Peierls distortion — where it appears
Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.
Conjugation, and its limits
Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.
The band limit
Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.
A chain cannot stay even
Diagonalise a half-filled chain, alternate its bonds slightly, and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever — because the gain is steeper than a parabola near the origin and a logarithm beats any constant.
Copper is never quite octahedral
A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.
The gap is not the band width
Two numbers describe a band and they answer different questions. The width is set by how many neighbours an atom has; the gap is set by how unequal they are. A structure can have a wide band and no gap, a narrow band and a large one, and changing one leaves the other alone.
The hexagon is the frame's doing
Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.
Hypervalency does not stop at three centres
The three-centre four-electron bond is written up everywhere as an arrangement peculiar to hypervalent molecules. It is the first member of a family — five centres and six electrons, seven and eight — and the family predicts alternating bond strengths that the polyiodide crystal structures have.
The distortion the filling chooses
A half-filled chain of equal bonds is unstable and alternates — long, short, long, short. That is the case everyone is shown, and it is one case. Fill the chain a third of the way instead and the alternation is worthless: what wins is a pattern that repeats every three bonds, and the period is one over the filling at every filling tried.
The distortion the ends decide
A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.
The exponent was the window's
A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.
A decay that keeps slowing down
A healing length fitted over the first twelve bonds of each relaxed chain goes as the gap to the power −0.60, against an argument that says −1. A fitted exponent can belong to its window, and this one does — but not because the window is too short for the chain. The profile is not an exponential at all, so there is no single length for an exponent to be of.
The amplitude the collapse left behind
Five Peierls curves became one curve when each was divided by the alternation its ring settles at cold, so that amplitude is the whole of what distinguished them — and it was five golden-section searches over diagonalisations with no formula anywhere. It has one, exactly, as a sum of square roots; and writing it down says that one of the five rings was never measuring a long chain.
The exponent was the floor
Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.
Five rings that were five different sizes
Five warmed rings have scaled alternation curves that lie on one another to 3.41 per cent, and the departure from the bulk amplitude turns out to be a function of the ring measured in its own alternations. The five cases span a factor of seven in that quantity. Choosing sizes that make them comparable halves the residual — and runs into a floor the lattice itself imposes.
The rule of thumb was on the flat part
Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.
The window that was not a plateau
The near and far ends of this fit both sit on plateaus, and it is tempting to expect the same of every window. The third choice — that the local decay rate is read from the first quarter of the chain — is not a plateau. It never bound the soft half of the series and it was setting the answer for the stiff half, where opening it moves an exponent by a seventh, always downward. And a profile allowed to end on its own always runs 13.26 of its own decay lengths, which is the ruler that shows three cases are still cut at the midpoint.
Three points, and they all go down
Matching five rings at one value of n·δ∞ tightens the temperature collapse from 3.41 per cent to 1.62, and what is left might be the even-site rounding rather than anything physical. At three targets the residual falls monotonically — and at the smallest one it is a third of what the rounding leaves, which the rounding cannot explain.
The other window was a plateau too
Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.
Named alongside it
The objects these essays reach for when they reach for this one.
Bond alternationTight-binding modelsBand gapModel limitEigenvalueExtrapolationApproximationDegeneracyDelocalisationDistortionHOMO–LUMO gapThermodynamic limit