Molecular Geometry & Bonding
what the shape decides
Concept

Scaling — where it appears

The observation that a family of systems collapses onto one curve when each is plotted in units it sets for itself. It is what lets measurements on small systems say something about large ones, and it fails whenever the variable held fixed is not the one that governs.

Named by 3 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 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 orderings in use move the splitting by 0.47 per cent between them. The change in NH₃'s ground inversion splitting under each ordering of the kinetic operator, relative to BenDaniel–Duke, with the bond-conserving mass throughout. The bars are exact solves and the ticks are first-order perturbation theory. The five span 0.469 per cent, from −0.407 to 0.060; the change from a constant mass to the bond-conserving one, in the same well and box, is 43.1 per cent, 92 times as large.

An ordering worth half a per cent

A mass that varies along a coordinate has no unique quantum kinetic energy, and the choice among the Hermitian orderings in use was the one thing left that could undo a forty-three per cent correction to ammonia's splitting. It cannot. The five orderings anybody uses span 0.47 per cent between them, a ninety-second of the correction, and the family only reaches the measurement at exponents three times larger than any of them.

shape · Inversion

Named alongside it

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

Bond alternationExtrapolationFinite-size effectModel limitPeierls distortionTight-binding modelsApproximationBand gapConventionDouble wellInversion splittingPerturbation theory

All concepts