A chain of 100 is more stable alternating

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

15 essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A chain cannot stay even

A hundred-site chain diagonalised at nine different bond alternations. The rising curve is what the electrons gain, the other is what the springs cost, and the dashed curve is their sum. The sum has its best value at an alternation of twelve per cent rather than at zero, which is the statement that the even chain is not the structure.

The same competition on a chain of forty. The optimum alternation is close to the long-chain answer already, because what the logarithm in the electronic gain depends on is the density of states near the Fermi level and that has converged long before the total energy has.

A softer chain, where the alternation the balance settles at is larger. Halving the elastic constant does not double the distortion: the electronic gain is steeper than a parabola near the origin, so the response to a change in stiffness is sublinear — which is the same logarithm showing up as a shape rather than as a magnitude.

The hexagon is the frame's doing

The extended case, where the same competition comes out the other way: the π gain against the elastic cost for a long chain, with a minimum away from zero. The quantity plotted is the same one as in this essay’s hero figure and the balance is different, which is the whole of the difference between benzene and polyacetylene.

The distortion the filling chooses

Why any spring constant permits it. The electrons’ gain near the origin is steeper than a parabola — it carries a logarithm — and the elastic cost is exactly a parabola, so the two curves cross at a non-zero distortion however stiff the springs are made.

Two distortions in one coordinate

The Peierls distortion of a chain: the electronic gain against the elastic cost, with the balance settling at a finite alternation. The first-order term here plays the part λ₁ plays above; a second-order term acting on the same coordinate would be a coupling to a band the chain has that this picture does not draw.

The distortion the ends decide

The Peierls competition itself: the electronic gain, the elastic cost, and the difference between them against the distortion. The value of δ used throughout is where that difference is largest, and holding it fixed is what turns the comparison below into a measurement of the ends alone.

The energy difference between the two patterns, multiplied by the number of sites. It settles on 1.08715 in units of the hopping from a hundred and twenty-eight sites onwards and does not move afterwards, which is the statement that the whole difference lives at the two ends.

The best distortion of both patterns and of the ring of the same length. The short-bond-first phase always distorts. The other does not pay for its own elastic cost until the chain reaches a hundred and twenty-eight sites, so below that length the molecule has one dimerised structure rather than two.

The chain distorts hardest where it stops

The alternation of each bond along a relaxed chain, counting from the end, at four elastic constants. The dashed lines are each chain’s bulk value — the number the end-energy calculation used everywhere — and every curve starts above its own and comes down to it.

The healing length against the gap. A chain with a large gap forgets its end in one or two bonds; a chain with a small gap remembers it for five. The two quantities come from the same calculation and are two views of the same thing.

The same competition on a chain twice as long. The optimum alternation is very nearly the same number, because it is a bulk quantity, and the total energy gained is twice as large — which is the check that what the profile above measures is an end effect rather than a length effect in disguise.

The exponent was the window's

The zero-temperature version of the same competition, on the larger ring. The alternation the chain settles at is one of the two scales the collapse divides by; the other is the temperature at which it goes.

A decay that keeps slowing down

Two lengths off the same ten relaxed chains. The lower line is the twelve-bond reading and the upper one has no window in it; the dashed line is a slope of exactly −1.

The excess alternation over the bulk, logarithmically, from the end inwards. An exponential is a straight line here. The dashed line is one, drawn through the first curve’s first point at the length a twelve-bond fit assigns it.

The local decay length, bond by bond, on one chain. The measured curve is a hyperbola: the rate falls as the reciprocal of the distance from the end, and the dashed fit is what that shape predicts.

The amplitude the collapse left behind

The five amplitudes the collapse divided by: what the search returns, what the closed form gives, and how far each ring is from an infinitely long chain. The last column is the finding.

The electronic energy per site of a ring of forty against the alternation: nine diagonalisations, the same nine as sums of twenty square roots, and the smooth curve an infinitely long chain would follow.

The alternation against the elastic constant, three ways: a ring of forty, an infinitely long chain, and the exponential expression. The three do not agree, and they fail to agree at opposite ends.

The exponent was the floor

The fitted power against the number of coherence lengths of tail that survive above the floor. The hollow marks have fitted powers far below this scale, in the negative direction.

How many coherence lengths of tail stay above the floor, against the gap. The stretch available to fit over shrinks from six and a half e-foldings to under four.

The same fit at three floors. Where the reach is comfortable, three decades of floor move the answer by about 0.02; where it is not, the floor decides everything.

Five rings that were five different sizes

Each case’s ring size measured in its own alternations. The published five span a factor of seven; the matched five are chosen so the quantity is one number.

The sites needed to hold the ring size fixed in its own alternations, against the stiffness, with the bulk alternation beneath it.

The same collapse twice: the published cases, and five chosen to be the same ring in the units that matter.

The rule of thumb was on the flat part

For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty.

How far the exponent moves across every start from six bonds outward, as a percentage of its value at six.

The exponent fitted from two bonds, from six, and from the start that gives the largest value, with the whole sweep’s span beside it.

The window that was not a plateau

How far the rate can still be read, against how much of the chain is read. Curves that flatten below the dashed cap ended on their own; curves that track it are being cut.

Every stiffness, with the reach at a quarter and at a half, and what stopped each profile.

The fitted exponent against stiffness, under the quarter rule and with the window opened. The curves coincide below K = 2 and separate above it.

Three points, and they all go down

The worst spread across the five scaled curves, at three values of the matched product n·δ∞.

The residual beside the spread the even-site rounding leaves un-matched, at each target.

How many times tighter the matched collapse is than the published one, at each target.

The other window was a plateau too

The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty.

The softest chain, whose profile gives local rates only out to twenty-three bonds, at every far end tried.

How far the exponent moves as the far end is opened, over the ends each profile actually reaches.