Delocalisation

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.

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.

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.

The number, computed #

The hedge in the previous section is honest and it is also an admission: the figure was quoted rather than produced, and the reference state it was measured from was never written down. Both can be fixed, and the calculation needed is smaller than the hedge.

A conjugated pi system’s Hückel matrix is its adjacency matrix. Two carbons that share a bond get a one, every other entry is zero, and the orbital energies are

$$E_k = \alpha + x_k \beta,$$

where the $x_k$ are the eigenvalues of that matrix of ones and zeroes. Nothing about carbon enters the calculation. Nothing about geometry enters it either — a benzene drawn as a badly squashed loop has the same eigenvalues as a regular hexagon, because the only thing the matrix records is which atom is next to which.

Diagonalising it gives $x = 2, 1, 1, -1, -1, -2$. Six electrons fill the lower three levels, so the total pi energy is $6\alpha + 8\beta$. Three isolated ethenes give $6\alpha + 6\beta$. The difference is exactly $2\beta$, and $\beta$ is negative, so it is a stabilisation.

Two things about that number are worth more than its value.

It carries its reference state. The comparison is against three isolated double bonds and says so; changing the reference changes the answer and the figure would print the new one. The 120-to-180 spread quoted above is not uncertainty in a measurement, it is four different comparisons being reported as though they were one.

It is in units of β, and β has no number here. Nothing here fits β to a thermochemical measurement, because doing so would turn a clean statement about a graph into a calibration with a hidden choice in it. An essay that wants kilojoules must supply β itself and say where the value came from.

4n+2, produced rather than recalled #

The same calculation gives the aromaticity rule as an output.

Run every ring size from three to ten, fill each with its own electrons, and ask whether the highest occupied shell came out full. The function that does it does not know the phrase “4n+2”; it fills computed levels and reports what it finds.

The pattern behind it is the degeneracy structure, visible in the figure as pairs of dots. A monocyclic system’s levels are $2\cos(2\pi k / n)$, and cosine takes the same value at $k$ and $n-k$, so every level except the lowest — and, for an even ring, the highest — comes twice.

Cyclobutadiene is the case that makes it a rule rather than a tendency. Four pi electrons, two in the lowest orbital and two in a degenerate non-bonding pair, and the computed delocalisation energy against two isolated double bonds is exactly zero. Hückel theory says the ring gains nothing at all from being a ring, and Hund’s rule then puts one electron in each of the degenerate orbitals, so the prediction is a triplet ground state.

That prediction is wrong, and being wrong is what makes it worth having. Square cyclobutadiene would be a triplet; real cyclobutadiene distorts to a rectangle, which lifts the degeneracy and gives a singlet. A theory that predicted “unstable” vaguely could not have been caught out. One that predicts a triplet can be, and the distortion that defeats it is a Jahn–Teller effect that becomes discussable rather than omitted.

What the eigenvectors give away #

Having diagonalised the matrix, the eigenvectors are already there and two more quantities cost nothing.

The pi bond order between two atoms is the sum, over occupied orbitals, of the product of their coefficients. The charge density on an atom is the sum of the squares.

Naphthalene is the better test of the two. Benzene’s uniformity could be attributed to symmetry alone and needs no theory at all; naphthalene’s bonds are not all equivalent, the theory ranks them, and the ranking is right.

What it costs #

Very little, which is the uncomfortable part.

The eigensolver is a cyclic Jacobi sweep: about sixty lines, repeatedly rotating the largest off-diagonal entry to zero until the off-diagonal norm falls below a tolerance. For a ten-by-ten naphthalene matrix it converges in six sweeps and runs in under a millisecond. Nothing about it is delicate, and it introduces no approximation beyond the ones Hückel theory has already made.

The check that it worked costs less still. A ring’s eigenvalues have a closed form, $\alpha + 2\beta\cos(2\pi k/n)$, so the solver can be compared against an exact answer at every size.

The two trace relations are worth more than they look. They follow from the graph rather than from the solution, they hold for systems with no closed form at all, and they are the check that catches the failure mode that matters here — an eigensolver converging quietly to the wrong place.

What all this actually costs is stated in what it leaves out. Hückel theory has no electron repulsion in it whatever. Sigma and pi are assumed separable, every carbon is assumed identical, one parameter carries every bond, and the geometry is thrown away before the calculation starts. It gets the symmetry and the degeneracies exactly right, because those come from the graph and the graph is exact. It gets the energies right only to the extent that a single parameter can carry them, and the honest form of every number above is therefore “in units of β”.

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, and now in two different senses. The experimental figure requires comparison with a hypothetical molecule, and depends on which hypothetical one is chosen. The Hückel figure of 2β is exact within its own theory and carries its reference state explicitly — but β is a parameter, so it is an exact statement about a model rather than a measurement of benzene.

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

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 $4n+2$ pi electrons is aromatic; one with $4n$ is antiaromatic and destabilised. Benzene has six, so $n=1$.

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 $2 + 4n$ 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 $4n+2$ electrons the shell closes; with $4n$ it does not, and two electrons sit in a degenerate pair with nothing gained.

The two-orbital version of the pattern is the ordinary bonding-and-antibonding pair. 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 #

Three limits, and the first has moved since this essay was first written.

The level pattern is computed; the energy scale is not. The eigenvalues, the degeneracies, the shell closures, the bond orders and the delocalisation energy are all produced here, from a matrix of ones and zeroes. What is not produced is β. Anywhere a kilojoule appears on this page it is quoted, and the boundary between the two is exactly the boundary between a graph and a measurement.

Hückel cannot see a distortion. The theory is handed a connectivity and returns the levels of that connectivity. Cyclobutadiene’s escape from its own prediction is a change of geometry, and geometry is the one thing this treatment discarded at the first step. A theory that could have predicted the rectangle would have to be given the freedom to move the atoms.

“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.

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. Everything in this essay is a statement about combinations of six such functions.

The π overlap between two neighbouring p orbitals is what the resonance integral stands for, and it is computed rather than assumed elsewhere in this collection. Here it is a single parameter with no number attached, which is what makes every energy in this essay a multiple of β.

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 to read on #

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 Hückel figures on this page draw a graph, and a graph has no lengths in it. They cannot show that benzene’s bonds are 1.397 ångström, only that the theory makes all six identical; they cannot show why cyclobutadiene distorts, because the coordinates were discarded before the matrix was built; and they cannot show an energy in any unit, because β is a parameter that this site declines to fit. What is drawn is the structure of the solution, which is exact, standing in front of a scale that is not there.