Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 13

Page 13 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

Bonding models

Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.

What a rotation group buys over a single angle. The Boys functional at the canonical orbitals, at the best mixing of the two bonds with the lone pair left alone, and at the maximum over the whole three-dimensional rotation group. The first step is 11.70 per cent and the second, which no two-orbital treatment can reach, is a further 2.50.

An interior maximum a third orbital allows

Two orbitals related by a mirror plane give a Boys functional with no linear term, so it has no interior maximum and no angle is ever searched for. Add the oxygen lone pair and the search over a three-dimensional rotation group finds one — eight and a half degrees of lone pair mixed into two bent bonds, worth a further two and a half per cent that no pair of orbitals could reach.

6 figures A heteroatom on ring 5, and the profile in both directions. The response of each ring to a heteroatom placed on ring 5 of 12, on a logarithmic scale. Two rings share the peak, because an interior ring's outermost carbon belongs to two rings at once, and the profile falls away at the same rate on both sides — which is the answer wanted: the reach is the molecule's and not the end's. Every point is a mean over the six bonds of its ring.

The reach is the molecule's

Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.

6 figures 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.

6 figures The criterion is the reciprocal of the p amplitude. The two lone pairs' mixing criterion — twice the dipole between them over the separation of their centroids — against the p fraction of the outward hybrid. The curve is 1/√(p fraction), written down rather than fitted; the marks are the numerical integrals. They agree to 15 parts in a million at every hybrid, and the criterion never falls to one, so the mixed description wins at every s character there is.

The five figures were an identity

Two lone pairs satisfy the two-orbital mixing criterion by a margin of 1.732128, against √3 = 1.732051 — agreement to five figures on a number assembled from three integrals over a numerical grid. It is exact. The criterion is the reciprocal of the p amplitude of the outward hybrid, and everything else in the molecule cancels out of it.

6 figures 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.

6 figures One kink, and the heteroatom's own ring notices only at the kink. The response on the heteroatom's own ring, on a straight chain of twelve and on one with a kink — two adjacent angular fusions — in the middle, as the heteroatom is moved along. The two agree to within one and a half per cent except on the kink's two rings: on the first the bent chain reads 18.9 per cent lower and on the second 3.1 per cent lower. An end halves the amplitude.

A bend is not an end

Moving a heteroatom from the end of a straight chain of twelve rings to its middle leaves the decay unchanged and halves the amplitude, which raises the question of whether a bend in the chain is a boundary of the same kind. It is not a boundary of the same kind: the kink — two adjacent angular fusions — leaves the fitted decay length within two per cent and costs the heteroatom's own ring a fifth only on the kink's first angular ring. It does multiply down the response on the rings beyond it, which a decay length does not see.

6 figures 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.

6 figures Five pairs, one boundary, and every case anybody asked about on one side. The mixing criterion's margin for every pair of orbitals available from the carbonyl and from ethene. Above one the mixed description wins. Three pairs were put to the criterion and all three clear the line; two more come from the same three orbitals and the same dipole matrices: π–n fails it, and σ–n lies outside it, because its dipole runs along the separation and its best rotation is a few degrees. The instrument discriminates — the selection did not.

The criterion that has never said no

The two-orbital mixing criterion is a tautology on a carbonyl's lone pairs, and it is tempting to contrast that with the double bond, where the criterion seems to discriminate. Both halves are wrong. The double bond's margin is wider, ethene's is infinite because symmetry puts both centroids at the same point, and the criterion has never returned a negative on any case it is usually put to — though the same three orbitals supply two pairs it refuses outright.

6 figures The A–C overlap changes by a fifth and the exact level does not move. The three levels of the trio as the overlap between A and C is raised from 0.25 by up to 0.2, with B's overlap to C held and both site energies at -13.6 eV. The two outer orbitals stop being equivalent at the first step. The lowest level falls by 0.80 eV and the highest rises by 5.24, and the middle one stays at -13.6 eV — its largest departure over the whole sweep is 2.7×10⁻¹⁴ eV, which is rounding.

A level no symmetry was protecting

A three-orbital trio keeps one level at the free-atom energy exactly, and the reason given was that its two outer orbitals are equivalent. Make them inequivalent by changing one overlap rather than one energy and the level does not move at all — not to first order, not to any order, at any energy of the third orbital. It was never the symmetry. It is allyl's non-bonding orbital, held by a count.

7 figures 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.

6 figures 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.

6 figures The verdicts hold and the number does not. For each predictor, what survives thirteen strictly increasing reparameterisations of its own scale. The count of exact ties, the share of the variation those ties leave unexplained, and the count of discordant pairs are identical under every one of them — to the last bit, because each asks only about the order of the predicted values and an increasing map preserves order. The slope floor asks for a ratio of differences, and an increasing map does not preserve differences.

A floor on models written in one scale

A tie asks whether two predicted values are equal and a discordance asks whether two differences have the same sign. Both are questions about order, and a strictly increasing change of scale preserves order — so both are exactly invariant under thirteen reparameterisations of all three predictors. The slope floor asks for a ratio of differences, and it moves by factors of nineteen, ten and eleven thousand.

6 figures