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 4 essays across 2 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.

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 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

Named alongside it

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

Model limitPeierls distortionApproximationBond alternationExtrapolationScalingTight-binding modelsBand gapBasis set superposition errorDelocalisationFragment methodGaussian basis

All concepts