Molecular Geometry & Bonding
what the shape decides
Concept

Finite-size effect — where it appears

The difference between a system of a stated size and the infinitely large one it stands for. It is governed by the size divided by a length the system itself sets, not by the size alone, so two systems with the same number of particles can be far apart in how nearly infinite they are.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

The same collapse, with the cases made comparable. The alternation divided by its own cold value against the reduced temperature, for the published five cases and for five chosen so that every ring is the same size in its own alternations. The published set agrees to 3.41 per cent and the matched set to 1.62 — so the residual left was finite size, as suspected.

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.

solids · Metal Half the series stops on its own; the other half stops where it is told. How far from the chain's end the local decay rate can still be read, against how much of the chain the reading is allowed to cover. The dashed line is the cap itself. A curve that flattens below it has ended on the noise floor and the window never mattered; a curve that tracks the cap is being cut, and has more to say. The soft chains do the first and the stiff chains do the second.

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 — four, once a stiff profile is counted only to the bond where its local rate turns back.

solids · Peierls distortion Three targets, and the residual keeps falling. The worst spread across the five scaled curves, at three values of the matched product n·δ∞. It falls from 2.51 per cent at 5 to 1.62 at 9.6, monotonically. The unmatched cases sit at 3.41 per cent throughout, because they are the same five rings whatever target is being aimed at — which is what makes the comparison a comparison.

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.

solids · Metal On an opened profile the local rate stops falling, and turns. The local decay rate at every bond, read to half the chain, for three stiffnesses. The soft chain's rate falls all the way to the last bond that produces one, where its excess meets the noise floor. The two stiff chains' rates reach a minimum and climb, because the excess is measured against a bulk value read from the chain's middle and the middle still carries both ends' healing. Every bond past the minimum is reading that reference, not the tail.

The rate that turned back

Opening the Peierls fit's window to half the chain added forty to eighty bonds to every stiff profile, and those bonds were read as tail. Most of them are not. The local decay rate on an opened stiff profile falls to a minimum and climbs again, because the bulk it is measured against is read from the chain's middle, which still carries both ends' healing — and a model of two ends and that reference, built from the quarter window alone, puts the minimum within two bonds of where it is. Read only to the turn, the fall of a seventh in the stiff exponents disappears, and one of the two cases the wider window was credited with resolving is not resolved.

solids · Peierls distortion Three bulks, and only one is the chain's. For each of the ten stiffnesses, how far three candidate bulk values sit from the bulk at the chain's own shift, on a logarithmic axis. A free infinite chain and a free ring of 320 sites are off by 10⁻⁵ to 10⁻³ at every stiffness — up to thirty thousand times the relaxation's floor of 10⁻⁸ — because neither has the shift the chain's fixed length imposes. The chain's own middle agrees with the shifted bulk to 10⁻¹¹ at every stiffness to 2.0 and then rises above it, by the two ends' tails, to 10⁻⁴ at the stiffest.

A ring is not the bulk

The local decay rate on a stiff Peierls chain turned back because its bulk was read from the chain's own middle. The remedy proposed was a ring of the same length, with no ends to heal from. A ring gets the soft chains wrong by thirty thousand times the floor — because the chain's relaxation holds its length fixed, and the ends' extra alternation is paid for by a shift in every bond that goes as one over the length. With that shift in the bulk, the soft middles agree to 10⁻¹¹, no stiff profile turns back, two tails describe every profile to its middle, and the stiff exponents keep rising where the published ones had turned down.

solids · Peierls distortion At a fixed number of coherence lengths, the curve still depends on the stiffness. The scaled alternation of twenty-five rings at reduced temperature 0.03, against each stiffness's bulk amplitude δ∞: five stiffnesses from 1.2 to 2.0, each at five sizes around seven coherence lengths, rings of 4m + 2 sites filled and rings of 4m sites in the warning colour. Size barely moves a point; stiffness moves it along a straight line, 1.63% from the softest to the stiffest, and rings of 4m sit 0.15% lower at every stiffness. The line runs back to 0.29646 at δ∞ = 0, against BCS's weak-coupling 0.29709 (open ring).

The residual was the stiffness

A matched Peierls collapse leaves 1.6 to 2.1 per cent between five rings that hold the same number of coherence lengths, and the question was how much of it the even-site rounding explains. Twenty-five rings answer it: a ring's curve moves by a few hundredths of a per cent per coherence length, so the rounding explains a hundredth of the residual. The rest is the stiffness itself — a term linear in the bulk amplitude, which no choice of sizes can match away — and its intercept at zero amplitude is BCS's gap curve to a fifth of a per cent.

solids · Metal Freeing the chain's length leaves every tail exponent where it was. The tail exponent p of a two-ended fit A·b^(−p)·e^(−b/λ) from bond 6, at ten stiffnesses, for the chain whose total length is held fixed and for the chain at zero tension. The fixed chain's middle is compressed by an amount that goes as one over its length; the free chain's is not compressed at all. The exponents differ by at most 0.0009 — the constraint was never in them.

The chain that pays for nothing

Held to a fixed length, a Peierls chain pays for its ends' extra alternation with a compression of every bond. Let the length go and nothing is paid: the middle is the free infinite chain to twelve figures, the ends adjust locally — and every tail exponent comes out where the fixed chain put it, to under a thousandth. The constraint was never in the exponents. Nor is the drift in them a missing 1/b term: fitting one makes the exponent wander three times further.

solids · Peierls distortion The consecutive triples fail for a uniform chain at every length. The share of three decades of accuracy line over which some basis size is usable, against the number of fragments from three to eight, for three arrangements and three ways of assembling the correction. On the uniform chain the consecutive-triple assembly covers less than pairs alone at every length from four, and the shortfall grows. On the heavy-inside chain it covers exactly what every triple covers. On the heavy-outside chain it covers more than every triple from five fragments on.

Length did not rescue the consecutive triples

On four fragments, keeping only the consecutive triples of a counterpoise assembly worked for two arrangements and did worse than pairs for the uniform chain, and a short chain was the obvious excuse. Carried to eight fragments the excuse fails: the uniform chain's consecutive assembly settles at 48 per cent of the line against 68 for pairs. And the heavy-outside chain turns the lesson over — from five fragments every triple together covers less than the consecutive ones alone.

orbitals · Basis Three terms in the numerator, and one place they cancel. The three contributions to the give-back's numerator on a six-site ring whose interaction falls as one over the distance round the ring — a half at two bonds, a third at the antipode — with their sum. The sum crosses zero once, at V = 5.9198, where the one-bond and antipodal terms are negative and the two-bond term positive: the two-bond enhancement is paying for depletions on either side of it.

A root that moves when nothing else does

Give the ring of six an interaction that falls off as one over the distance and the give-back's numerator has three terms. It still has one root, and at that root every separation's pair change has the opposite sign from the two-bond case. But the root has a weakness the earlier essays never tested: lower the interaction by the same amount at every separation and not one state changes, while the root moves — by as much as the reach of the interaction did, and out of existence altogether if the zero is put at one bond.

bonding · Correlation

Named alongside it

The objects these essays reach for when they reach for this one.

Peierls distortionModel limitBond alternationRelaxationApproximationExtrapolationLeast-squaresScalingDelocalisationThermodynamic limitTight-binding modelsBand gap

All concepts