Molecular Geometry & Bonding
what the shape decides
Beyond the octet

Hypervalency without d orbitals

Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre four-electron, and the textbook account has been known to be wrong for fifty years.
10 min read 7 figures Taught confidently, and wrongA basis is not a thing

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.

sulfur hexafluoride — OhThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.FFFSFFFOhprincipal axis C49 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates7 atoms
Fig. 1 Sulfur hexafluoride, point group Oₕ. Six equivalent bonds, an octahedral arrangement, and twelve electrons in the valence region — none of which requires d orbitals to explain.

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.

Splitting goes with overlapFor each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.1s-1sσ overlapS = 0.39002pz-2pzσ overlapS = 0.45852px-2pxπ overlapS = 0.75291s-2pxsymmetry forbids itS = 0 exactlyno splittingoverlaps computed at 2.8 bohr, levels in proportionone electron
Fig. 2 The two-centre version of the same construction, drawn from computed overlaps. The three-centre case adds a middle level that sits where it started, and the non-bonding character of that level is what lets four electrons occupy the system without populating anything antibonding.

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.

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 machinery.

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 this site. 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.

phosphorus pentafluoride — D3hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.FFFPFFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms
Fig. 3 Phosphorus pentafluoride, D₃ₕ. The axial pair forms one three-centre system and the three equatorial bonds are ordinary two-centre ones — which predicts both the longer axial bonds and the preference of electronegative substituents for those positions.

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.

6 sites, minimisedThe arrangement of 6 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.octahedral90.00° × 12180.00° × 3repulsion minimised, angles measured off the result6 sites
Fig. 4 The octahedral arrangement six regions produce, from minimising repulsion. The geometry is the same whichever bonding account is used, which is why it cannot decide between them — and why the discriminating evidence has to be the charge distribution and the computed d participation.

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.

Nothing here computes it. This site’s machinery is one-electron and hydrogenic. The d-participation numbers quoted come from the literature, and the figures show geometries and symmetry rather than the electronic structure that settles the question.

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.

methane — TdThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms
Fig. 5 A second-period case where the rule holds absolutely. Carbon has four orbitals, four bonds, eight electrons, and no possibility of more — the constraint is the orbital count and it is not negotiable.

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.

The 2pz orbitalThe 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one.2pzencloses 90% of the densitycontour at |ψ| = 9.48e-30 radial nodes1 angular nodecontour solved for by integrating the density1 shell · one electron
Fig. 6 One p orbital on the central atom, with lobes of opposite sign pointing along an axis. That single orbital, plus one orbital on each of two neighbours, is the whole of a three-centre system.
2pz with 2pz at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.2pz · 2pzS = 0.45852sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 7 The interaction at one end of it. Two such interactions, one on each side, share the central p orbital between them — which is what makes the system three-centre rather than two.

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.

Where the ladder goes next

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.

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 this site does not perform, and none of these drawings is evidence either way.

Where this goes next

1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron Overlap decides Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it. axial — 2 sitesneighbours at 90°, 90°, 90°, 180°equatorial — 3 sitesneighbours at 90°, 90°, 120°, 120°the two are not equivalentneighbour angles measured, not assumed5 sites Five sites are not alike Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it. HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms Delocalisation Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.