A chain cannot stay even

Every chain in this field so far has had equal bonds. That assumption is convenient, it makes the closed forms available, and it is wrong.

A half-filled chain of equally spaced atoms is not a stable structure. It lowers its energy by alternating — long, short, long, short — and it does so for any value of the elastic stiffness resisting the change, however large. The result is Peierls’s, and it can be computed here rather than quoted.

The result is worth stating carefully because its strength is easy to underestimate. It is not that an even chain is usually unstable, or unstable for reasonable spring constants, or unstable in the models people happen to use. It is that no spring constant, however large, holds it — and the reason is a property of a logarithm rather than of any particular number.

The setup, which is three lines of arithmetic #

Take a chain of $n$ sites at half filling — one electron each, so the lower half of the levels is occupied. Let the bonds alternate: every other one is $\beta(1 + \delta)$ and the rest are $\beta(1 - \delta)$.

Two things change as $\delta$ grows from zero.

The electrons gain. Alternating opens a gap in the middle of the band. The levels below the gap go down and the ones above go up, and only the ones below are occupied — so the total electronic energy falls. This is computed here by rebuilding the matrix at each $\delta$, diagonalising it, filling the lower half and summing, with no formula assumed.

The springs cost. Moving atoms from their evenly spaced positions costs elastic energy, and for a small displacement that cost is quadratic: $K\delta^2$ per site, with $K$ a stated parameter and not a fitted one. Quadratic is not an approximation chosen for convenience — any smooth energy has a minimum where its first derivative vanishes, so the leading term in a displacement from equilibrium is always the square. Whatever the σ framework of a real chain actually does, it does this for small enough displacements.

The question is which wins for small $\delta$, and the ordinary answer would be the springs — because a quadratic that starts at zero and a gain that starts at zero should be comparable, and whichever has the larger coefficient wins.

Why the ordinary answer is wrong #

The electronic gain is not quadratic. It goes as $\delta^2 \log(1/\delta)$, and a logarithm diverges — slowly, but without limit — as $\delta$ approaches zero.

So the ratio of gain to $\delta^2$ does not settle on a coefficient. It rises as $\delta$ falls, without bound, and therefore exceeds any fixed $K$ once $\delta$ is small enough. There is no spring constant large enough to hold the even chain, and that is the whole theorem.

The computed numbers show the rise directly. Dividing the computed gain by $\delta^2$ at three small alternations gives $2.564$ at $\delta = 0.08$, $3.194$ at $\delta = 0.05$ and $5.330$ at $\delta = 0.02$. A quadratic gain would give the same number three times; this one has doubled by the time the alternation has fallen to a quarter, and it keeps going.

That test can fail: run on a sequence that really is quadratic, the same comparison comes back flat, so a flat answer here would have meant a quadratic gain.

Where the logarithm comes from #

The divergence is not an accident of one-dimensional arithmetic; it is a consequence of what a half-filled band looks like at the Fermi level, and the reason is worth following.

Opening a gap of size $\Delta$ moves each level near the Fermi energy down by an amount that falls off with distance from it. The total gain is therefore a sum over levels, weighted by how close each is to the Fermi level and by how many levels lie there — which is the density of states.

For a one-dimensional band the density of states diverges at the band edges, and after the gap opens the new band edges sit exactly where the Fermi level was. So the sum is over a region where the levels are densest, and the integral picks up a logarithm from the divergence.

This is the same fact that makes one dimension special throughout the subject: a one-dimensional band edge has an infinite density of states and a three-dimensional one does not. In three dimensions the corresponding gain is genuinely quadratic and a stiff enough lattice does hold, which is why three-dimensional metals exist in abundance and one-dimensional ones almost do not.

What the distortion produces #

The consequences are large and all of them are observable.

A gap opens. At the best alternation for a stiffness of $1.6$, the computed gap is $0.49\beta$, from a chain whose gap was falling towards zero before the distortion. The metal has become an insulator.

Bond lengths alternate. A chain of carbon atoms with one π electron each is polyacetylene, and its bonds do alternate — about $1.36$ and $1.44$ ångströms rather than the $1.40$ an even chain would give. That alternation is measured and is not small.

The symmetry drops. The even chain has a repeat of one atom and the alternating one a repeat of two. Every symmetry argument applies: what was one level per repeat becomes two, and the descent is exactly the kind of thing descent in symmetry computes for finite groups.

It is the Jahn–Teller argument at the limit #

The same argument works on a molecule.

Cyclobutadiene is square, has four π electrons, and puts two of them into a degenerate non-bonding pair. Distorting it to a rectangle splits that pair; the lower half drops and holds both electrons; the energy falls linearly in the distortion while the elastic cost is quadratic, so a small enough distortion always wins. That is the Jahn–Teller theorem, and delocalisation is stabilising, and other things that are false in general computes the balance.

The Peierls argument is the same argument with the degeneracy replaced by a continuum. Instead of two levels at exactly the same energy there are many levels within an arbitrarily small energy of each other, which is what a half-filled band is. The distortion splits them, the occupied ones go down, and the only difference is that the gain is $\delta^2\log(1/\delta)$ rather than linear in $\delta$ — because there is no exact degeneracy to break, only an arbitrarily close approach to one.

Both are cases of the same principle: a system with states at the Fermi level and a distortion available that splits them will take the distortion. The molecular version has a name from 1937 and the extended one from 1955, and they were arrived at independently.

What the numbers actually are #

It is worth putting the computed quantities in one place, because the argument is easier to check than to follow.

At a stiffness of $K = 1.6$ and a hundred sites, the scan runs over nine alternations from zero to thirty per cent. The net gain — electronic minus elastic, per site — is best at $\delta = 0.12$, where it is worth $0.0078\beta$ per site. That is a small number and it is a per site number: for a chain of a hundred it is nearly $0.8\beta$ in total, which is a substantial fraction of a bond.

The gap at that alternation is $0.49\beta$. If $\beta$ is taken as the usual $2.5$ electronvolts, that is about $1.2$ electronvolts — squarely in the range where a material is a coloured semiconductor rather than a metal or a transparent insulator, which is what polyacetylene is.

The three ratios above deserve a second look as well. The gain divided by $\delta^2$ has risen from $2.56$ to $5.33$ as the alternation fell from eight per cent to two, which is a factor of two over a factor of four in $\delta$ — consistent with a logarithm and inconsistent with any power law with a fixed exponent. Extending the scan to smaller alternations continues the rise until the chain’s own level spacing cuts it off, which is the finite-size limit of the argument rather than a limit of the physics.

Benzene, which is the case that does not distort #

The comparison that makes the whole argument legible is why benzene keeps its equal bonds.

Benzene has six π electrons in a ring of six, and its π shell is exactly closed: the highest occupied level is a degenerate pair, both members full, with a clear gap to the empty pair above. There are no states at the Fermi level to split.

Alternating benzene’s bonds therefore gains nothing to first order. The occupied levels move by an amount quadratic in the distortion, the elastic cost is also quadratic, and the elastic term wins for a stiff enough frame — which the σ framework is. So benzene stays regular, and it stays regular for a reason that is about its electron count rather than about resonance.

That is a considerably better account than the usual one. “Benzene’s bonds are equal because the two Kekulé structures resonate” explains nothing about why cyclobutadiene’s are not, since cyclobutadiene has two such structures too. The shell-closure account explains both cases with one argument, and aromaticity as a computed shell closure does the counting.

The chemist’s version of the same statement #

There is a way of saying all of this in the vocabulary a chemist already has, and it is exact rather than an analogy.

A chain of carbon atoms with one π electron each, drawn as a structural formula, has to be drawn with alternating single and double bonds — there is no way to draw a chain of carbons with one π electron each and give every bond the same order. The Peierls result says that this drawing is right, and that the temptation to average the two Kekulé-like structures into a chain of equal one-and-a-half bonds is what is wrong.

Benzene is the case where the averaging is right, and the reason it is right there is that the ring closes a shell. The general rule is therefore the reverse of what the resonance picture suggests: equal bonds are the exception and need a reason, and alternation is what happens by default.

That reverses a piece of standard teaching. Conjugation is usually presented as a stabilising influence that tends to equalise bonds, with benzene as the archetype and long polyenes as more of the same. The arithmetic says long polyenes are not more of the same at all — they are the case where the stabilisation is not enough to hold the structure together, and they alternate. Conjugation and its limits works through where the extrapolation from benzene stops being safe.

A six-carbon chain’s highest occupied π orbital has unequal coefficients along it, and the bond orders that follow are unequal too — an alternation already present in the eigenvectors before any geometry is allowed to respond. The distortion computed here is what happens when the geometry is allowed to respond to it.

The parameters, and what does not depend on them #

Three numbers appear in this calculation and it is worth separating what turns on each.

The chain length is a hundred. Nothing important depends on it: the same computation at forty and at two hundred gives the same shape and the same conclusion, and the length matters only in that the gain’s logarithm is cut off by the level spacing, so a very short chain shows a weaker effect.

The spring constant is $1.6$ in $\beta$ units, and it is stated rather than fitted. Changing it moves where the best alternation is — a stiffer frame distorts less — and does not move whether there is one. That is the theorem: the best $\delta$ is non-zero for every $K$.

The half filling is essential and is the one assumption that cannot be relaxed. At any other filling the distortion that helps is a different one, with a repeat length set by the filling rather than by two, and at a filling near zero or one the effect vanishes because there are no states at the Fermi level to speak of.

So the claim being made is narrow and checkable: a half-filled one-dimensional chain of equally spaced atoms is not at a minimum of its energy. Everything else in this essay is context around that.

The instability no perturbation theory could have found #

A logarithm beating a constant is the mechanism, and it has a consequence for how the result had to be discovered — because a gain that is steeper than a parabola at the origin is a gain that no expansion in the coupling can see.

Solve for where the two terms balance and the alternation the chain settles at goes as

$$\delta \sim \exp\!\left(-\frac{1}{\lambda}\right),$$

with $\lambda$ the dimensionless strength of the coupling between the electrons and the lattice. So does the gap that opens.

That expression is not analytic at zero. Every derivative of it vanishes there: expand it in powers of $\lambda$ and every coefficient is zero, to all orders. A perturbation treatment in the coupling — the standard way of asking what a weak interaction does — returns exactly nothing at first order, at second, and at every order after.

So the instability is invisible to the method a physicist would reach for first. It is not small in the coupling; it is zero in every term of the expansion and non-zero in the sum, which is a distinction perturbation theory cannot draw.

Finding it requires doing what is done here: writing down the total energy as a function of the distortion and comparing two terms, rather than expanding in the interaction that produces one of them.

The same functional form appears in one other place in physics and the coincidence is not one. The superconducting gap has the identical shape — an exponential of minus the reciprocal of a coupling — and for the identical reason: a logarithmic divergence in a sum over states near a Fermi surface, beaten against a constant cost. Two phenomena discovered decades apart, in different materials, by different arguments, sharing an expression because they share a divergence.

Which is worth carrying as a warning about a class of results rather than about this one. An effect whose size is an exponential of a reciprocal coupling is an effect that arbitrarily careful perturbation theory will report as absent, and the only way to find such a thing is to sum something rather than to expand it.

It also explains a feature of the result that would otherwise look like a weakness. The alternation depends on the spring constant through an exponential of its reciprocal, so a factor of two in the stiffness changes the distortion by far more than a factor of two — and the conclusion that some distortion always wins is robust while any prediction of how much is extremely sensitive. That is the honest division: the instability is a theorem and the amplitude is a number that no quoted parameter is accurate enough to fix.

What it does not establish #

It does not establish that real polyacetylene alternates because of this mechanism, though the evidence is good. Electron repulsion, which this model omits entirely, also favours alternation, and separating the two contributions experimentally is genuinely difficult.

It does not establish anything about three dimensions, where the density of states at a band edge does not diverge and the gain really is quadratic. Three-dimensional metals are stable and this argument does not threaten them.

It does not establish the size of the alternation in any real material, and the figure’s twelve per cent should not be read as a prediction. That number is set by the stated spring constant, which was chosen to make the balance legible rather than fitted to anything. What the calculation predicts is the existence of a best alternation and its non-zero value, not where it falls.

And it does not establish that a distortion is always available. The argument shows that if a distortion exists which splits the states at the Fermi level, it will be taken; whether the structure admits one is a separate question about geometry, which is the sort of question where the atoms go handles at molecular sizes and which this field cannot compute at all.