The same ring, three charges

Hückel’s rule is usually stated about molecules: a planar conjugated ring with $4n+2$ π electrons is aromatic. Read carefully, the subject of that sentence is a count and not a substance, and the difference can be made concrete by holding a skeleton fixed and changing only how many electrons are put into it.

The Hückel matrix of a ring is its adjacency matrix, as Hückel theory and what it gets right sets out, so it depends on the connectivity alone. Its eigenvalues are therefore the same for the cation, the radical and the anion of a given ring; what changes is the filling. So the whole family can be computed at once, and the results are more striking than the rule suggests.

One skeleton, three fillings #

Three carbons in a ring give a Hückel matrix whose eigenvalues are $2$, $-1$, $-1$ in units of β above α — one strongly bonding level and a degenerate antibonding pair.

Two electrons. Both go into the level at $2$. The π energy is $4.0000$β and the highest occupied shell is full — a closure computed rather than recalled, in the manner of aromaticity as a computed shell closure. The gap to the next level is $3$β, and the delocalisation energy against a reference of one isolated double bond is exactly $2$β. Aromatic on every count.

Three electrons. The third goes into the degenerate pair. The π energy is $3.0000$β, there is one unpaired electron, and the gap is zero because the shell it sits in is half-filled.

Four electrons. Two electrons in the degenerate pair, one in each by Hund’s rule. The π energy is $2.0000$β, there are two unpaired electrons, and the delocalisation energy is exactly zero.

Three carbons in a ring, and the description runs from aromatic through radical to antiaromatic and paramagnetic depending entirely on a charge that lives outside the calculation.

The chemistry follows: the cyclopropenyl cation is a stable, isolable species and one of the smallest aromatic systems known, while the cyclopropenyl anion is famously difficult and behaves as an antiaromatic one. Nothing in the skeleton distinguishes them.

Adding electrons can subtract binding #

The second row of that comparison contains something worth stating on its own, because it contradicts a plausible intuition.

The cyclopropenyl cation’s π energy is $4.0000$β. The anion’s, with two more electrons in the same orbitals, is $2.0000$β. Adding a pair of π electrons reduced the π binding energy by half.

The mechanism is not subtle once the levels are on the page: the added electrons occupy a level at $-1$β, so each of them subtracts a β from the total, and two of them subtract two. But the intuition it violates — more electrons, more bonding — is common enough to be worth naming, and it is the same intuition the antibonding level goes up more disposes of in the two-orbital case.

The cycloheptatrienyl skeleton makes the same point at a larger size: $8.9879$β with six electrons, $8.5429$ with seven, $8.0978$ with eight. Its non-bonding-ish levels sit at $-0.445$β, so each added electron costs less than in the three-ring, and the direction is the same.

Where the electrons go when the shell is open #

The filling rule used here is the one every shell structure uses: fill the lowest levels, and within a degenerate set place one electron in each orbital before pairing any. That is Hund’s rule, and it is worth being explicit that the Hückel Hamiltonian does not contain it.

The one-electron matrix has no electron-electron term at all, so it is indifferent between putting two electrons in one orbital of a degenerate pair and putting one in each. The choice is imposed on the filling from outside, on the strength of a repulsion the model cannot see.

That is exactly the gap the smallest many-electron calculation fills. For square cyclobutadiene the one-electron model predicts a triplet ground state, which is wrong; the exactly diagonalised Hubbard calculation shows the singlet and triplet exactly degenerate at zero repulsion — the one-electron model has no preference to express — and the singlet dropping below for every repulsion above zero.

So the open-shell rows in these tables should be read with care. The level occupations are right, the unpaired counts are what the model gives, and whether the real molecule is a triplet is a question the model is not equipped for.

Energy per electron, which ranks them differently again #

Total π energy is the wrong quantity for comparing rings of different sizes, since a bigger ring has more electrons and more bonds. Dividing by the electron count gives a figure that can be compared, and it reorders the family.

The smallest aromatic ring has the highest π energy per electron and the smallest π energy in total. Both electrons sit in the one strongly bonding level at $2$β, which is the deepest level any ring has — every ring’s lowest level is at exactly $2$β, as the closed form $E = \alpha + 2\beta\cos(2\pi k/n)$ requires at $k = 0$ — and there is nothing else to dilute it.

The ordering by total energy and the ordering by energy per electron therefore disagree, and neither is the ordering by chemical stability, which also involves ring strain, the σ framework and the reference the comparison is made against. Delocalisation treats the reference question and conjugation, and its limits treats what happens when the ring is opened out into a chain.

Cyclobutadiene, whose delocalisation energy is zero three times over #

The four-membered ring is the case where the bookkeeping is worth doing slowly, because a single number does three different jobs.

Its four levels are $2$, $0$, $0$, $-2$. With four electrons the π energy is $4.0000$β. With two it is also $4.0000$β. With six it is again $4.0000$β. The two levels in the middle sit at exactly zero, contributing nothing whether they are occupied or not, so the total is the same for all three counts.

And the delocalisation energy — the π energy less that of a reference state of isolated double bonds — is exactly zero at each. Two isolated ethenes give $4$β, and the ring gives $4$β.

Three things follow, and the third is the useful one.

A zero delocalisation energy is not an absence of π bonding. Both terms in the difference are $4$β. The ring is π-bonded; it is simply no better bonded than the localised alternative.

A delocalisation energy is a difference against a stated reference, and quoting one without the reference is quoting half a number. A delocalisation energy for a system that does not state how many isolated double bonds it is compared against is not a number at all.

Exactly zero is a different kind of statement from small. It arises because the two non-bonding levels are exactly at α by symmetry — a consequence of the ring’s fourfold symmetry rather than of a cancellation — and the same exactness appears wherever symmetry rather than arithmetic is doing the work — most sharply in exactly zero, where a forbidden overlap comes out at arithmetic noise.

What the rule looks like when the count is held fixed instead #

The comparison can be run the other way: fix the electron count and vary the ring size. That is the arrangement in which Hückel’s rule was originally noticed and it produces a different-looking table.

Six electrons close a shell in the six-ring, leave the three-ring badly over-filled, and leave the seven- and eight-rings with room to spare. So the same count is aromatic in one skeleton and not in another, which is the mirror image of this essay’s argument.

Both readings are the same statement. Aromaticity is a relation between a skeleton and a count, and neither half decides it alone. What makes the electron-count reading the more useful of the two in practice is that a chemist can change the count — by oxidation, reduction or deprotonation — far more easily than the skeleton.

What the bond orders say #

The energies are the usual currency and the eigenvectors carry a second answer worth having.

Uniform bond orders in a symmetric ring, computed the way bond order from the eigenvectors computes them, are a consequence of the filling being closed or symmetrically open, and they hold in every ring here. That uniformity is what a picture of alternating single and double bonds denies, and it is the computed content of the claim that benzene has no long and short bonds.

What the model cannot decide, and what happens next #

The Hückel treatment of an antiaromatic ring stops in a specific place, and it is a place with a name.

A square cyclobutadiene has a half-filled degenerate shell. A degenerate electronic state in a non-linear molecule is unstable against a distortion that removes the degeneracy — the Jahn–Teller theorem, whose group-theoretic half degeneracy is a group theorem establishes — and the ring escapes by becoming rectangular, splitting the two zero levels and dropping the occupied one.

A ring with a half-filled degenerate shell has an alternation available to it that a closed-shell ring does not: the π energy gained by alternating is set against an elastic cost, and the minimum is where the two balance. That escape is what the open-shell members of each family above actually do, and it is why they are not observed with the geometry drawn here.

Cyclooctatetraene, with eight π electrons in eight levels including a degenerate pair at zero, escapes twice over: it alternates its bonds and it folds into a tub, which removes the conjugation altogether. Its delocalisation energy in the planar arrangement is $1.657$β, and the molecule declines to collect it.

The vibration that lowers the symmetry computes the distortion; what this comparison adds is that the distortion is triggered by an electron count meeting a degeneracy, and that the same skeleton with two electrons fewer has no degeneracy to be unstable about.

Which two combinations get drawn within a degenerate shell is arbitrary — any rotation of the pair is equally an answer — so the individual pictures carry a convention that the level diagram does not. That is why the arguments here are made from the levels and the occupations rather than from the shapes of particular orbitals.

The check the closure has to pass #

A rule reported as an output is only worth having if the report can come out wrong, so the shell closure is tested rather than described.

For each ring, the levels are filled with the stated electron count, the highest occupied shell is examined, and the question did that shell come out full is answered from the occupations. The answer is then compared against the $4n+2$ predicate evaluated on the same count. The two agree for every ring and every filling drawn, and a single disagreement would have falsified the rule.

That check has bite in both directions. It fails if the filling routine pairs electrons where it should not, since a half-filled degenerate shell would be reported as closed. It fails if the eigenvalues lose a degeneracy, since two levels that ought to coincide would be filled one at a time. And it fails on cyclobutadiene at four electrons unless the arithmetic is done honestly, because that is the case where a shell is open and the total energy is nevertheless the same as if it were not.

The closed form is the second half of the same discipline: every ring’s levels are checked against $E = \alpha + 2\beta\cos(2\pi k/n)$ at every size, so a numerical diagonalisation that had drifted would be caught by an expression rather than by a table.

Who found it, and when #

Erich Hückel’s papers appear in 1931 and 1932, and the $4n+2$ pattern in them is a result rather than a premise: he solved the ring problem, saw which counts closed a shell, and reported the arithmetic. The rule was named after him later and hardened into a mnemonic in the process.

The charged rings are mostly post-war experimental chemistry. Ronald Breslow made the cyclopropenyl cation in 1957 and coined antiaromatic in 1967 for the systems that are destabilised rather than merely unstabilised; William von Eggers Doering made tropylium salts in 1954; the cyclopentadienyl anion was in use long before either, since cyclopentadiene is unusually acidic and the reason is exactly the shell closure computed above.

The historical order is worth noticing. The theory predicted which counts would be special before most of the species existed, the species were then made, and the prediction held. That is a stronger record than the theory’s crude approximations would suggest, and the reason is that the parts of it doing the predicting — the degeneracies and the shell closures — are consequences of the ring’s symmetry rather than of the numerical values it assumes.

The measurement that puts a number on a count #

Every quantity above is in units of β, and the charge that decides the whole story sits outside the calculation. Both are limits of the model rather than of the subject: the ordering it predicts is measurable, the measurement is a routine one, and comparing the two says which part of the model is carrying the result.

For the anions the measurement is acidity. Deprotonating a hydrocarbon makes the anion whose π count this essay has been varying, so the equilibrium constant for losing that proton prices the count directly. Cyclopentadiene, whose anion has six π electrons in five orbitals and a closed shell, has a pKa near 16 — comparable to an alcohol, and a startling number for a hydrocarbon. Take an ordinary allylic C–H, where the anion is delocalised over three centres and closes nothing, and the pKa is in the low forties. The difference of some twenty-seven units is about 150 kJ per mole at room temperature.

The control that makes it an argument is the ring one size and one count away. Cycloheptatriene’s anion has eight π electrons — an open degenerate shell, the antiaromatic case — and its pKa is near 36. Twenty units, or about 110 kJ per mole, separates two hydrocarbons that differ by two carbons and by whether the resulting shell closes. The count is worth more than the ring size by a wide margin, which is the prediction of the table above expressed in a quantity somebody measured.

For the cations the corresponding measurement is the equilibrium between the cation and its alcohol in water, reported as $\mathrm{p}K_{R^+}$. Tropylium sits near $+4.7$, which means the bare cation survives in water — extraordinary for a carbocation. The triphenyl-substituted cyclopropenyl cation is close behind. The triphenylmethyl cation, which has three benzene rings to spread its charge over and no ring closure at all, sits near $-6.6$. Eleven orders of magnitude, or about 65 kJ per mole, in favour of the ring that closes a shell over the one that merely delocalises.

So the model’s ordering survives contact with measurement, and it survives on the discriminating comparisons rather than only on the easy ones.

What does not survive is the per-electron table read as a stability ranking. The cyclopropenyl cation tops that column at $2.0000\beta$ and is not the most easily formed of these cations; the parent cyclopropenyl cation is difficult, and it is the triphenyl derivative that is bottled. The reason is the thing the adjacency matrix threw away first: a three-membered ring has internal angles of 60°, and the strain in that σ frame is worth tens of kilojoules per mole against a five- or seven-membered ring. A π energy per electron and a total molecular stability differ by everything the σ framework contributes, and in the smallest ring the σ framework contributes a great deal.

Which is the honest summary of what the count buys. It predicts the ordering within a series of similar rings and it does not predict the ordering across ring sizes, because across sizes a term the model does not contain is changing faster than the term it does. The two measurements above are chosen to sit inside that limitation: cyclopentadiene against cycloheptatriene compares two counts at nearly equal strain, and tropylium against triphenylmethyl compares a closed shell against an open-chain delocalisation at no ring strain at all.

Still open: energies in kilojoules, and a heteroatom #

The shell closure computed rather than recalled, the cases where delocalisation is not stabilising, and the distortion an antiaromatic ring uses to escape all treat aromatic as one variable. This comparison separates the two variables that the word runs together.

What remains uncomputed here is any energy in kilojoules. Every number the model produces is in units of β, a parameter fitted to spectra by other people, and the kilojoules in the section above are measured equilibrium constants quoted against it rather than anything derived here. The model still has no σ framework, no strain and no repulsion in it, which is why the comparison had to be arranged so that none of the three was changing. The next step — a heteroatom put into the ring — keeps all of those limitations and adds the one ingredient that makes the arithmetic interesting again: atoms that are not all alike.