The hexagon is the frame's doing

Delocalisation opens with benzene’s structure: one molecule, not two Kekulé forms alternating, and a π energy of $8\beta$ against $6\beta$ for three isolated double bonds. The difference, $2\beta$, is the delocalisation energy, and it is computed here rather than quoted — against a reference state written down explicitly as a count of isolated double bonds.

That number is then read as the reason benzene’s bonds are all the same length. Delocalisation stabilises; alternating the bonds would localise; therefore the bonds do not alternate.

The second sentence does not follow from the first, and this essay is about what happens when the question is asked directly.

Ask the π system what it wants #

Hold the ring together and let its bonds alternate: long, short, long, short. In a Hückel model that means alternating resonance integrals, $\beta(1+\delta)$ and $\beta(1-\delta)$, since a shorter bond has a larger $\beta$.

Diagonalise and fill — the machinery of Hückel theory and what it gets right, with the one option nobody had passed it: a resonance integral that varies from bond to bond. For benzene at $\delta = 0.05$ the π binding rises from $8.000$ to $8.015$ in units of $|\beta|$ — rises, meaning the energy falls, meaning the alternated ring is more stable in its π electrons than the regular one.

At $\delta = 0.1$ it is $8.060$; at $\delta = 0.2$, $8.233$. The gain is quadratic in $\delta$ with a coefficient of $5.96$, fitted over five distortions with a residual of $6 \times 10^{-5}$.

And it is not a peculiarity of six. A ten-membered ring gains with a coefficient of $13.19$; cyclobutadiene, which nobody expects to be stable, gains with $45.97$ — the molecule delocalisation is not always stabilising uses to make the same point about the word delocalised. Every ring computed here is lowered by alternating, whether its electron count is $4n+2$ or $4n$.

So delocalisation energy and resistance to alternation are different quantities, and the first does not imply the second. Benzene really is $2\beta$ more stable than three isolated ethylenes; it is also less stable than a version of itself with unequal bonds.

Why the two statements are compatible #

The apparent paradox comes from a reference state, and setting the two comparisons side by side dissolves it.

Delocalisation energy compares benzene with three ethylenes at the same bond length, all with the same $\beta$. That is a comparison between a connected ring and a disconnected set of dimers, and connectivity wins: $8\beta$ against $6\beta$.

The distortion question compares benzene with a version of itself in which three bonds have shortened and three lengthened. That is a comparison between two connected rings at different geometries, and here the alternated one wins.

Nothing is inconsistent. The first says a ring is better than fragments; the second says a ring with unequal bonds is better than one with equal bonds. The delocalisation energy is not a measure of resistance to anything — it is the energy of one comparison, and every reference state gives a different number, which is what conjugation, and its limits is careful about when it quotes one.

The exponent, which sorts two kinds of instability #

Fitting a power to the gain rather than assuming one separates two mechanisms that produce the same-looking curve.

Second order means the flat geometry is a genuine stationary point — no first-order push away from it — and whether it survives is a competition with whatever resists. That is benzene and the ten-ring, both closed shells.

First order means it is not stationary at all and the distortion is unavoidable. Two quite different things produce it. Cyclobutadiene has a degenerate half-filled shell, so alternation splits it at first order: that is a Jahn–Teller distortion and the molecule has no choice, which is the subject of the vibration that lowers the symmetry. A chain is first order for an entirely different reason — its two ends make its two alternation patterns inequivalent, because the short bonds prefer to be at the ends, so there is a preferred sign of $\delta$ and therefore a linear term.

A ring has no ends, and its two patterns are related by a rotation. That is why a ring is the case where the question is interesting: the symmetry makes the linear term vanish, and what is left is a genuine competition.

What has to resist, and how hard #

The two energies can be put in the same units.

If the resonance integral falls off exponentially with bond length, $\beta(u) = \beta_0 e^{-u/d}$, then $\delta = u/d$ for a displacement $u$, and the π gain is $A\lvert\beta_0\rvert u^2/d^2$ per ring. The σ frame’s cost for the same displacement is $\tfrac12 k u^2$ per bond, and every bond moves, so it is $\tfrac12 k n u^2$.

The ring stays regular only if the second beats the first, which puts a critical force constant on the frame:

$$k_c = \frac{2A\lvert\beta_0\rvert}{n\,d^2}.$$

Both $\beta_0$ and $d$ have to be quoted — they are the two numbers a Hückel model cannot supply — so the answer is a range rather than a number.

And that is the result: they are the same size. With $\lvert\beta_0\rvert = 2.4$ eV and a decay length of $0.30$ Å, $k_c$ comes out at $8.5$ mdyn/Å against a measured $7.6$. With a decay length of $0.35$ it is $6.2$ and the frame wins comfortably; with $0.25$ it is $12.2$ and the π system wins.

So this model cannot settle whether benzene should alternate. What it settles is the shape of the question: the regular hexagon is a near-run thing between two large opposing effects, and the π system is on the wrong side of it.

What is known independently #

The model’s inability to decide is not the end of the matter, and two facts from outside it are worth setting down, marked as what they are.

Benzene does not alternate. Its bonds are $1.397$ Å, all of them, and the symmetry of its spectrum requires D6h rather than D3h — which is exactly what how many frequencies, not how many modes counts for benzene and would count differently for an alternating ring. That is a measurement and it is not in doubt.

And the current view of why is the one this calculation points at. The argument that the π system of benzene is distortive and the σ frame is what holds it regular is Shaik and Hiberty’s, from the 1980s, and it was contentious for a long time. The evidence for it is of the kind this calculation produces — the π energy’s response to a distortion, computed at levels far beyond Hückel — together with the observation that as rings get larger the balance does tip: the annulenes above about $[18]$ do show bond alternation, which is what a competition between a π term that scales one way and a σ term that scales another looks like when the size is varied.

What is supplied here is the arithmetic underneath that argument, in a model small enough that every number in it is visible.

The same effect where it does win #

The extended version of this competition is a central subject in solids, and the connection is worth making explicit.

A chain cannot stay even computes exactly this for an infinite half-filled chain and finds the π gain wins: the chain dimerises, opens a gap, and becomes an insulator. That is the Peierls distortion, and polyacetylene is the molecule.

The difference between the two cases is in the table above. A chain of twelve gains $5.045$ per bond and benzene gains $0.994$ — five times as much — because a chain’s levels are dense near the Fermi energy and a small ring’s are not. So the same physics gives opposite answers for a ring of six and a chain of many, and the crossover is the ring size at which the annulenes begin to alternate.

What the per-bond number is measuring #

One column of the table above is the one that carries the physics, and it is worth separating from the raw coefficient.

The gain per bond is $0.994$ for benzene, $1.319$ for a ten-ring and $5.045$ for a chain of twelve. Those are the numbers that matter for the competition, because the σ cost is also per bond — every bond stretched or compressed pays, so the comparison is between two quantities that both scale with the ring size.

The ordering says the effect grows with the system. A larger ring has more levels packed into the same band width, so more of them sit near the top of the filled set where alternation does its work, and the gain per bond climbs. That is the trend that ends in a chain cannot stay even: the infinite half-filled chain has a continuum of levels at the Fermi energy and the gain per bond is at its largest there.

So there is a ring size at which the σ frame stops being able to hold. That is a prediction with a number in it, and this model is not the place to compute the number — but the direction is unambiguous and the annulene chemistry agrees with it.

What the argument replaces #

Something has to be said about the picture this displaces, because it is one of the most widely taught explanations in organic chemistry.

The usual account has three claims stacked on each other: benzene is unusually stable; the stability comes from delocalisation; and the delocalisation is why the bonds are equal. The first is a measurement and is not in dispute. The second is a comparison with a reference state and is fine as long as the reference is stated. The third does not follow, and the computation above says it is the wrong way round.

What replaces it is a competition, and the competition is between things of comparable size. The π system pulls towards alternation with a strength this model measures; the σ frame pulls towards regularity with a strength a vibrational spectrum measures; the regular hexagon is what wins by a margin of tens of per cent.

That is a less satisfying story and it makes better predictions. It says the balance should shift with ring size, and the annulenes above about eighteen carbons do alternate. It says an open shell should have no choice, and cyclobutadiene distorts. It says a chain should alternate more readily than a ring, and polyacetylene does. The delocalisation account predicts none of those, because it has only one direction in it.

The general shape is one that keeps recurring: an explanation that gets a fact right by an argument that would get the neighbouring facts wrong. Testing it means asking what else it implies, and the way to do that here was to stop asking whether benzene is stable and start asking what its π electrons would do if nothing else were holding them.

The experiment that makes benzene’s bonds unequal #

If the hexagon is the σ frame’s doing rather than the π system’s, then weakening or distorting the σ frame should let the bonds alternate — in benzene itself, not in an annulene four times its size. That is a prediction with a sign and a target, and it has been tested.

The test is to fuse small strained rings onto the benzene. A ring fused across two adjacent carbons pulls on the σ framework there, and a small enough fused ring pulls hard: the internal angles it demands are far from the 120° the benzene wants, so the σ bonds inside the six-membered ring are forced away from their preferred lengths. Fuse three such rings, on alternate bonds, and the distortion is applied with the ring’s own symmetry — which is exactly the pattern a bond alternation has.

The molecules exist and their structures have been measured. Benzenes annelated with three small strained rings show clearly alternating bonds, with the two sets differing by something approaching a tenth of an ångström — against benzene’s own difference of exactly zero. The π system is unchanged: six carbons, six π electrons, a closed shell, and the same 2β of delocalisation energy against three ethenes. What changed is the frame, and the bonds followed the frame.

That is the cleanest available confirmation of the argument, and it is worth being precise about what it confirms and what it does not.

It confirms the direction of the competition. The π system is not holding the hexagon regular, because if it were, a strained σ frame could not have overcome it — a delocalisation energy of 2β is a large quantity to be defeated by fusing three small rings.

It confirms that the margin is thin. The critical σ force constant computed here lands within thirty per cent of the measured one, which says the competition is close; a molecule in which a modest σ distortion flips the answer is what a close competition looks like from the outside.

It does not show that benzene is secretly alternating. Unstrained benzene’s bonds are equal to the accuracy of any measurement, and the arithmetic here says why: the σ frame wins, narrowly, when nothing is pulling on it.

The general form is one worth carrying. A property attributed to one part of a molecule can be tested by changing a different part — and a property that survives every change to the part it was attributed to, while collapsing under a change to a part it was not, was attributed to the wrong thing.

Where the model stops #

Hückel, one π electron per site, no repulsion, which is the model the hole that is not repulsion is about the absence of. Electron repulsion opposes bond alternation — it prefers uniform density — so a model with it in would put the π system’s preference for distorting somewhat lower than this one does. The direction of that correction is known and its size is not, in this model.

The σ term is a harmonic spring per bond. A real σ frame has angle terms, cross terms and anharmonicity, and its stretching constant is not cleanly separable from the π contribution to the same stretch — which is part of why the comparison above is a range rather than a number.

$\beta_0$ and $d$ are quoted. They are the two parameters that make this a comparison of energies rather than of dimensionless numbers, and both are conventional.

Nothing here computes a bond length. The competition decides whether the symmetric geometry is a minimum; the actual bond lengths would need the full potential.

A closing note on what the word aromatic survives all this. Benzene’s shell closure is real, its 4n+2 count is an output of a filling rather than a rule, and its stability relative to three ethylenes is $2\beta$ and computed. None of that is touched by the finding here.

What is touched is one inference from it. Aromaticity is a statement about an energy relative to a reference, and the regularity of the bonds is a statement about a second derivative with respect to a distortion. Those are different quantities, they are computed differently, and a molecule can have the first without the second — which is what the arithmetic here says benzene’s π system does, and what the annulenes above eighteen carbons demonstrate by having the first and losing the second.

What separating the two quantities adds #

The standard Hückel account establishes that benzene has one structure rather than two, computes its levels from a matrix of ones and zeroes, gets every bond order out of the eigenvectors, follows the gap down a conjugated chain, and puts a heteroatom into the ring.

This one asks what the delocalisation energy does not say. It does not say the ring resists alternation: the π energy falls when the bonds alternate, at every ring size, by a coefficient measured here. What resists is the σ frame, and putting the two in the same units gives a critical force constant that lands within thirty per cent of the measured one — close enough to say the question is a real competition and not close enough to decide it. The exponents sort the rings from the chains and the closed shells from the open ones, and the same arithmetic run on a long chain gives the opposite answer, which is what polyacetylene is.

The open question is the ring sizes where the balance actually tips — the larger annulenes, where alternation is observed and the crossover can be located rather than argued about.