Molecular Geometry & Bonding
what the shape decides
Concept

Extrapolation — where it appears

Reading a quantity off beyond the range it was measured in, by fitting a form to the measurements and evaluating it outside them. Its worth depends entirely on whether the fitted form is known to hold there, which the fit itself cannot say.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The same three fillings, on a ring and on a chain. The intensity contrast against the on-site repulsion at two, four and six electrons, for both geometries. Two of the ring's three curves flatten onto exactly one and stay there — the hollow marks, where the rank cut falls between two degenerate lines. The chain's corresponding curve approaches the same value from above without reaching it, because a chain of six has no exactly degenerate removal lines at any repulsion at all.

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

spectra · Photoelectron The exponent against how far the tail can be seen. The fitted power against the reduced reach — how many coherence lengths of the decay survive above the floor before the excess is numerical noise. The six cases with a reach past six give a power between 0.41 and 0.57 and are drawn solid; the rest are hollow and their fitted powers are off this scale in the negative direction. The reach is not a choice — it falls as the gap closes, because the excess the tail starts from falls with it.

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.

solids · Peierls distortion 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 The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

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.

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 far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

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.

solids · Peierls distortion

Named alongside it

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

Bond alternationPeierls distortionTight-binding modelsApproximationCoherence lengthModel limitNumerical precisionBand gapFinite-size effectScalingDegeneracyExact diagonalisation

All concepts