Hückel theory and what it gets right

Write down a conjugated molecule. Put a one wherever two carbons share a bond and a zero everywhere else. Diagonalise the result.

That is Hückel theory in its entirety, and the astonishing part is how much survives such a violent simplification.

What the matrix is #

The Hückel matrix of a pi system is its adjacency matrix, up to two parameters.

The diagonal entries are all α, the energy of an electron in an isolated carbon 2p orbital. The off-diagonal entries are β where two carbons are bonded and zero otherwise. So the matrix is

$$H = \alpha I + \beta A,$$

where $A$ is a matrix of ones and zeroes recording which atom is next to which. Since $\alpha I$ shifts every eigenvalue equally and $\beta$ scales them, the orbital energies are

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

with the $x_k$ the eigenvalues of $A$ and nothing else.

Three approximations got the problem to this point and each is drastic. The sigma framework is assumed separable from the pi system, so the two can be treated independently. All overlap between neighbouring p orbitals is set to zero in the normalisation, though not in the interaction — an inconsistency Hückel accepted knowingly. And every carbon is assumed identical, so one α and one β carry the whole molecule.

What the matrix cannot know #

It is worth being blunt about what has been thrown away, because it explains exactly which results are trustworthy.

Geometry is gone. A benzene drawn as a regular hexagon and a benzene drawn as a badly squashed loop have the same adjacency matrix and therefore the same eigenvalues. Bond lengths, bond angles and planarity are all invisible. The only geometric fact the theory retains is the assumption of planarity, and it retains that implicitly by assuming the p orbitals are parallel.

Electron repulsion is gone entirely. There is no term anywhere for one electron avoiding another. The theory is a one-electron model in the strictest sense.

The atoms are all the same. A nitrogen in a ring is handled by adjusting α for that site, which is a patch rather than a derivation, and the adjustment is fitted.

Anything the theory predicts that depends on geometry, repulsion or atomic identity is therefore not being predicted. What is left is the connectivity, and the results that survive are exactly the ones the connectivity determines.

What survives, and why #

Three classes of result come out right, and the reason is the same in each case: they are properties of the graph.

Degeneracies. Benzene’s two pairs of degenerate levels are exact. They arise because the ring’s symmetry group has two-dimensional representations, and a symmetry of the graph is a symmetry of the matrix. Nothing about the approximations can affect that, because a better theory of benzene still has a sixfold axis.

Node counts and orbital shapes. The eigenvectors of the adjacency matrix have the nodal structure the exact orbitals have, for the same reason. Nodes are a count, and a count fixed by symmetry does not move.

Shell closure. Whether the electrons fill complete shells is a question about degeneracies and an electron count, and both are integers.

The chains, where the pattern is easier to see #

Rings are the celebrated case and linear chains are the instructive one, because a chain’s levels are non-degenerate and the level pattern can be read straight off.

A linear chain of $n$ conjugated carbons has eigenvalues $2\cos(k\pi/(n+1))$ for $k$ from one to $n$. Every level is distinct, the node count goes up in ones rather than twos, and the whole sequence is the particle in a box with the box made of atoms.

Comparing a chain with a ring of the same size makes the difference visible. Both have the same number of levels; the ring’s come in pairs and the chain’s do not, and the reason is a boundary condition. A wave on a ring can run either way round and the two directions have the same energy. A wave on a chain has ends, which pick out one standing pattern per energy.

That single distinction is what makes aromaticity a property of rings rather than of conjugation in general.

What survives with a parameter attached #

Energies come out in units of β, and β has no number.

That is a deliberate discipline rather than a shortcoming. β can be fitted — to a resonance energy, to a spectroscopic transition, to a bond length correlation — and the values obtained from those three fits differ by a factor of two. A single number quoted for β is a number with a fitting procedure hidden inside it, and quoting one here would turn a clean statement about a graph into a calibration with an invisible choice in it.

So every Hückel energy reads as a multiple of β, and any argument wanting kilojoules must supply β and say where it came from.

The convention is worth stating once because it reads backwards. β is negative, so a level with a larger $x$ lies lower in energy, and the figures accordingly put the most bonding orbital at the top. That is the chemists’ convention and it catches everyone at least once.

What was computed, and how #

The eigensolver is a cyclic Jacobi sweep: sixty lines that repeatedly choose the largest off-diagonal entry and rotate it to zero. Each rotation disturbs the entries already cleared, but never by as much as it removes, so the off-diagonal norm falls monotonically and the matrix converges to its diagonal of eigenvalues.

It is not the fastest method known and for a ten-by-ten naphthalene that is irrelevant. What it is, is short enough to read and to believe — and correct for degenerate eigenvalues with no special handling, which matters here more than anywhere, because the degeneracies are the results the theory gets exactly right.

Believing a solver is not the same as checking one, and there are three independent checks available.

The closed form, for rings and for chains. A ring gives $2\cos(2\pi k/n)$ and a linear chain of $n$ atoms gives $2\cos(k\pi/(n+1))$, both exactly, so two whole families can be checked against an expression.

The trace relations, which follow from the graph rather than from the solution. The trace of an adjacency matrix is zero, because no atom is bonded to itself, so the eigenvalues sum to zero. The trace of its square counts every edge twice, so the squares sum to twice the bond count. Both hold for systems with no closed form, including naphthalene.

The pairing theorem, and this is the check worth having because it rejects things. A graph whose atoms can be two-coloured so that no edge joins two of the same colour has eigenvalues in plus-and-minus pairs about zero. Benzene, butadiene and naphthalene are such graphs and their level diagrams are symmetric about α. Odd rings are not, and the cyclopropenyl, cyclopentadienyl and tropylium systems accordingly have unsymmetric diagrams — so the check runs in both directions, and refuses a non-alternant system whose levels came out paired anyway.

A check that has never rejected anything proves nothing. This one rejects three of the eleven systems in the table, correctly.

The surprise: geometry never enters #

The fact worth carrying away from this essay is the one that sounds like an error.

Benzene’s eigenvalues do not depend on its bond lengths, its bond angles, or whether it is regular. They depend on the fact that six carbons form a cycle. Change every bond length by a different amount and the Hückel answer is unchanged.

That is not a defect being confessed. It is what makes the theory’s successes meaningful: when Hückel gets a degeneracy right, it has got it right from the connectivity alone, which means the degeneracy is a consequence of connectivity and not of anything finer. Aromaticity, on this account, is a property of a graph, and the extraordinary thing is how far that gets.

It also predicts precisely where the theory must fail. Anything that depends on geometry — the fact that cyclooctatetraene is a tub rather than a planar ring, the fact that cyclobutadiene distorts to a rectangle — is invisible to a theory that discarded the coordinates at the first step.

What the eigenvectors are worth #

Having diagonalised the matrix, the eigenvectors are sitting there and two further quantities cost nothing to extract.

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 theory than benzene is, and the bond orders read off those eigenvectors are where it shows. Benzene’s six identical bonds could be attributed to symmetry alone and need no theory at all. Naphthalene’s are not equivalent: the computation gives 0.725 for the bonds adjacent to the ring fusion, 0.603 across the top and only 0.518 for the shared bond itself, the ordering matches the measured bond lengths, and no set of equally weighted Kekulé structures produces it. That ranking is taken up in its own essay, where it is drawn rather than quoted.

What it costs #

A Jacobi sweep on a matrix this size costs microseconds, which makes the cost question an unusual one to ask and the answer worth having anyway.

The expensive part is not the arithmetic; it is the modelling decision that came before it. Reducing a molecule to a graph is a choice about which of its properties matter, and the choice has to be made before any computation begins. Every subsequent number inherits it.

That is the pattern with cheap theories generally. Their cost is not paid in computation, it is paid in the assumptions that made the computation cheap — and the assumptions are invisible in the output, which arrives as a set of clean numbers with no reminder attached. So every figure on this page carries one: Hückel, no repulsion.

The one genuine expense is the checking. Verifying against closed forms, trace relations and the pairing theorem takes more work than solving does, and that ratio is the same one computing an orbital settles at. A solver that converges to a plausible wrong answer is the failure mode, and it is invisible from inside.

One prediction that is an integer, and where it comes from #

Most of what survives the simplification survives with a parameter attached, and is therefore a shape rather than a number. There is a class of prediction that survives as an integer, and integers are the one kind of output a model with no fitted content can be held to absolutely.

The clearest is the count of non-bonding orbitals — levels at exactly $\alpha$, contributing nothing to the binding, and carrying the radical and ionic chemistry of a conjugated system. For a conjugated skeleton with no rings in it, that count is

$$\text{non-bonding levels} = N - 2\nu,$$

where $N$ is the number of carbons and $\nu$ is the largest number of double bonds that can be drawn on the skeleton at once, with no carbon used twice.

The right-hand side is something a chemist does with a pencil. Draw the best classical structure available; count the atoms it leaves over; that is how many non-bonding orbitals the molecule has, exactly, with no diagonalisation.

The cases check out immediately. Butadiene’s four carbons take two double bonds and leave none over, so it has no non-bonding level. Allyl’s three carbons take one double bond and leave one carbon over, so it has exactly one — which is the orbital that makes the allyl cation, radical and anion three species with the same skeleton and different chemistry. Trimethylenemethane’s four carbons admit only one double bond however they are drawn, leaving two over, and it duly has two non-bonding orbitals — which is the degenerate pair that makes it a ground-state triplet.

And every odd conjugated chain has an odd $N$, so $N - 2\nu$ is at least one: an odd alternant hydrocarbon always has a non-bonding orbital, which is why the odd radicals are a family rather than a list of coincidences.

The restriction to skeletons without rings is not a technicality and it is instructive. Once a ring is present the expression becomes a lower bound rather than an equality, and cyclobutadiene is where it fails: four carbons, two double bonds drawable, so the formula says none — and the molecule has two, sitting at exactly $\alpha$ and carrying everything interesting about it. The cycle supplies a degeneracy that no pencil-and-paper pairing can see.

So the theory hands back two things at once here. A count that a chemist could have obtained by drawing, exactly, whenever the skeleton is a tree — and a specific, identifiable class of molecule where drawing is not enough and the matrix has to be diagonalised. The second class is small, it contains the rings, and it is where every surprise in aromaticity lives.

Where the model stops #

Four limits, in order of how badly each bites.

No electron repulsion at all. This is the largest. Hückel cannot describe anything where two electrons avoiding one another matters, which includes excited-state energies, ionisation potentials beyond the crudest ordering, and every singlet–triplet gap. The cyclobutadiene prediction above is wrong for exactly this reason as well as the geometric one.

No geometry. A theory handed a connectivity returns the levels of that connectivity. It cannot predict a distortion, and every case where a molecule escapes a Hückel prediction by changing shape is a case it could not have seen coming.

Energies carry an unfitted parameter. Every number is in units of β, and β differs by a factor of two depending on what it is fitted to.

Only pi systems, and only planar ones. The separation of sigma from pi is exact only when a mirror plane makes the two sets of orbitals belong to different symmetry species. Twist the molecule out of plane and the separation is an approximation whose quality nothing in the theory reports.

Who found it, and when #

Erich Hückel published the treatment in 1931 and 1932, in a series of papers on unsaturated and aromatic compounds. He was a physicist by training working in a chemical problem, and the reception was correspondingly cool: the method was ignored by most chemists for two decades.

Part of the delay was presentational. Hückel wrote in the language of quantum mechanics for an audience that did not read it, and the reformulation into the graph-theoretic language above — which is what makes it teachable in an afternoon — came later, largely through Coulson and Longuet-Higgins in the 1940s.

The rest of the delay was Pauling. Resonance theory arrived at the same time, answered the same questions in a vocabulary chemists already had, and was promoted vigorously. That the two approaches are two bases for the same thing took a further twenty years to become common ground.

The vindication came from an unexpected direction. In 1965 Woodward and Hoffmann showed that the symmetry of the frontier orbitals — the highest occupied and lowest unoccupied, which Hückel theory gives correctly for the reasons above — governs whether a pericyclic reaction proceeds. A theory that gets no energy right to better than a factor of two turned out to predict reaction stereochemistry exactly, because the prediction depends only on the parts the graph determines.

Where to read on #

The result this theory makes computable is delocalisation, where benzene’s stabilisation becomes exactly 2β against a stated reference.

The rule it produces as an output is aromaticity as a shell closure.

The quantity its eigenvectors give away for nothing is bond order.

And the general lesson about which parts of a model to trust is orbitals are not where the electron is.

A matrix of ones and zeroes, and a great deal of chemistry.

What the pictures here cannot show. Every Hückel figure on this page draws a graph, and a graph has no lengths in it. The positions of the carbons in these drawings are a layout convenience and carry no information whatever — moving one would change the picture and not the answer. Nothing here can show a bond length, a bond angle, or the distortion that defeats the cyclobutadiene prediction, because all three were discarded before the matrix was built.