The band limit

A ring of six carbons has six π levels. A ring of sixty has sixty. The natural expectation is that the sixty are spread over a wider range than the six, because there is more molecule for them to spread over.

They are not. Every level of every ring lies between the same two values, and the only thing that changes is how many of them are packed inside.

The closed form, and what it says #

A ring of $n$ atoms has Hückel levels

$$x_k = 2\cos\!\left(\frac{2\pi k}{n}\right), \qquad k = 0, 1, \ldots, n-1,$$

in units of $\beta$ above $\alpha$. That comes from the eigenvalue problem rather than from a table, and every computed ring is checked against it — Hückel theory and what it gets right is where the derivation sits.

Read the formula as $n$ grows. The cosine takes values between −1 and 1 whatever its argument, so every level lies in $[-2, 2]$ for every $n$. Benzene’s lowest level is at $+2$ and so is the lowest level of a ring of a million.

What changes is the spacing. For six atoms the levels are at 2, 1, 1, −1, −1, −2 — a gap of one unit between neighbours. For sixty they are separated by a few hundredths. In the limit they become a continuum, and a continuum of levels inside a fixed interval is a band.

The width of that band is $4|\beta|$, and $\beta$ is a property of two neighbouring atoms rather than of the crystal. That is the point the picture makes most sharply: how wide a band is depends on how strongly a neighbouring pair interacts, and not at all on how many neighbours there are in total.

Where the levels crowd #

The levels are not spread evenly through the band, and the unevenness has a closed form worth deriving.

The number of levels per unit energy is $\mathrm{d}k/\mathrm{d}x$, and differentiating $x = 2\cos(2\pi k/n)$ gives, per atom,

$$g(x) = \frac{1}{\pi\sqrt{4 - x^2}}.$$

That expression diverges at $x = \pm 2$. At the band edges the cosine is turning over, so $\mathrm{d}x/\mathrm{d}k$ vanishes and a whole range of $k$ maps onto almost the same energy. Physically: the levels pile up at both edges and thin out in the middle.

The divergence is integrable — the total number of states is finite — but it is real, and it is the one-dimensional van Hove singularity. In two and three dimensions the same construction gives weaker singularities at the band edges and kinks in the interior, and the whole vocabulary of solid-state spectroscopy is built on locating them.

Everything in that paragraph is visible in the figure at the top of this page, computed from the same ring formula the six-atom case uses.

Watching the gap close #

The approach to the limit is worth following one ring at a time, because it is not fast and it is not uniform.

The gap does not close smoothly, and the way it fails to is the more interesting result. Rings with $4m$ atoms have a zero gap at every size: their highest occupied and lowest empty levels are a degenerate pair sitting exactly at the band centre, half filled, which is cyclobutadiene’s problem repeated at every multiple of four. Rings with $4m+2$ atoms have a gap, and it is that series which closes as $1/n$:

$$2.000,\; 1.236,\; 0.890,\; 0.695,\; 0.418,\; 0.203,\; 0.123$$

for rings of 6, 10, 14, 18, 30, 62 and 102 carbons. Halving the gap means roughly doubling the ring. A ring with a gap of a hundredth of a $|\beta|$ needs of the order of a thousand atoms.

So “a big enough molecule is a metal” is a statement about a limit that real molecules approach very slowly. It is also the reason a conjugated chain of ten or twenty carbons is coloured rather than metallic: the gap has come down into the visible, which is a few electron volts, and stopped.

The levels converge faster than the gap does. A ring of sixty already has a density of states visually indistinguishable from the infinite one across most of the band; it is only near the centre, where the gap lives, that the finite size still shows. That is a general feature of taking this kind of limit — bulk properties converge quickly and the states nearest the Fermi level converge last, which is unfortunate, because the states nearest the Fermi level are the ones that decide almost everything.

Where the electrons go #

A band is only half the story; the other half is how many electrons there are to put in it.

Each carbon contributes one π electron, so a ring of $n$ carbons has $n$ electrons for $n$ levels, and fills exactly half of them. The highest occupied level sits at the middle of the band, where $x = 0$, and so does the lowest empty one.

For a finite ring of the closed-shell kind there is a gap between them, and the sequence above closes it as $1/n$. In the limit the gap is zero, the highest occupied and lowest empty states touch, and the system is a metal — and the rings whose atom count is a multiple of four have got there already at any size, which is why they are not closed-shell molecules at all.

That is the conclusion Hückel theory reaches, and it is wrong for the molecules it is about. Long polyenes are insulators with a gap of about 1.5 electron volts that does not close however long the chain gets. The escape is a distortion: the bonds alternate, the ring’s symmetry drops, and the degeneracy at the band centre splits — a Peierls distortion, which is the extended version of the Jahn–Teller argument in descent in symmetry. Conjugation, and its limits computes what the gap does and where the model parts company with the measurement.

The bottom of the band is the one combination with the same sign on every atom, and it looks the same in a ring of six as in a ring of a million. That is the sense in which a band is a catalogue of phase patterns rather than of places: the patterns are indexed by how many sign changes they have, and the count runs from zero to the number of atoms whatever that number is.

The same arithmetic, drawn as a chain #

A ring is periodic and a chain is not, and the difference shows up in exactly one place.

A chain of $n$ atoms has levels $x_k = 2\cos\!\left(k\pi/(n+1)\right)$ for $k = 1$ to $n$, which lie in the same interval and crowd in the same way. The band is identical in width and nearly identical in shape. What differs is that a chain has no degeneracies — every level is single, because a chain has no rotational symmetry to enforce a pair — while a ring’s levels come in degenerate pairs except at the very top and bottom.

That difference is worth a moment, because it is the source of everything aromaticity does. A ring’s pairing is what makes a shell close at 2, 6, 10 electrons rather than at every even number, and it is a consequence of the ring’s symmetry rather than of anything energetic. Aromaticity as a computed shell closure is that observation run through every ring size from three to ten.

In the large-$n$ limit the distinction washes out: a band computed from a ring and a band computed from a chain differ by terms of order $1/n$, and the boundary condition stops mattering. Which is the formal statement of something a chemist knows anyway — the middle of a long molecule does not know where the ends are.

What was computed, and how #

The levels come from the closed form for a ring, which is itself checked against the eigenvalues of the adjacency matrix for every ring from three to twelve, to a part in $10^9$. Neither is trusted on its own: the closed form is a formula that could be mistyped and the eigensolver is a numerical method that could converge somewhere wrong, and the agreement of the two is what makes either quotable.

The density of states is a histogram of the computed levels of a ring of two thousand, in twenty-eight bins across the band. The comparison curve is the closed-form density integrated across each bin rather than sampled at its centre — a detail that matters at the outermost bins, where the centre value is finite and the true average is not, so sampling would quietly under-report the very divergence the figure is about.

The agreement is checked. Away from the two edge bins, the computed density must match the closed form to better than 0.02, and it comes out at 0.008 for a ring of two thousand. The check excludes the edges deliberately and says so, because the closed form diverges there and a histogram cannot.

Every level is required to lie inside the band. For each ring drawn, the smallest and largest computed levels must fall within ±2 to within a rounding error. A version of the ring formula with a wrong factor would break that immediately, and it is the cheapest possible check on the central claim of the essay.

Where the model stops #

This is a one-dimensional band and the world is three-dimensional. The $1/\sqrt{4-x^2}$ divergence is specific to one dimension. A three-dimensional band has a density of states that goes as $\sqrt{E}$ near a band edge and has kinks rather than divergences inside, and getting those shapes right is what band-structure calculations are for.

One orbital per atom. Real solids have several bands, which overlap or do not, and whether they overlap is the difference between a metal and an insulator in most real cases. Nothing in the ring model can produce a second band.

No electron repulsion at all. A half-filled band of one orbital per site is exactly the case where repulsion matters most: the Hubbard model, which adds a cost for putting two electrons on one atom, turns the same half-filled band into an insulator for a large enough cost. That transition is invisible here and is the subject of a large literature.

The geometry is fixed. The ring is assumed regular, and the Peierls distortion mentioned above is precisely the failure of that assumption. The model that predicts the metal is the model that cannot see what stops it being one.

And $\beta$ still has no number. Every energy here is in units of it. Turning the band width $4|\beta|$ into electron volts requires a value fitted to something, and the values fitted to different observables differ by a factor of two.

The generalisation #

The useful transfer is that a limit can be taken on the level structure without taking it on the calculation.

Nothing in this essay is a solid-state calculation. It is the same adjacency matrix, the same eigenvalues and the same filling rule used for benzene, evaluated at a larger $n$ — and the band, the density of states and the van Hove singularity all fall out of arithmetic that was set up to describe a six-membered ring.

That continuity is the argument where two-centre bonding stops makes in the other direction: rings, clusters and metals are not a different subject from molecules, and the two-centre bond is the special case rather than the general one. A band is what delocalisation looks like when there is nothing to stop it.

It also puts a limit on how much can be learned from the small cases. Benzene contains every qualitative feature of the band — the edges, the crowding, the half-filling — and none of the quantitative ones, because six levels cannot show a density of states. Knowing which features survive the limit and which are artefacts of smallness is most of what taking a limit is for.

What a band is not #

Two things are worth ruling out, because the word carries associations the arithmetic does not support.

A band is not a set of orbitals spread over the crystal in the sense a molecular orbital is spread over a molecule. Every level in the band is spread over every atom equally — that is what the ring formula says — so “delocalised” does not distinguish one level from another. What distinguishes them is the phase pattern: the lowest has every neighbour in phase, the highest has every neighbour out of phase, and the levels in between change sign at intervals set by their position in the band. A band is a catalogue of phase patterns, not of locations.

A band gap is not a bond energy. The gap in the sequence above is a difference between two one-electron levels, and turning it into an excitation energy assumes the removal of an electron from one and the addition of it to another cost nothing in repulsion — the same assumption orbitals are not where the electron is takes apart. In real conjugated systems the optical gap and the one-electron gap differ substantially, which is why a Hückel gap in units of $|\beta|$ can be calibrated against absorption spectra and still fail to predict conductivity.

Both cautions belong to the same family as this site’s standing one: the model is a one-electron model, and every quantity it returns is a one-electron quantity wearing the name of something measurable.

Why matter has a finite binding energy per atom #

The levels crowding into a fixed interval rather than spreading out is stated here as an arithmetic fact, and it is worth reading forwards, because the alternative would make the world impossible.

Suppose the band did widen as the ring grew — that the lowest level went on descending as atoms were added. Then the binding energy per atom would grow without limit with the size of the sample, a large crystal would be more stable per atom than a small one by an amount that never stopped increasing, and there would be no such thing as a cohesive energy of a material.

The fixed interval is what forbids that. Every level lies between $-2\beta$ and $+2\beta$ whatever $n$ is, so the sum over occupied levels grows in proportion to how many there are and the energy per site converges. For a ring at half filling it converges to $4/\pi$ times β, which is a number rather than a trend.

That convergence is the reason a cohesive energy is a property of a substance rather than of a sample, and it is why a table of them can be printed at all. It is also what makes the whole apparatus of thermochemistry possible: enthalpies of formation are quoted per mole because the quantity per mole stops changing, and it stops changing because the levels stop spreading.

So the crowding is not a curiosity of the arithmetic on the way to a band. It is the statement that energy is extensive, derived rather than assumed, on the smallest system where the derivation can be watched — and a model in which it failed would be a model of a universe with no materials in it.

Who found it, and when #

Bloch’s 1928 theorem is the general statement: in a periodic potential the eigenfunctions are plane waves times a periodic function, labelled by a wavevector, and the energies form bands. The ring formula here is Bloch’s theorem for a one-dimensional lattice with periodic boundary conditions, arrived at by a route — diagonalise the adjacency matrix — that makes no reference to periodicity at all.

The tight-binding method, which is what Hückel theory is when a physicist writes it, is Bloch’s too, and the fact that chemists and physicists reinvented the same arithmetic independently and gave it two names is a small historical embarrassment that Roald Hoffmann spent much of the 1980s dismantling. His extended-Hückel treatments of surfaces and solids are the standard bridge between the two vocabularies.

Van Hove’s paper on the singularities in the density of states is from 1953. Peierls’s argument that a one-dimensional metal is unstable to a distortion that opens a gap is from 1955, and the compounds that demonstrate it were not made until the 1970s.

Still open: three centres and four electrons #

This essay takes delocalisation to its limit and finds a band. The other direction — the smallest system in which bonding is genuinely not a two-centre affair — is three atoms and four electrons, which turns out to explain a family of molecules the octet rule declares impossible.