The criterion that has never said no

The five figures were an identity found something uncomfortable about a standard instrument, and it is tempting to say something about it that is not true.

The instrument is a two-orbital criterion for whether a mixed description of two orbitals beats the canonical one — the question two pictures, one plane poses and arithmetic has been answering ever since. Rotating a pair increases the Boys functional exactly when the separation of their centroids is smaller than twice the off-diagonal dipole between them, so the margin — twice the dipole over the separation — is one at a tie and larger when mixing wins. The reason is short. The functional rewards centroids that sit far apart, and rotating a pair by forty-five degrees replaces their centroid separation with twice the off-diagonal dipole between them — so the canonical pair wins when it is already further apart than a full mixing could make it, and the mixed pair wins otherwise.

The uncomfortable finding was that on the carbonyl’s two lone pairs that margin is 1/√s, where s is the oxygen’s s fraction, and 1/√s exceeds one for every s less than one. The criterion cannot say no to that case at any physical polarisation. A test that returns the same answer for every input it can be given is not measuring the case in front of it, and that essay says so.

The tempting next step is to say the criterion discriminates elsewhere: that on the double bond it refuses to produce bent bonds, and that the refusal is the content of the σ–π analysis.

Neither half of that is so #

The essay cited is two bent bonds, or a σ and a π, and it contains no refusal. Its two verdicts are that the two descriptions are one occupied space in two bases — the density matrix unchanged to 10⁻¹⁵ at every mixing angle, and changed by seventeen per cent as soon as the mixing is made non-orthogonal — and that the bent pair are sp⁵ hybrids at 50.768° to the axis with their charge 0.235 Å off the plane. It is an essay about two pictures being the same picture, quantitatively. Nothing in it rejects anything.

The criterion was not applied there at all. Attributing a verdict to a neighbouring argument rather than reading it is a small failure and an instructive one: the sentence supplies a contrast the argument wants, and a contrast an argument wants is exactly the kind that gets assumed.

It is worth being precise about what such a sentence does, because the shape recurs. A genuinely awkward result — an instrument that returns the same answer whatever it is given — is much easier to state if it can be localised. Saying “the criterion is a tautology here and discriminating there” turns a problem with the tool into a problem with one application of it. The sentence is load-bearing for the argument’s comfort rather than for its content, and that is precisely the kind of sentence nobody checks.

So the question left open — what the criterion does on the double bond — had never been answered. This essay answers it.

The double bond clears by more, not less #

The carbonyl’s σ and π give a margin of 2.455. The lone pairs give 1.732. So the case held up as discriminating clears the boundary by more than the case called a tautology.

And a wider margin is a weaker test, not a stronger one. If 1.732 is close enough to one to make the lone-pair verdict a foregone conclusion — and it is, because nothing in the physical range brings it lower — then 2.455 is more foregone, not less.

A polarisation sweep should settle whether the σ–π case has a boundary inside the range where molecules actually live. The sweep is done here.

Across s fractions from 0.05 to 0.99 the margin has a floor of 2.412 and a ceiling near 39. It is above the lone pairs’ own margin at every polarisation. There is no boundary in the physical range, and there is not one anywhere near it.

Ethene’s verdict was never at risk #

The third case is stronger still, and it needs no numbers.

Ethene’s σ and π centroids both sit at the midpoint of the carbon–carbon bond, because the molecule’s symmetry puts them there. The separation is not small; it is zero. What the quadrature returns is 2.5 × 10⁻¹⁹ Å, which is the grid’s own noise, and the margin is whatever a positive number divided by that comes to. It is the same kind of exactness as the lone pairs’ 1/√s — a number that is what it is for a reason, not because a calculation happened to land there.

So one of the three cases usually put to the criterion could not have come out any other way. A verdict forced by a point group is not evidence that an instrument works — it is evidence that the instrument was pointed somewhere it had no choice. The same distinction between a result and a symmetry runs through every argument that has had to separate the two.

But the instrument is not vacuous #

This is where the lone-pair diagnosis needs correcting in the other direction, and it is the reason the essay is not simply a retraction.

The carbonyl trio gives three pairs, not one. Besides σ–π there are σ–n and π–n, from the same three orbitals and the same dipole matrices, requiring no new computation at all.

The criterion refuses both. σ–n scores 0.111 and π–n scores 0.739 — the first by a factor of nine, the second clearly.

So the criterion discriminates. It is capable of saying no, it says no to two of the five pairs in front of it, and it says no by wide margins. What is one-sided is not the instrument but the sample.

And the reason for that is worth stating plainly, because it is not an accident. Every pair usually put to the criterion is a pair somebody had already proposed as rival descriptions of the same thing — two lone pairs, or a double bond drawn two ways. Two orbitals get proposed as rival descriptions precisely when they occupy the same region and can be rotated into each other, which is precisely the condition for a large off-diagonal dipole and a small centroid separation. The pairs nobody proposes mixing — a bond and a lone pair on different atoms — are the ones with distant centroids.

The selection criterion and the measurement criterion were measuring the same thing. That is why the answers agreed on every row.

What the gap between 0.74 and 1.73 means #

There is a fact about the five margins that is easy to pass over and is the sharpest thing here.

They are 0.111, 0.739, 1.732, 2.455 and infinite. Nothing lies between 0.739 and 1.732 — a gap containing the boundary, with a factor of two and a third of empty space around it.

A criterion is only informative near its boundary. Far above it every reasonable variation of the inputs leaves the verdict alone; far below it the same. The interesting cases are the ones where a small change in the molecule, the basis or the polarisation could flip the answer, because those are the cases where the criterion is doing work rather than confirming what the geometry already made obvious.

The cases available here include no such pair. Every pair it can construct is comfortably on one side or comfortably on the other, and the polarisation sweep — the one continuous parameter available — moves the σ–π margin between 2.4 and 39 without ever bringing it down. So the criterion has never been tested at the only place a threshold can be tested.

That is a stronger statement than “the sample was one-sided”, and it survives the correction that the sample was. Widening the sample to include σ–n and π–n gives the instrument a demonstrated negative and still leaves the boundary untouched: the refusals are as far below one as the acceptances are above it. A test with an untested boundary is not refuted by anything on this page, and it is not supported by anything either.

What was computed, and how #

Five pairs and one sweep, using calculations that already existed. The carbonyl’s σ, π and lone pair are Löwdin-orthogonalised Slater functions, the dipole matrices are the same numerical integrals used for the lone pairs, and the criterion is applied exactly as published — a separation along the bond axis, a dipole magnitude over the three components, and their ratio.

The three pairs from the trio cost one call each and no new integrals. The lone-pair margin is unchanged. Ethene’s comes from the same construction one dimension across.

Six checks, and the third and fourth are the ones that matter most. One says every asked case mixes, which is the finding. Two are the control: that some pair is refused, and that the nearest refusal is clearly on the other side rather than a hair below the line — because a criterion whose refusals all sat at 0.99 would be a different and much weaker result. One says the σ–π margin exceeds the lone pairs’, which is the correction. One says ethene’s separation is zero to machine precision rather than merely small, so that the word “forced” is earned. And the last says the sweep’s floor is above two.

The control assertion deserves a note. It was not in the first draft, which asserted only that every asked case mixes — and that draft would have supported the conclusion “the criterion is vacuous”, which is false. The two unasked pairs were computed to test the opposite possibility and they changed the essay’s finding from a retraction into something more specific.

Where the model stops #

One molecule and one neighbour. The trio is a carbonyl at one geometry and ethene is a single extra point, so “every pair available” means five, and five is what is available rather than a survey.

The criterion itself is a two-orbital statement inside a problem that is not two-dimensional. Three orbitals have a Boys maximum a sequence of two-by-two rotations cannot reach — the finding the trio’s rotation group produced alongside the closed form, so a pair-by-pair verdict is not the same as a verdict about the best description — and none of the margins here should be read as saying what the localised orbitals of a carbonyl actually are.

And the σ–n and π–n refusals are not being offered as chemistry. Nobody proposes mixing a bond with a lone pair on another atom — the two are not rival descriptions of one region — so the criterion refusing them is the expected answer to a question nobody asks. Their value here is entirely as a control: they establish that the instrument has an output on the other side, which is the thing the lone-pair finding could not establish from the cases it had.

The generalisation #

The transferable point is about how a test comes to look vacuous.

The lone-pair diagnosis was that the criterion is a tautology — that it returns the same answer whatever it is given. That is a claim about the instrument, and the way to test it is to give the instrument something else. Doing so shows the claim is false: it returns a different answer for two of five inputs.

What was actually true is a claim about the sample. Every case put to the criterion was chosen by a rule that correlates almost perfectly with the criterion’s own verdict, so the criterion could only ever agree. That is a much more ordinary failure than a vacuous test and a much harder one to see, because nothing about the instrument looks wrong and every individual application is correct.

The diagnostic is cheap and general: before concluding that a test always says yes, find a case it should say no to and check that it does. If it does, the problem is the sample; if it does not, the problem is the test. Those call for entirely different repairs — a wider set of cases in the first, a better instrument in the second — and taking the lone-pair result at face value would have made the second repair to a problem of the first kind.

The second point is about citing a neighbouring result. The mistaken contrast is caught by opening the analysis it names, and it gets written instead from a memory of what that analysis was for. In any chain of arguments where each builds on the last, that is the failure mode with the longest reach: an attributed verdict is inherited by everything downstream and carries no marker saying it was never read.

What the correction costs the lone-pair result #

It is worth totalling what changes downstream, because the answer is less than the length of this essay suggests.

The lone-pair headline is untouched. Its finding is that the lone-pair margin equals 1/√s exactly, agreeing with the numerics to five figures, and that the criterion therefore cannot say no to that case at any physical polarisation. Every word of that survives; nothing here bears on it.

What changes is one piece of interpretation: the contrast with the double bond, which does not exist, and the calculation proposed on the premise that it did.

And the correction improves the lone-pair conclusion rather than weakening it. Its point was that a criterion which cannot say no is not measuring the case in front of it. That point is now supported by three cases rather than one, and by the observation that the usual way of choosing cases guarantees the answer. It was a better finding than it looked, and the false contrast was the only thing standing between it and the stronger version.

That is the ordinary shape of such corrections and it is worth saying so. Finding a mistake in an earlier argument is usually not overturning it; it is usually removing something added to make the result look more modest than it was.

Who found it, and when #

Boys localisation is Foster and Boys’s, the two-orbital rotation condition is standard, and the rabbit-ear-versus-σ-π question is as old as hybridisation. The criterion in the form used here, the closed form for the lone pairs, and the audit are new.

The real finding belongs to the lone-pair analysis. Noticing that an instrument might not be measuring anything, in the middle of a result whose headline is an exact agreement to five figures, is the harder thing to do, and it is what made the error beside it worth finding.

Still open: a fourth orbital, and a case near the boundary #

The obvious open question is the fourth orbital. Adding the second lone pair makes the group six-dimensional and lets the rabbit-ear mixing and the bent-bond mixing compete, and the two-orbital criterion cannot see a competition because on each pair separately it has already decided. This essay sharpens the reason to expect something: the criterion says yes to every pair anybody would propose, so on a four-orbital problem it says yes to all of them at once, and the question of which mixing actually wins is one it cannot even express.

The nearer question is whether the criterion has a case that is genuinely close to the boundary. Every margin here is either well above one or well below it — 0.111, 0.739, 1.732, 2.455, infinite — and nothing sits between 0.74 and 1.73. A criterion with a boundary but no cases near it is untested at the only place its value is decided, and finding a pair whose margin can be tuned through one would be the first real test this instrument has had. The carbonyl’s polarisation does not do it; a heteroatom series might, and it can be built — the carbonyl trio is already parameterised by the heteroatom’s charge as well as by its polarisation.