Conjugation, and its limits

Ethene absorbs in the far ultraviolet. Butadiene absorbs at longer wavelength. Hexatriene longer still, and by the time a chain has eleven conjugated double bonds it absorbs blue light and looks orange, which is why carrots are orange.

That progression is the most familiar consequence of conjugation, it has a closed form, and the closed form makes a prediction that does not come true.

The gap, in closed form #

A chain of $n$ conjugated carbons has Hückel eigenvalues

$$x_k = 2\cos\!\left(\frac{k\pi}{n+1}\right), \qquad k = 1 \ldots n,$$

and with $n$ electrons filling $n/2$ orbitals, the highest occupied is $k = n/2$ and the lowest empty is $k = n/2 + 1$.

The gap is therefore

$$\Delta = 2\beta\left[\cos\!\left(\frac{(n/2)\pi}{n+1}\right) - \cos\!\left(\frac{(n/2+1)\pi}{n+1}\right)\right],$$

which for large $n$ behaves as $2\pi\beta/(n+1)$ — inversely proportional to the chain length.

So the gap closes as the chain grows, the absorption moves to longer wavelength, and an infinitely long polyene would be a metal.

The prediction, and what happens instead #

Real polyenes do not follow the $1/n$ law indefinitely. The absorption shifts to longer wavelength as predicted for the first ten or so double bonds and then saturates: beyond about fifteen, adding more conjugation stops moving the absorption.

Long polyacetylene is not a metal. It is a semiconductor with a gap of about 1.5 electronvolts, and no amount of extending the chain closes it.

The reason is a distortion that Hückel theory cannot see, and it is the same kind of failure as cyclobutadiene’s escape from its own prediction.

A chain of equally spaced carbons is unstable with respect to alternating its bond lengths. Alternating opens a gap at the point where the levels would have crossed, and the energy gained by lowering all the occupied levels exceeds the elastic cost of moving the atoms. So the chain buckles into alternating short and long bonds, and the gap stops closing.

That is the Peierls distortion, and it is invisible here for a reason stated plainly in Hückel theory: the geometry was discarded before the matrix was built. A theory handed a connectivity returns the levels of that connectivity, and cannot report that the connectivity would prefer to be a different shape.

The alternation is already visible in the bond orders #

What the theory can do is show why the distortion is available, and the evidence is in the eigenvectors rather than the eigenvalues.

Those numbers are the interesting part. In a chain of equally spaced carbons, Hückel theory still gives unequal bond orders, largest at the ends and converging toward the middle. A bond with a higher pi order would be shorter if the atoms were free to move, so the eigenvectors are already pointing at the distortion the eigenvalues cannot see.

The theory therefore contains the seed of its own correction, in a quantity nobody usually looks at.

Where a conjugated chain reacts, read off a coefficient #

The gap is the property conjugation is famous for. The eigenvectors carry a second one, and it is the property organic chemistry actually uses.

Drag either slider up and down the ladder and the readout gives energy and node count together. The pattern is invariable: zero nodes at the bottom, one more at each step, and the highest orbital with a node across every bond. That is the particle in a box with atoms in it, and the terminal-coefficient rule falls out of the boundary condition — a standing wave on a segment has its antinodes at the ends.

The 1,4-addition of bromine to butadiene, the Diels–Alder reaction’s regiochemistry, the sites at which a conjugated ketone is attacked: all three are read off which carbons carry the large coefficients in the frontier orbitals, and Hückel theory gives those coefficients correctly for the reason it gives degeneracies correctly — they are fixed by the connectivity, and the connectivity is exact.

Colour, and how much of it this explains #

The opening claim about carrots deserves scrutiny, because it is stated far more confidently than it should be.

What the calculation gives is a HOMO–LUMO gap in units of β. Turning that into a wavelength requires two things it does not supply: a value for β, and the assumption that the lowest absorption corresponds to promoting one electron from the highest occupied orbital to the lowest empty one.

The first is a fit. The second is wrong for polyenes, and interestingly wrong. Long polyenes have a low-lying excited state — the so-called $2\text{A}_g$ state — that is a doubly excited configuration and lies below the singly excited one that this argument describes. It is also forbidden by symmetry, so it does not absorb, which is why the story survives: the state the argument predicts is the one that shows up in an absorption spectrum, and the lower state is invisible there.

So the account is right about the trend, right about the transition that absorbs, and wrong about the lowest excited state. That combination is characteristic of one-electron models: the ordering of orbitals is robust and the ordering of states requires electron repulsion, which is the term Hückel theory does not have at all.

What was computed, and how #

The eigenvalues come from a cyclic Jacobi sweep on the adjacency matrix, and for chains there is a closed form to check them against: $2\cos(k\pi/(n+1))$ for every $k$. The worst disagreement across chains from two to eight carbons is a few parts in $10^{16}$.

Two further relations hold without any closed form. The eigenvalues sum to the matrix trace, which is zero, and their squares sum to twice the bond count. Both are asserted for every system on this site, and both would catch a solver converging quietly to the wrong place.

The gaps quoted above are differences of computed eigenvalues, in units of β. β has no number here, deliberately — fitting it to an absorption maximum would make the gaps look like wavelengths and hide the fit inside them. The wavelengths quoted in the opening paragraph are experimental.

The bond orders are sums over occupied eigenvectors, and they carry one piece of care: they are invariant under any rotation within a fully occupied degenerate set, which is what makes them safe to quote. The individual orbital coefficients in a degenerate set are not.

The surprise: the same arithmetic gives a band #

Push the chain to its limit and the level pattern does something worth watching.

The eigenvalues $2\cos(k\pi/(n+1))$ for large $n$ fill the interval from $-2$ to $+2$ densely. Discrete levels become a continuum — a band — with a width of $4\beta$ that does not depend on the chain length at all.

That is the tight-binding model of a solid, arrived at from a molecule by taking $n$ large, and it is the same arithmetic throughout. A molecule’s discrete orbitals and a solid’s bands are not two theories; they are one calculation at two sizes.

The band’s width being independent of $n$ is the detail that makes the connection useful. Adding atoms to a chain adds levels without widening the range they occupy, so the levels crowd together, and the crowding is what a band is. There is no point along the way where a molecule stops and a solid starts.

Which is why the question of where two-centre bonding stops has no sharp answer, and why the same treatment covers benzene and graphite.

What it costs #

The arithmetic costs nothing: a small eigenvalue problem per molecule.

What conjugation costs as an idea is the thing worth pricing, and the bill is a specific one.

It is not additive. Two isolated double bonds and one conjugated diene do not differ by a constant, and the difference is not proportional to anything simple. The delocalisation energy of butadiene against two ethenes is 0.472β; hexatriene against three is 0.988β; benzene against three is 2β. Three numbers, no pattern a rule of thumb would capture.

It requires planarity, and planarity is not free. The p orbitals must be parallel for the treatment to apply at all. Twisting one double bond out of plane by ninety degrees cuts the conjugation entirely, and a twist of thirty degrees cuts it by the cosine — so a substituent that forces a twist can turn a conjugated system into two isolated ones without breaking a bond. Biphenyl is the standard case, and its two rings are twisted by about forty-four degrees in the gas phase.

And it does not survive the first heteroatom without a fitted parameter. A nitrogen or an oxygen in the chain is handled by adjusting α at that site, and the adjustment is fitted rather than derived.

Where the model stops #

Four limits.

The geometry is not there. The Peierls distortion, the twist in biphenyl, the bond alternation in polyacetylene — all three are the molecule moving, and none is visible to a matrix built from a connectivity.

Energies carry an unfitted parameter. Every gap above is in units of β. Converting one to a wavelength requires a value for β, and values fitted to different observables differ by a factor of two.

Excited states are not ground states. An absorption maximum is a difference between two states, and a HOMO–LUMO gap is a difference between two orbitals. Equating them ignores the change in electron repulsion on excitation, which for polyenes is large enough that the lowest excited state of a long polyene is famously below the one this argument predicts. The correction requires configuration interaction, which is a many-electron treatment, and none exists here.

Sigma and pi are assumed separable. Exact only when a mirror plane puts them in different symmetry species, which requires the planarity the first limit says is not guaranteed.

Cross-conjugation, where the length argument breaks #

One further limit belongs here because it is a limit on the concept rather than on the theory, and it is the one most often missed.

Everything above concerns a linear chain. Branch it, and the count of double bonds stops predicting anything.

Cross-conjugated systems — where three double bonds meet at a central carbon rather than running end to end — have gaps that do not follow the chain formula, because their eigenvalue spectrum is not $2\cos(k\pi/(n+1))$. The graph is a star rather than a path, and a different graph has different eigenvalues.

The practical statement is that “number of conjugated double bonds” is not a well-defined variable. It counts edges in a graph while the gap depends on the graph’s shape, and the two coincide only for the path graph the chain formula assumes. A branched system with six double bonds and a linear one with six are not comparable, and the habit of quoting a conjugation length as though it were a single number quietly assumes a linearity that many real chromophores do not have.

That is the same lesson as everywhere in this treatment, met once more: the connectivity is the input, and any summary of it — a count, a length, a number of bonds — throws away part of what the calculation uses.

A last observation about the vocabulary. “Extended conjugation” is used as though it were a single lever, and this essay has now separated it into three: the graph’s shape, which fixes the level pattern; the chain’s length, which fixes the spacing; and the planarity, which decides whether the treatment applies at all. Changing a substituent can move any of the three, and the phrase records none of them.

Who found it, and when #

The link between conjugation length and absorption wavelength was empirical long before it was theoretical. Dye chemists in the nineteenth century knew that extending a conjugated chain deepened a colour, and the rule was used commercially for decades with no account of why.

Hückel’s treatment of 1931 supplied the account, and the free-electron model of Kuhn in 1948 supplied a cruder version that gets the same $1/n$ scaling from a particle in a one-dimensional box — which is, on reflection, the same physics with the atoms smoothed away.

Peierls’s theorem on the instability of a one-dimensional metal dates from 1955 and was a result in solid-state physics, arrived at without reference to polyenes. That it explains why carotene’s absorption saturates was recognised later, and the connection is the clearest instance in this subject of a molecular puzzle being solved by a result from a different field entirely.

Polyacetylene’s semiconducting behaviour, and the discovery that doping makes it conduct, won the Nobel Prize in Chemistry in 2000. The gap that the doping overcomes is the gap Peierls predicted and Hückel could not.

Where the ladder goes next #

The theory throughout is Hückel theory.

The quantity that reveals the coming distortion is bond order.

The cyclic case, where a count decides instead of a length, is aromaticity as a shell closure.

And what happens when the chain becomes a solid is where two-centre bonding stops.

The conclusion the essay reaches is therefore double, and both halves are worth keeping. Conjugation does what it is said to do, for the reason it is said to do it, over the range where the theory applies. And the range ends, at a length the theory cannot itself predict, for a reason that lives entirely in the coordinates it threw away.

The practical version, for somebody reading a spectrum rather than a level diagram: an absorption that has stopped moving as the chain is extended is not a failure of the sample. It is the chain having found its own bond alternation, and the wavelength at which it stops is a measurement of how strongly it did so.

That saturation point is, incidentally, the one genuinely quantitative thing a reader can take from this essay to a laboratory: an absorption that has stopped shifting is reporting that the chain has alternated, and roughly by how much.

What the pictures here cannot show. Every figure on this page is drawn from a graph with equally weighted bonds, and the alternation that real polyenes show is precisely what such a graph does not contain. The bond thicknesses in the bond-order figures encode a computed number, not a distance, and a reader looking for the short-long-short pattern of a real polyene will not find it drawn anywhere here — because the calculation that produced these figures assumed it away.