The chain distorts hardest where it stops

An open chain’s two ends are worth 1.08715 of end energy, fixed, independent of length, with the per-site difference between two dimerisation patterns falling at an exponent of −0.99986. Every number in that result depends on one assumption, and it is stated.

The alternation was held uniform. Every bond in the chain was given the same δ, which is what makes the end energy a clean constant and is not what a real chain does. An end has less to lose than the middle does, so it should distort further — and letting each bond find its own value turns a one-parameter problem into a minimisation over as many parameters as there are bonds.

Doing that produces a profile, and a profile contains something the energy arguments never had: a length.

That is worth saying about an argument whose every earlier result was an energy. The chain cannot stay even is an energy argument; the gap is not the band width relates two energies; the distortion the filling chooses is an energy against a period. A distance is a different kind of output, and it is the one a structural measurement reports.

The condition, which is one line #

With hoppings $1 + \delta_b$ and an elastic cost $K\sum_b \delta_b^2$ under a fixed total length, the derivative of the electronic energy with respect to $\delta_b$ is twice the bond order there. Setting the derivative of the whole thing to zero and imposing the constraint gives

$$\delta_b = \frac{P_b - \bar{P}}{K}$$

with $P_b$ the bond order of bond $b$ and $\bar P$ their mean. The subtraction is the constraint: the chain is not allowed to stretch, so only the variation of the bond order across the chain drives a distortion.

That is a fixed point rather than a search. The bond orders come from the hoppings, the hoppings come from the bond orders, and the two are iterated under a mixing until nothing moves — a hundred and fifty to two hundred and thirty passes depending on the stiffness, with the convergence checked rather than assumed.

The check that the answer is right is variational and cannot be otherwise: the relaxed chain is more bound than the uniform one at the same bulk alternation, because the uniform chain is one point in the space the relaxation searches. It is checked at two stiffnesses, against a uniform array with its mean subtracted so that both obey the same constraint — an alternating array on an odd number of bonds does not sum to zero, and the leftover is a stretch the relaxation was never allowed to buy.

The profile #

Two things are worth reading out of that table.

The uniform assumption is worst where it matters most. A stiff chain is barely disturbed by its end — an eleven per cent enhancement over one or two bonds — and a soft one is disturbed by more than a factor of three over five. The clean end-energy constant was computed in a regime where the assumption is fair, and it becomes unfair exactly in the limit where the chain is nearly undistorted.

And the excess decays. It is not that the chain has two alternations, one at the end and one in the middle. It has a profile that approaches the bulk value exponentially, and the rate is a number.

The length, and what it is a length of #

Fitting the excess over the bulk to an exponential gives a healing length that runs from 1.51 bonds in the most strongly dimerised chain to 5.29 in the weakest.

Set against the gap the bulk chain has opened — computed from the same chains, one as an energy and one as a distance — the length goes as the gap to the power −0.60 over the measured range.

That is the relation the received theory expects, and the exponent is not the one it expects. A gap $\Delta$ and a band width set a coherence length that goes as the reciprocal of the gap, so the exponent should be −1; the measured −0.60 is an effective exponent over a finite range, fitted through five points on chains of ninety-six sites with lengths between one and five bonds.

It is quoted as a description of the measurement rather than as a law, and the reason it is not the asymptotic value is visible in the numbers: a healing length of 1.5 bonds is not much longer than a bond, so the exponential the fit assumes is being fitted to two or three points of genuine decay. What the measurement supports is the direction and the order of magnitude, and it supports those firmly.

What the uniform calculation got right #

None of this overturns the earlier result, and it is worth being explicit about why.

The end energy is still a constant. It is a total, and a total is insensitive to how the distortion is distributed as long as the distribution is confined near the end — which is what a healing length of a few bonds means. The 1.08715 was measured on chains from sixty-four to a few hundred sites, where five bonds of profile is a small correction at both ends and is length-independent.

And the exponent of −0.99986 for the per-site difference survives for the same reason. A fixed amount of energy divided by a growing length gives a reciprocal, whatever the energy is made of.

What changes is what the structure is. The distortion the ends decide found that a chain’s two ends pick which of two dimerisation patterns it adopts; this essay finds that the pattern they pick is not uniform, and that the region where it is not uniform has a size that grows as the chain’s own gap shrinks.

What it says about a real chain #

A measured bond length near a chain end is not the bulk one. The alternation is larger there, so the short bond is shorter and the long bond longer, by up to a factor of three in a weakly dimerised material.

And the region this applies to is set by the gap rather than by the chemistry of the end. Nothing about the end appears in the healing length: it is the same number for any perturbation that breaks the chain’s translational symmetry, because it is a property of how far the electrons carry information. That is why the same length turns up as the extent of a defect state — a level in the gap lives on the same number of sites, for the same reason.

A short oligomer is all end. A chain of ten bonds with a healing length of five has no bulk at all, which is the regime most of the measured molecules are in — and a bulk alternation extrapolated from a series of short ones is being extrapolated through a length scale rather than towards one. That is the same trap where a molecule stops being one is about, from the structural side rather than the spectroscopic: the size at which a finite system starts behaving like an infinite one is a measurement, and here it is five bonds rather than a hundred sites.

What a length is worth having #

The earlier results are all energies, and an energy is compared against a thermochemical measurement. A length is compared against a structure, and structures are what diffraction reports.

It predicts a bond-length profile. In a material whose chains have identifiable ends — an oligomer series, a chain interrupted by a defect — the first bond should be more alternated than the fourth by an amount this calculation gives, and the deviation should die away over a number of bonds set by the material’s own gap.

It is a second route to the gap. Two quantities computed here from the same chains, one an energy and one a distance, and a measurement of either constrains the other. That is worth having in a subject where the gap is often the harder of the two to measure cleanly.

And it says which materials the uniform picture is safe for. A strongly dimerised chain heals in a bond and a half, so a uniform alternation is a good description of almost all of it. A weakly dimerised one heals in five, so a uniform alternation describes the middle of a long chain and nothing else.

What is quoted, and what is computed #

Nothing is quoted. There is no material named and no measurement anywhere. The chain, the elastic constants and the filling are the model’s parameters; every bond order, every alternation, every gap and both lengths are computed from matrices that were written down.

The two quantities the essay relates — the healing length and the gap — come from the same relaxed chains but from different operations on them: the length from a fit to a spatial profile and the gap from a diagonalisation of the uniform chain at the same bulk alternation. Neither is the other read back to itself.

What the healing length is a length of #

The distortion being largest at the end and decaying inwards is a profile, and the length it decays over is a physical quantity with a name and a measured value in the material these chains describe.

A chain that has dimerised has two equally good patterns, and any place where one gives way to the other is a boundary. The end of a chain is such a place — the pattern has to start somewhere — and so is the interface between two regions of opposite alternation in a long chain. Both are boundaries, and a boundary between two degenerate patterns is not sharp: the alternation reverses over some number of bonds, and that number is the boundary’s width.

So the healing length measured here is the width of the defect whose energy is priced separately. One calculation gives its energy, this one gives its size, and together they describe an object rather than an effect.

The measured value in polyacetylene is around seven repeat units — a defect spread over roughly fourteen carbons rather than localised on one bond — and it was obtained by fitting spectroscopic and magnetic measurements rather than by looking at a structure, because a defect that wide and that mobile does not sit still for a diffraction experiment.

The dependence measured here is the informative part. The width is one and a half bonds in a strongly dimerised chain and five in a weak one, so a larger gap makes a narrower defect: the more strongly a chain prefers its pattern, the less room it needs to switch between the two. That is the expected direction — a stiffer commitment heals faster — and it makes the width a report on the gap rather than a separate parameter.

Which gives the two calculations a shared conclusion. A dimerised chain’s boundary has an energy that does not depend on the chain’s length and a width that depends only on the gap, and both are properties of the boundary rather than of the sample — which is what makes such a defect a thing rather than a feature of a particular calculation.

The direction of the dependence is worth turning into a rule, because it settles a question a reader might otherwise have to compute. The gap and the width move oppositely, so a material that is strongly dimerised has narrow, well-localised defects and a weakly dimerised one has wide, diffuse ones — and the second kind overlap each other at much lower concentrations than the first. A chain with a small gap therefore stops behaving as a set of isolated defects at a much lower doping level, which is a statement about where an experiment leaves the dilute regime and it follows from a length rather than from anything about the chemistry of the dopant.

And it says why the uniform-alternation assumption was reasonable in the middle and wrong at the ends. The profile is flat wherever it is more than a few widths from a boundary, so a long chain is uniform over almost all of its length and structured only near its two ends — which is exactly the condition under which an end energy is a clean constant, and exactly why relaxing the assumption changed the ends and not the interior.

What this cannot say #

Half filling only. Everything is at one electron a site, which is where the alternation is a Peierls instability at all. A different filling wants a different period — the distortion the filling chooses is where that was computed — and the profile question would have to be asked again for each.

A quadratic elastic energy and a linear coupling. Both are the leading terms of an expansion, and a strongly distorted chain — the small-K end of the table, where δ reaches 0.32 — is where they are least reliable.

No electron repulsion. A real conjugated chain’s alternation is set by an argument with correlation in it, and the same relaxation with an on-site repulsion would give a different bulk value and a different length. The insulator band theory cannot see is the boundary of what this file can say.

And the ends are bare. A real chain terminates in something — a hydrogen, a substituent, another molecule — and the model’s end is simply where the matrix stops. The end is the hardest place to bind computed what that costs for a state; the profile here is what it costs for a structure, and both are properties of the truncation rather than of any chemistry.

And the exponent is fitted over a factor of eight in the gap with lengths between one and five bonds. It is a comparison between chains rather than a law, and the essay says so above rather than only here.

What was checked #

Every profile converged, with the iteration count recorded — a fixed point that was still moving would produce a profile and a length and neither would mean anything.

The relaxed chain does not stretch: its alternations sum to zero to eight decimal places, which is the constraint the stationary condition was derived under.

The end bond alternates more than the middle at every elastic constant, which is the claim the uniform calculation made and could not check.

And the excess decays, so there is a length to measure — a profile that was merely noisy would satisfy the previous check and fail this one.

A chain with a larger gap heals in fewer bonds, at every step of the table, which is the relation the essay is about.

And the length goes as a power of the gap between a half and the reciprocal. Measured: −0.60.

The relaxed chain is more bound than the uniform one at the same bulk alternation and under the same constraint, at two stiffnesses. That is variational and cannot be otherwise, which makes it the check on the minimisation rather than a result.

And the refusal is an elastic constant of nothing. With no cost to distorting there is no scale in the fixed point at all, and a chain with K = 0 is refused rather than relaxed.

And why a half-filled chain is where all of this happens: the levels at the Fermi energy crowd as the chain grows, so there is always a pair close enough for a small distortion to separate. The profile is what that separation looks like when the chain has ends to accommodate as well.

Still open: the soliton’s width, and the exponent #

The obvious open question is the soliton. Two dimerisation patterns exist above a hundred and twenty-eight sites, so a long chain can hold one of each with a boundary between them — and the relaxation here is exactly the calculation that boundary needs, because a soliton is a place where the alternation passes through zero and the width it does so over is the healing length measured above. Whether the width the relaxation finds matches the length fitted here, on the same chain at the same stiffness, is a comparison between two independent measurements of one quantity.

The nearer question is about the exponent. Five gaps over a factor of eight, with lengths of one to five bonds, gave −0.60 where the argument says −1. Running the same relaxation on chains of a few hundred at stiffnesses that give lengths of ten to thirty bonds would say whether the exponent is approaching the reciprocal or settling somewhere else — and the answer matters, because a length that goes as the gap to the power −0.6 rather than −1 would mean the received relation is missing a term at the sizes real oligomers have.