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 7 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 Runs of four saturate in a grain; runs of eight never saturate at all. The chain length at which half the runs of a given length sit in a resonant pair, against the run length, at three concentrations of low sites. The scale is logarithmic in the logarithm's own sense: each extra site in the run roughly doubles the number of decades required. At half filling runs of four reach half at 10¹⁵ sites — a grain thirty micrometres across — and runs of eight at 10¹²³, which is forty orders of magnitude past every atom in the observable universe.

The commonest runs reach it in a grain

Half the runs of six low sites in a disordered chain sit in a resonant pair only at 10⁴¹ sites, which no piece of matter reaches. Runs of four are four times as common and bind less deeply, and both of those shorten the answer, so they reach the same state at 10¹⁵ sites — a grain thirty micrometres across. What the essay before it found was not a fact about the resonant share but a fact about half filling and runs of six: reachability is a diagonal line across run length and concentration, and at seven tenths of the sites low it passes above runs of seven.

solids · Defect 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 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 Two pyramids, and only one of them has ever shown a line. What ammonia and phosphine each bring to the same one-dimensional model. The barrier is six times larger and the apex sits twice as far out, so the action under the barrier is seven times larger and the splitting is twenty-four orders of magnitude smaller. What is not six times anything is the transverse zero-point difference between a molecule and its deuterated twin: the modes that carry it are the ligands' own, the ligands are the same three hydrogens, and the two numbers are within thirty per cent of each other.

The same fraction at six times the barrier

Ammonia's two fitted barriers leave a gap that an isotope-dependent term would have to fill, and the term was sized at five per cent of the transverse zero-point difference. Phosphine can be asked the same question and cannot answer it the same way: its splitting is twenty-four orders of magnitude below anything the solver can represent, and the solver does not fail cleanly — it reports values that rise as the barrier rises. Calibrated where a splitting is still visible, the action carries the question across, and the same fraction moves phosphine's isotope ratio by sixteen per cent against ammonia's twenty-one.

shape · Inversion Two patterns, one shape, and a slope between them. How far the logarithm of each splitting falls as one hydrogen at a time is replaced by deuterium, for the ground doublet and for the first excited one. Neither falls in equal steps — the first step is half again the last — and the two patterns are the same shape: the excited step is a fixed fraction of the ground step at every stage, within a few per cent. The well is fitted to both of NH₃'s measured lines and nothing is fitted to any deuterated molecule.

The steps were a square root of a mass

Replacing ammonia's hydrogens one at a time does not lower its splitting in equal steps, and the obvious readings are the potential's shape or the tunnelling arithmetic. Both lines — the ground doublet and the first excited one, which nothing in this argument has ever been fitted to — come out straight in the square root of the reduced mass and bent against the count, by a factor of eight, so the unequal steps are the shape of a square root drawn against the wrong variable. And the excited line is the one test the missing correction cannot pass by construction: it improves it threefold and over-shoots it.

shape · Inversion

Named alongside it

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

Model limitBond alternationExtrapolationFinite-size effectInversion splittingPeierls distortionReduced massTight-binding modelsTunnellingDouble wellIsotopologueWavenumber

All concepts