Molecular Geometry & Bonding
what the shape decides
Beyond the octet

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.
11 min read 6 figures A basis is not a thingTaught confidently, and wrongSymmetry settles it

Benzene’s six carbon–carbon bonds are all the same length. That is a measured fact, and it is the whole of what needs explaining.

benzene — D6hThe 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.HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms
Fig. 1 Benzene, point group D₆ₕ. Six equivalent carbons, six equivalent bonds, and a sixfold axis — recovered here from the coordinates. A structure with alternating single and double bonds would be D₃ₕ, and it is not.

What the symmetry settles

Before any bonding argument, the group settles something.

A structure with alternating short and long bonds has a threefold axis, not a sixfold one: rotating by 60 degrees would take a short bond onto a long one. Its point group is D₃ₕ.

Benzene’s group is D₆ₕ, recovered from its coordinates by finding the operations that permute its atoms. So the alternating structure is not benzene’s structure, and no amount of bonding argument is needed to establish it.

That is a good example of symmetry doing work that is usually done less cleanly by a discussion of resonance.

The resonance picture, and how it is misread

The valence bond description writes benzene as a superposition of two Kekulé structures, drawn with a double-headed arrow between them.

The mathematics is correct: the wavefunction is a linear combination of the two, plus smaller contributions from other structures. The arrow means “is a superposition of”.

The misreading is nearly universal. Students take it to mean the molecule alternates between the two forms — flickering, oscillating, spending half its time in each. It does not. There is one structure, one wavefunction, and the Kekulé forms are basis functions in the expansion of it.

The analogy that helps: a mule is not a horse half the time and a donkey the other half. It is one animal, and the two parents are how it is described rather than states it visits.

Why the misreading persists

Three reasons, and the third is the interesting one.

The notation invites it. A double-headed arrow between two pictures looks like a process.

The word invites it. “Resonance” was borrowed from a physical phenomenon in which something does oscillate, and Pauling later said he regretted the choice.

Nothing corrects it. A student who believes benzene alternates makes almost no wrong predictions in a first course, because the observable consequences of the correct and incorrect readings coincide until quite late.

That is the same pattern as the d-orbital account of hypervalency and the hybridisation story: a wrong picture survives because the syllabus never tests it.

The molecular orbital account

The other framework describes benzene without any superposition at all.

Six p orbitals perpendicular to the ring combine into six molecular orbitals, spread over all six carbons. Three are bonding and hold the six pi electrons; three are antibonding and are empty. Every occupied orbital has amplitude on every carbon, so every bond is identical by construction.

No resonance, no arrow, no superposition — the delocalisation is in the orbitals rather than in a combination of structures.

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 case, from computed overlaps. Benzene’s pi system is the six-centre version of the same construction: more orbitals, the same principle, and combinations that extend over the whole ring rather than over two atoms.

Which account is right

Neither, in the sense the question intends.

Both are expansions of the same wavefunction in different bases, and a basis is not a thing. Push the valence bond treatment to include enough structures and it reproduces the molecular orbital answer; push the molecular orbital treatment to include enough configurations and it reproduces the valence bond one.

What is real is the total density, the bond lengths, the energy and the spectrum. Both descriptions get those right, and asking which is true is asking which coordinate system nature uses.

The stabilisation, and how it is measured

Benzene is more stable than a hypothetical molecule with three isolated double bonds, and the difference is quoted around 150 kilojoules per mole.

The number needs a caveat, because the hypothetical molecule does not exist. It is estimated from the hydrogenation enthalpy of cyclohexene multiplied by three, compared with benzene’s actual hydrogenation enthalpy — so the comparison is with a construct, and different constructs give different numbers between about 120 and 180.

That is worth stating rather than quoting a single figure with confidence. The stabilisation is large, real, and its value depends on what it is measured against.

What is measured, and what is inferred

Being careful about the evidence, because “benzene is delocalised” is supported by different observations of different strengths.

The bond lengths are measured. All six are 1.397 ångström, by diffraction and by rotational spectroscopy. That is direct and it is not in doubt.

The symmetry follows from them. D₆ₕ, recovered from the coordinates, which rules out the alternating structure without any bonding argument.

The stabilisation is inferred. It requires comparison with a hypothetical molecule, and the number depends on which hypothetical one is chosen.

The orbital picture is a model. Neither the six delocalised pi orbitals nor the two Kekulé structures is observable.

ethene — D2hThe 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.HHCCHHD2hprincipal axis C23 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms
Fig. 3 Ethene for comparison: a genuine localised double bond, D₂ₕ, with a C–C length of 1.334 ångström. Benzene’s bonds at 1.397 sit between this and a single bond at 1.54, which is the measurement the whole discussion is about.

Sorting the evidence that way is worth doing, because the strongest claims in this essay rest on the first two items and the interesting ones on the last two.

Aromaticity, and what it actually requires

Delocalisation alone is not aromaticity. The extra requirement is a count.

Hückel’s rule: a planar cyclic system with 4n+24n+2 pi electrons is aromatic; one with 4n4n is antiaromatic and destabilised. Benzene has six, so n=1n=1.

The rule comes from the pattern of orbital energies in a ring: the levels come in a low singlet and then degenerate pairs, so a closed shell needs 2+4n2 + 4n electrons. Cyclobutadiene has four, cannot close its shell, and is spectacularly unstable — which is a strong test, since the naive expectation from “delocalisation is stabilising” would be that it should be fine.

That cyclobutadiene is destabilised is the best evidence that the effect is a genuine consequence of the orbital structure rather than a general tendency of conjugated systems.

Where the localised picture stops

Delocalisation is where the whole apparatus of localised bonds becomes awkward, and it is worth naming the cases.

Conjugated systems. Butadiene, allyl, and every extended pi system: the bond lengths are intermediate and no single localised structure is right.

Charged fragments. The carboxylate ion has two identical C–O bonds, and localised structures place a double bond on one of them.

Metals and clusters. Bonding is spread over many centres and the two-centre picture is simply the wrong starting point.

Hypervalent molecules. Three-centre four-electron bonding is delocalisation over three atoms.

In every case the delocalised description is easier, and the localised one requires a superposition to reach the same answer.

Delocalisation is not always stabilising

A correction to a claim that follows naturally from the benzene case and is false in general.

The instinct is that spreading electrons over more centres lowers their energy, so delocalisation is stabilising. Cyclobutadiene contradicts it: four pi electrons in a four-membered ring, delocalised in exactly the same sense, and the molecule is so unstable that it was not isolated until it could be trapped in a matrix at very low temperature.

The reason is in the level pattern. A cyclic system’s pi orbitals come as a low singlet followed by degenerate pairs. With 4n+24n+2 electrons the shell closes; with 4n4n it does not, and two electrons sit in a degenerate pair with nothing gained.

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. 4 The two-orbital version of the pattern. In a ring the same construction gives degenerate pairs above the lowest level, and whether the count closes a shell is what separates an aromatic system from an antiaromatic one.

So the useful statement is not “delocalisation is stabilising” but “the level pattern of a cyclic system is stabilising for particular electron counts”. That is a sharper claim, it predicts cyclobutadiene correctly, and it is why Hückel’s rule is a rule rather than a tendency.

Where the model stops

Two limits.

Nothing here computes it. This site’s machinery is one-electron and hydrogenic. Bond lengths, stabilisation energies and Hückel counts quoted here come from elsewhere; the figures show geometry and symmetry, which is what the site’s own computations reach.

“Delocalised” is a description too. The canonical molecular orbitals of benzene can be transformed into localised ones by a unitary transformation, giving a picture with three localised pi bonds and the same total density. That the molecule is delocalised is a statement about which description is natural, not about which is true.

Where the idea came from

Kekulé proposed the ring in 1865 and the alternating structures in 1872, the second explicitly to account for the fact that the six positions behave identically — so the problem was identified within a decade of the structure.

Hückel’s treatment came in 1931, and Pauling’s resonance account through the 1930s. The molecular orbital picture of benzene as a six-centre pi system belongs to the same period.

The Soviet denunciation of resonance theory in the early 1950s, on the grounds that a molecule cannot be a superposition of structures that do not exist, is a curious footnote — the objection was philosophical rather than scientific, and it was made against a formalism whose meaning had been widely misdescribed by its own proponents.

The pi system, drawn

Benzene’s six pi orbitals are built from six p orbitals perpendicular to the ring, and the building blocks are worth looking at.

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. 5 One of the six. Each carbon contributes a p orbital perpendicular to the ring plane, with a positive lobe above and a negative one below — and the sign is what decides how they combine.
2px with 2px 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.2px · 2pxS = 0.75286pi interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 6 Two of them side by side: a pi interaction, with overlap above the plane and below it. Six such interactions around a ring give the six molecular orbitals, three of them bonding.

The lowest of the six has no nodes among the carbons at all — every p orbital in phase — and it is the one that gives benzene its ring current and its unusual magnetic behaviour. The next two are degenerate with one node each, and together the three hold the six pi electrons.

That the levels come in a singlet and then pairs is the pattern behind Hückel’s rule, and it is a property of the ring’s symmetry rather than of carbon.

Where the ladder goes next

The frameworks this sits between are molecular orbital and valence bond theory.

The general point about descriptions is hybrids are a basis.

And the related multi-centre case is hypervalency.

What the pictures here cannot show. The figures on this page draw nuclei and symmetry. Delocalisation is a property of an electron distribution that this site does not compute, and the argument here rests on the geometry — which the figures do establish — plus results quoted from elsewhere.

Where this goes next

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 Molecular orbital and valence bond Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet. sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron Hybrids are a basis An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers. HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms Point groups from coordinates A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.