantity the two accounts disagree about, which is a smaller contribution than settling the question and is what makes the question settleable.

What it costs #

The reduction costs a group and a division.

Generating Oh is the larger part: candidate axes drawn from the structure, each tried as a rotation at every order up to eight, as a mirror and as an improper rotation, then closed under multiplication to forty-eight elements and sorted into classes by conjugating each by every other. On a seven-atom molecule that is a few tens of thousands of coordinate transformations, and it takes a few milliseconds.

The reduction itself is ten multiplications and a division by forty-eight, per representation.

What that buys is a claim that can fail in a way the d-orbital argument cannot. The multiplicities must come out as whole numbers, and there is nothing a fractional multiplicity could mean — so a miscounted character, a class matched to the wrong column or a wrong class size all produce an answer the arithmetic refuses. Nothing else in the pipeline would notice any of the three.

The reduction is also tested in the direction that can fail. A span claimed as something the group does not give is refused; a forbidden transition declared allowed is refused; a linear molecule, whose group has no finite order to divide by, is refused rather than approximated. A check that has never rejected anything proves nothing, and the six fluorine orbitals spanning a₁g ⊕ eg ⊕ t₁u is worth exactly as much as the reduction’s willingness to say otherwise.

Counting electrons honestly #

A last point about the octet rule, since this essay has been about a molecule that appears to break it.

The rule is a statement about second-period elements, where it is essentially inviolable: carbon, nitrogen, oxygen and fluorine have four valence orbitals — one 2s and three 2p — and four orbitals hold eight electrons. There is nowhere else to put any.

Below the second period the rule is a guideline. Not because d orbitals become available, but because the atoms are larger, can accommodate more neighbours geometrically, and can form the multi-centre bonds that spread electrons over more atoms than orbitals.

So “expanded octet” is a misleading phrase. Nothing expands: sulfur in SF₆ uses four orbitals, exactly as carbon does, and the extra bonds come from those orbitals being shared over more centres. Counting bonds and counting orbitals are different operations, and the octet rule is about the second.

Framed that way there is no exception to explain, which is a more satisfying position than having a rule and a list of things that break it.

What the three-centre system looks like #

The construction can be drawn, and it is simpler than the d-orbital alternative it replaces.

One p orbital on the central atom, with lobes of opposite sign along an axis, plus one orbital on each of two neighbours, is the whole of a three-centre system. Two such interactions sharing one central p orbital is what makes the system three-centre rather than two, and it is what the count needs.

The interaction at one end of it is an ordinary σ overlap of two p functions. Nothing about it is unusual; what is unusual is that the central orbital is shared between two of them, which is the arrangement the accounting requires and the one a two-centre picture has no room for.

Three atomic orbitals give three molecular orbitals: bonding, non-bonding, antibonding. Four electrons fill the first two. The non-bonding one has a node at the centre, so its electrons sit on the outer atoms, which is why the terminal atoms carry negative charge and why they need to be electronegative.

One central orbital, two bonds, four electrons — and no d orbital anywhere.

The evidence that seemed to support it, and why it supported nothing #

The forty years are worth accounting for, because the d-orbital story did have a quantitative argument behind it and the argument is a good illustration of a trap met elsewhere too.

The argument was that adding d functions to sulfur’s basis set lowers the computed energy of sulfur hexafluoride substantially — by tens of kilojoules a mole, far more than a rounding. That looked like a measurement of d participation: put the orbitals in, and the molecule uses them.

It is not, and the reason is that the same experiment succeeds everywhere. Adding d functions to oxygen lowers the energy of water. Adding them to carbon lowers the energy of methane. Nobody proposes that water is d-hybridised or that methane uses sulfur-like expanded valence, and the energy falls all the same.

What such functions actually do belongs to the theory of basis sets rather than to bonding. A d function on a second-row atom is a polarisation function: it lets the s and p density distort away from spherical symmetry towards the bonds, which every molecule wants and which no combination of s and p alone can produce. Its job is to improve the shape of the electrons that are already there, not to hold new ones.

The measurement that distinguishes the two readings is the occupancy, and it settles it. Population analyses of sulfur hexafluoride put a few tenths of an electron in sulfur’s d functions — enough to matter for the energy, nowhere near the two electrons per bond a d²sp³ description requires, and comparable to what the same analysis finds on atoms nobody argues about.

The energetic support is also available from the other direction. Sulfur’s 3d orbital in the free atom is diffuse and lies far above the 3p, high enough to be barely bound; the fluorine 2p functions it would have to interact with lie well below the 3p. Two orbitals separated by that much interact weakly whatever their symmetry permits.

So the case for d participation rested on a quantity that responds to any added flexibility, and it was never checked against the quantity that would have distinguished flexibility from occupation. That is the same failure the essay’s closing paragraph names, in its most concrete form: a number was measured, it moved in the expected direction, and it was a number the rival account moves too.

Where to read on #

The related case of electrons spread over several centres is delocalisation.

The geometry that goes with five-coordination is five sites are not alike, where the axial preference appears.

And the general lesson about descriptions promoted into claims is hybridisation does not explain.

The general shape of the correction is worth restating in one sentence, because it recurs. An account that was built to explain a fact, that reproduces the fact, and that is never tested against anything else, has not been confirmed by the fact it was built for. The d-orbital story reproduced sulfur hexafluoride’s octahedral geometry, which every account reproduces, and the geometry was the only evidence anybody asked it for.

The positive form of the same lesson is that the argument was settled by finding a quantity the two accounts disagreed about — the occupancy of one representation — and then measuring it. That is what a real test looks like, and it took forty years of the geometry being cited as evidence before anybody constructed one.

What the pictures here cannot show. The figures on this page draw nuclei and symmetry. Whether d orbitals participate is a question about an electron distribution, settled by calculations not performed here, and none of these drawings is evidence either way.