Molecular Geometry & Bonding
what the shape decides
Concept

Measurement uncertainty — where it appears

How far a measured value may sit from the true one, from the apparatus, the method and the analysis together. A conclusion drawn from measurements is only as sharp as theirs, and a quoted number with no uncertainty attached does not thereby have none.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

What it would cost to destroy each refutation. For every discordant pair in the collection, the measurement error that would be needed to reverse it, as a fraction of that predictor's own spread of measurements. Nothing is quoted: the question is not what the uncertainties are but how large they would have to be. The cheapest to destroy needs 3.4 per cent of the range and the dearest needs 51.6.

The error bar that would be needed

Two things a pair of measurements can say without any model both treat their numbers as exact. The quoted measurements carry no uncertainties, and inventing some would be worse than having none — so the question is asked the other way round. Not what the errors are, but how large they would have to be. The three refutations that seemed most worrying turn out to be the sturdiest of the lot.

bonding · Models The bracket has no upper end. Every price is multiplied by √2 over √(1 + r² − 2ρr), where ρ is the correlation between the two measurements' errors and r is the ratio of their sizes. For equal precision the factor is 1/√(1 − ρ), which is one at independence and unbounded at perfect correlation. The other extreme the question asked for is not a number.

The other end of the bracket is not a number

Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.

bonding · Models Adding one measurement, and six of fifteen change verdict. Each atom's chemical capacity from a cubic fitted through four electron counts and from a quartic fitted through five, on a logarithmic axis, with an unbounded capacity drawn at the right-hand margin. Five atoms go from a finite capacity to an unbounded one and one goes the other way — and the five are exactly the five largest the cubic reported. The seven that stay finite keep their order and change their values.

Six of fifteen change verdict

The chemical capacity is finite or infinite according to the sign of a coefficient that is one sixth of a second difference of three measurements, and the pair test asked how much of its reported ordering is the quantity and how much is the fit's resolution. Adding a fifth point changes the verdict on six of fifteen atoms — and the five it unbounds are the five it had largest.

wrong · Electronegativity The ranking is fragile and the headline is not. The correlation required to disturb two different things. Reordering the easiest neighbouring pair needs ρ = 0.154, which is weak enough to expect. Displacing the collection's most fragile claim outright needs ρ = 0.954, which is near-perfect correlation. So a per-predictor structure scrambles the middle of the ordering and leaves the top of it alone.

The ranking moved and the headline did not

One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.

bonding · Models The ceiling moves by 8.7-fold across the gauche energy's own reported range. The rotamer ceiling against butane's gauche energy, for the rotor counts each convention assigns to a five-membered and a six-membered closure, with the three measured accelerations drawn as horizontal lines. Across the reported range the ceiling moves by up to 8.70-fold, and two of the nine verdicts cross a measurement during the sweep. The natural prediction is that this input would be the smaller lever of the two; it is the larger.

The lever that was supposed to be smaller

Sweeping the rotor conventions leaves every verdict unmoved, and suggests that the other quoted input will be a smaller lever. It is a larger one. Sweeping butane's gauche energy over its reported range moves the ceiling by 8.7-fold, flips the six-membered verdict at 3.630 kilojoules a mole — inside the quoted error bar — and takes the five-membered refutation with it at 4.422.

shape · Strain One ranking, three chains. The collection's priced claims in order of fragility, with each claim placed in its own predictor's column. Two claims sharing a predictor share its error correlation, so their prices carry the same factor and their relative order cannot be changed by any correlation structure whatever — a column is rigid. Claims in different columns can be reordered at a price. So the ranking is not one ordering but three chains interleaved, and only the within-column statements need no assumption about anybody's errors.

Three chains and ninety orderings ruled out

Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

bonding · Models Two prices, and which is cheaper is a property of the claim. For each slope floor, what it costs to halve it two ways. The measurement price is a standard error on the measured rise, as a fraction of the measured range — the currency every price in this argument has been quoted in. The model price is the change in the computed run, as a fraction of the predictor's own range. They are different currencies and their ordering differs between claims: the angle strain's floor is three times cheaper to move through its model, and the spin-only moment's cannot be moved through its model at any price at all.

A denominator needs three currencies

Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.

bonding · Models Ammonia's E verdicts across the fields its spectrum allows. The concentration of each distortion on one occurrence of ammonia's E species — the share of its E projection on the favoured vibration — for every field in the walk, with the stretch–bend constant held and the rest refitted to both isotopologues. The band is where the fit cannot tell the fields apart and the dotted line is the published fit. Across the band the angle under the B row reads 0.502 to 0.579, the angle at least cost 0.840 to 0.923, the bond at least cost 0.890 to 0.957.

The worst case was the loosest constant

A symmetry label's verdict on a distortion is read off mode shapes, and mode shapes are set by a fitted force field. Walked along the direction its own frequencies see least, ammonia's field moves the verdict on its bond angle from 0.840 to 0.923 without the fit getting measurably worse — and boron trifluoride, the census's worst case at 0.573, has a whole curve of fields that reproduce its spectrum exactly, along which the same verdict runs from a coin to 0.862.

symmetry · Point group Ammonia's two Coriolis constants along the family of fields the spectrum cannot tell apart. ζ₃ and ζ₄ of NH₃ on every member of ammonia's family of force fields, each refitted to the frequencies of NH₃ and ND₃ with its stretch–bend constant held, against that constant. The shaded band is where the fit is no worse than its own noise, -1.000 to -0.634. Across it ζ₄ runs from -0.807 to -0.641 and ζ₃ from 0.418 to 0.584, while their sum stays at -0.222 on every member.

Two Coriolis constants are one number

Ammonia's frequencies leave its force field free along a band, and across that band a symmetry label's verdict on the bond angle changes. The measurement proposed to settle it was the Coriolis constant of each E vibration, which reads mode shapes directly — and it would settle it, sixteen times more sharply than the frequencies do. But the two constants are not two measurements. Their sum is fixed by the geometry and the masses, ζ₃ + ζ₄ = I∥/2I⊥ − 1 (the moments of inertia about the axis and across it), on every field there is, so the spectrum's Coriolis constants carry exactly one number about the field between them.

symmetry · Point group

Named alongside it

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

Error propagationModel limitPredictorUnderdeterminationClosed formModel selectionMonotonicityRing strainApproximationConventionCorrelationForce constant

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