A stabilisation is measured from somewhere

Every number this field has produced for benzene rests on a subtraction. The π energy of the ring is computed — eight β, from six eigenvalues of a matrix of ones and zeroes — and something is taken away from it, and what is left is called the delocalisation energy. The first half of that sentence can be made careful: benzene’s 2β is computed rather than quoted, against a reference state written down as a count of isolated double bonds.

Being careful about the first half is one step better than most sources manage and one step short of the question worth asking. The subtraction has a second operand, somebody chose it, and nothing so far has asked what happens if it is changed.

The answer is that the choice is worth about a factor of two for benzene, a factor of twelve for cyclooctatetraene, and the sign for cyclobutadiene. It also changes which of six rings is called the most aromatic.

The usual reference #

The classical Hückel delocalisation energy compares a conjugated system against a hypothetical molecule with the same atoms and the same electrons in which the π electrons are confined to isolated double bonds. Each of those bonds is worth exactly 2β — the π energy of ethene, from a two-by-two matrix — so the reference is a count: three double bonds for benzene, two for cyclobutadiene, four for cyclooctatetraene.

Benzene’s six π electrons give 8β, three ethenes give 6β, and the difference is 2.0000β. The number is exact and it is not a fit. It comes straight from diagonalising the ring, and it is checked against a tripwire that would reject a wrong value.

The reference state has a name in the literature — the Kekulé structure, cyclohexatriene, the localised structure — and a property worth pausing on: it is not a molecule. Nobody has ever made a hexagonal C₆H₆ with three double bonds fixed in place, and if they had, its bonds would not all be the same length. The reference exists to be subtracted from and has no other duties.

That is not an objection. Every reference state in chemistry is like that; a heat of formation is measured against elements in states that are conventions as much as substances. The objection is only that a reference invented for a subtraction can be invented in more than one way, and the differences do not cancel.

A second reference, made of the same parts #

The alternative used here is the ring cut open: the same atoms, the same number of π electrons, one bond deleted. Benzene becomes hexatriene, cyclobutadiene becomes butadiene, the tropylium cation becomes the heptatrienyl cation.

This reference has two properties the first one lacks. It is a real molecule — hexatriene exists, is conjugated, and has a measurable heat of hydrogenation. And it isolates the thing the word aromatic is supposed to name: the extra stabilisation a cycle has over an open conjugated chain of the same length. Conjugation itself is not in dispute; every polyene has it. What is in dispute is whether closing the ring buys anything beyond it.

The right-hand column of that figure is the reference, diagonalised rather than assumed. Hexatriene’s six levels sum to 6.9879β for its six electrons — already 0.9879β below three isolated ethenes, which is the entire conjugation of an open chain and is the part the classical reference charges to the ring. Higher up the page is more stable there and in every level diagram here, because β is negative.

That last number is the whole mechanism in one line. The open chain is already 0.9879β better than three ethenes. The classical reference does not know that, so it hands the ring credit for an effect that has nothing to do with the ring — and hands it in an amount that depends on how long the chain is.

Nothing new has been introduced to compute this. The open system is the same matrix with two entries deleted, diagonalised by the same solver, filled by the same aufbau, and its energies are checked against the closed form for a chain in the same way the ring’s are checked against the closed form for a ring.

Where the two answers part company #

For benzene the two references differ by a factor of two, which is embarrassing and survivable. For cyclobutadiene they differ in sign.

Cyclobutadiene’s π energy is exactly twice ethene’s, a coincidence with a reason behind it: two of its four electrons sit in a pair of levels at exactly the energy of an isolated p orbital, contributing nothing at all, which is the argument about what delocalisation is worth. Against isolated double bonds that comes out as neither stabilised nor destabilised.

No chemist means that by antiaromatic. The word is a claim that the ring is worse off for being a ring, and the classical reference is structurally incapable of expressing it: with the reference set at two ethenes, the best cyclobutadiene can score is zero, and zero is what it scores. The cut-open reference says the ring costs 0.4721β against the open chain of the same four carbons, which is the statement the word was invented for.

The comparison that looked safest is the one that breaks #

A reader who has followed the argument this far might reasonably try to escape it by refusing to quote either number and quoting a ratio instead. Ratios are the standard defence against a convention: β cancels, the unit cancels, the zero of energy cancels, and what is left ought to be a statement about the molecules rather than about the bookkeeping. Benzene is so many times as delocalised as cyclobutadiene sounds like the kind of claim that cannot depend on where the subtraction started.

It depends on it more sharply than either number does.

Against isolated double bonds the denominator is 0.0000β — not small, exactly zero, for the reason the section above gives — and a ratio with zero underneath it is not a large number or an infinite one. It is not a number. Against the ring cut open the two values are +1.0121 and −0.4721, so the ratio is −2.144, and a negative ratio is the arithmetic saying the two systems are on opposite sides of their reference rather than differing in degree.

That is worth sitting with, because it inverts the usual intuition about which quantities are robust. A ratio is immune to the two changes that are exact symmetries of the model — shifting α and rescaling β — and it is more exposed than either operand to the one change that is a convention, because the convention enters the numerator and the denominator differently and cannot cancel. Being dimensionless is protection against units. It is no protection at all against a chosen origin.

The ordering changes, which a scale factor could not do #

If the two references disagreed by a constant factor there would be no argument here — one would be the other in different units, and every conclusion drawn from one would survive.

Benzene is fourth of six on the classical reference and second on the other. The tropylium cation goes the other way. And the three rings that all carry six π electrons — the cyclopentadienyl anion, benzene and the tropylium cation — come out spread over fifty per cent of their own value on the classical reference and within eight per cent of each other on the cut-open one.

The second result is the one worth having, because those three rings are alike. Each has six electrons in a closed shell of a monocyclic system; each is aromatic by every experimental criterion available; and no chemist believes tropylium is half as aromatic again as benzene. The cut-open reference puts them at 0.1687, 0.1680 and 0.1555β per electron and the classical one puts them at 0.3333, 0.4120 and 0.4980.

The figure above is the same six-ring calculation with three rows removed rather than a separate one, which matters for what it is evidence of. Nothing was refitted and no reference was retuned to make the lower bars agree; the agreement is what the cut-open reference already said about those three rings in the table at the top of this essay, made visible by taking away the three rings whose electron counts differ. A convention that gets the easy comparison right is not much of an achievement, and this is the easy comparison — three rings that differ only in size, with the same closed shell of six electrons in each.

The cause is a counting artefact and it is visible as soon as it is stated. The classical reference is a count of double bonds, and an odd ring with a charge does not have a whole number of them. The tropylium cation has seven carbons and six π electrons, so three double bonds and one empty p orbital — and the reference charges nothing for the empty orbital, because an empty orbital holds no electrons and contributes no energy. So the ion is compared against a reference that is missing an atom’s worth of structure, and the missing structure is scored as a gain.

Cyclooctatetraene, where the disagreement is largest #

The worst case is the eight-ring, and it is worth its own section because a reader who has met only the classical number has been badly served.

Planar cyclooctatetraene has eight π electrons, four double bonds’ worth, so its classical reference is 8β and its π energy is 9.6569β. That yields 1.6569β, which reads as a strongly delocalised molecule, and it is nothing of the kind: the same structure sits 0.1393β above octatetraene, its own open chain — a fortieth of a beta per electron.

The left-hand column of that figure is why the eight-ring scores so well on a reference made of double bonds: it has four of them to be credited with, and eight levels over which its electrons are spread. Two of its eight electrons sit in a degenerate non-bonding pair on the α line, contributing exactly nothing to the sum — the same feature cyclobutadiene has, and what makes both of them 4n rings.

Real cyclooctatetraene is not planar, and the tub shape it adopts is usually explained by ring strain and by the antiaromaticity it escapes by twisting. That explanation is available in this language on the cut-open reference and not on the classical one: on the classical reference the planar ring is 83 per cent as delocalised as benzene and there is nothing to escape from.

A rule that separates by size rather than by sign #

The obvious thing to expect of the cut-open reference is that it reproduces Hückel’s rule as a change of sign: every 4n+2 ring stabilised by closing, every 4n ring destabilised.

It does not, and the failure is informative. Planar cyclooctatetraene has eight electrons, is a 4n ring, and comes out at +0.1393β — small, and on the wrong side of zero for the rule as stated. Only cyclobutadiene is actually destabilised by closing its ring.

The reason is a fact about ring size rather than about electron count. Closing a large ring changes its level pattern very little: the two levels the closure moves are already nearly degenerate with the ones next to them, so the cost of a bad electron count is spread thin. Closing a four-ring rearranges everything it has.

So what survives contact with the arithmetic is this: every 4n+2 ring here is stabilised by closing, and the weakest of them beats the strongest 4n ring by more than eight times per electron. The rule separates the two families by an order of magnitude rather than by a sign, and saying so is more useful than a rule that is wrong for the eight-ring.

What this does to β #

There is a practical consequence, and it is the reason this matters outside the arithmetic.

β is a parameter with no number attached. It is fixed, when anybody fixes it, by matching a computed delocalisation energy to a measured stabilisation — and the commonest match is benzene’s, whose empirical resonance energy from heats of hydrogenation is quoted at about 150 kJ mol⁻¹ against three cyclohexenes.

Match that to 2β and β comes out at 75 kJ mol⁻¹. Match it to 1.0121β and β comes out at 148. The same measurement, the same theory, and a factor of two in the parameter — carried thereafter into every other system the parameter is used on.

This is not an argument that either fitted value is wrong. It is an argument that a value of β quoted without the reference state that produced it is not a number anyone else can use, and the same is true of every resonance energy quoted in kilojoules. The literature’s own spread makes the point independently: the thermochemical figure of about 150 kJ mol⁻¹ and Dewar’s resonance energy of about 84 kJ mol⁻¹ for the same molecule differ by very nearly the factor the two references here differ by, and for the same reason — Dewar’s reference is a polyene rather than a set of isolated double bonds.

The measured number has a reference in it too #

It would be easy to read all of this as a complaint about calculations, with the implication that a measurement settles the matter. It does not, and the reason is that the measured resonance energy is a subtraction with a chosen second operand in exactly the way the computed one is.

The classic measurement is a hydrogenation cycle. Adding hydrogen to cyclohexene releases 120 kilojoules a mole; to benzene, 208. If benzene were three isolated double bonds it would release three times 120, or 360, so the shortfall is

$$3 \times 120 - 208 = 152\ \text{kJ mol}^{-1},$$

and that is where the figure usually quoted for benzene’s resonance energy comes from.

Every step of that is a real measurement, and the reference is nonetheless a choice: three cyclohexenes is the thermochemical spelling of three isolated double bonds, which is the first of the two references this essay compares.

Choosing the other one changes the answer, and the same experiment supplies the ingredient. Hydrogenating 1,3-cyclohexadiene releases 232 kilojoules a mole where two isolated double bonds would give 240 — so a pair of conjugated double bonds is already stabilised by 8 kilojoules a mole before any ring closes. A reference that credits the open conjugated chain with the stabilisation it demonstrably has therefore starts from a lower number, and benzene’s excess over it is correspondingly smaller.

That is the same move, made with calorimetry instead of eigenvalues, and it moves the answer in the same direction and for the same reason. The 120-to-180 kilojoule range that different sources quote for one molecule is not measurement scatter. The hydrogenation numbers are known to a kilojoule or two. The range is the reference changing.

Which sharpens the essay’s conclusion rather than softening it. A computed delocalisation energy has an obvious second operand, written down in a system definition where anybody can see it. A measured one has its second operand hidden inside the choice of which molecule to hydrogenate alongside, and that molecule is real, so nothing about the procedure announces that a convention was applied.

A reference chosen from a table of matrices is visible. A reference chosen from a bottle is not, and it is exactly as much of a choice.

What the model cannot supply #

Everything above is Hückel theory, so the standing limits apply with full force. There is no electron repulsion, one parameter carries every bond, the σ framework is assumed separable, and every carbon is assumed identical.

Three limits are specific to this argument.

The σ frame is not in the subtraction. Closing a ring makes a σ bond as well as a π one, and the reference chains here are short by exactly that bond. The comparison is therefore a π-only comparison between systems whose σ frameworks differ, which is legitimate at this level of theory only because the σ contribution is assumed to be transferable — the same assumption the classical reference makes, at a different place.

The geometry is not optimised. Both the ring and the cut-open chain are drawn with every resonance integral equal, which is the model’s standard idealisation and is not what the π system would choose if it were allowed to alternate. Letting the open chain alternate would lower its energy and lower the cyclic figure further, so the numbers here are, if anything, generous to the ring.

The hydrogen count differs between a ring and its open chain. Hexatriene is C₆H₈ and benzene is C₆H₆. Dewar’s construction handles this by pricing bonds thermochemically; this one ignores it, as the classical reference also ignores it. What is being compared is the π electron system, which has the same size in both, and every conclusion above is a comparison of π energies.

Who did this, and when #

The classical delocalisation energy is Hückel’s, from 1931. The objection is Dewar’s, from the 1960s: he argued that the correct reference for a cyclic conjugated system is an acyclic conjugated one rather than a set of isolated double bonds, defined resonance energies against bond-energy increments fitted to polyenes, and found that they came out at roughly a third of the classical values — with several systems that the classical treatment called aromatic coming out at essentially zero.

The numbers here are not his. They come from cutting a ring open rather than from fitted increments, which is cruder and needs no parameters at all; the point of computing them was to see whether the qualitative conclusion depends on his increments, and it does not. Two matrices, the same solver, and the disagreement is already visible.

What has changed since is that the argument stopped being about which resonance energy is correct. Modern treatments define several — vertical, adiabatic, against various references — and quote which one they mean, which is the whole of the lesson and is what this essay is asking for.

Still open: a reference state of the same molecule #

The obvious next question is what happens when the reference is not a smaller molecule at all but a different state of the same one. A vertical resonance energy compares a molecule against itself with its bonds held at the delocalised geometry and its wavefunction constrained to a single structure — a comparison that has no second molecule in it and therefore none of the counting artefacts above. This model cannot make that comparison, because it has no way to constrain a wavefunction; that needs a model with repulsion in it.

The nearer question is whether the same reference sensitivity infects the other quantities this field quotes. A bond order is a sum over occupied orbitals with no subtraction in it, so it is immune. A charge is a difference against a neutral atom, so it is not. And a stabilisation per electron, which is what the comparison here had to use to be fair across rings of different sizes, is a ratio of a difference to a count — which is two conventions rather than one, and the count is the one nobody argues about.