Two systems a model cannot tell apart

A fit of π ionisation energies sets six measured π ionisation energies against six computed Hückel eigenvalues, scans a third parameter over its whole range, and finds that the third parameter buys twelve per cent and has no interior optimum. The reason it can say that without exhausting every functional form is a tie: ethene and benzene have the same highest occupied eigenvalue, exactly, and their measured ionisations differ by 1.27 eV. Any model of the form $\text{IE} = f(x)$ gives them the same answer, whatever $f$ is and however many parameters it carries.

That closes the question about the third parameter. It also produces an instrument: two systems with the same predictor value and different measurements is a specific and reusable test that turns a model into a claim with no fitting anywhere.

This essay turns it on three other predictors.

What a tie proves, and what it costs #

A model that assigns one number to a system predicts that two systems with the same number have the same property. So if two really do coincide:

• both get the same answer from every model of that form;
• the gap between their measurements is a lower bound on the error of all of them;
• and none of this depends on the parameters, the functional form, or how the model was fitted.

The last point is what makes it cheap, and it is the same economy a refusal that costs nothing has: a claim that can be settled by arithmetic on what is already known needs no new experiment. A test that needs data the model has not seen needs new data; a tie needs only that two systems in the data already in hand coincide, and coincidences of this kind are common because most predictors in chemistry are functions of small integers.

The quantity worth reporting is the largest within-tie spread as a fraction of the whole set’s range: the share of the variation the predictor demonstrably cannot see. It is a floor rather than an estimate — the real share is at least that and possibly more.

VSEPR, which reads two integers #

VSEPR assigns an arrangement from a count of bonding pairs and a count of lone pairs. Two molecules with the same two counts get the same arrangement, and the lone-pair clause says the angle is less than the arrangement’s without saying how much less — so the model’s output for a pair of such molecules is one output.

Four molecules with measured angles:

The four angles span 15.7°. The worst tie spans 14.5° of it.

VSEPR accounts for at most 7.6 per cent of what separates these four molecules, and the number needs no fit, no correlation coefficient and no argument about whether the model is being applied properly. The two pairs are the same model applied to the same arrangement, and the measurements are what they are.

This is not news about VSEPR in the sense that everybody knows heavier central atoms have smaller angles. It is news about how large the effect is relative to everything VSEPR can say: the model’s whole resolving power on this set is a degree and a bit, and the thing it cannot see is worth twelve times that.

Pushing VSEPR at a transition metal finds it failing for a reason with a name — a d shell that is not spherical. The failure here has no name and needs none: it is the ordinary failure of a model that reads two integers when the answer depends on something else.

The spin-only moment, which reads one integer #

$\mu = \sqrt{n(n+2)}$ is a function of the number of unpaired electrons and of nothing else. Nine measured moments, five of them with a partner sharing their count:

Four ties from nine ions, and the worst is 0.940 μB out of a range of 4.17 — 22.5 per cent. That is a larger share than a fit to a whole susceptibility curve leaves undetermined in the parameter it is best at, from a model nobody would call underdetermined.

The interesting part is which member of each pair is the outlier. Chromium(III) sits on the formula to two decimal places and cobalt(II) is nearly a whole Bohr magneton above it; the same asymmetry appears in every pair, and always the same way round. That is the orbital contribution this collection identified by a different route — and the tie shows it without needing to know what it is, because the tie only asks whether two systems the model equates were measured to differ.

Accuracy and resolution are different quantities. The spin-only formula is usually described as good to about a tenth of a Bohr magneton, which is true of five of these nine. Its resolution — the finest distinction it can draw — is nothing at all between two ions with the same count, and the measurements differ by nine times the quoted accuracy.

The ionisation tie, reproduced #

The general instrument, applied to the ionisation predictor, returns exactly the original number: ethene and benzene tied at an eigenvalue of 1 and measured 1.27 eV apart, butadiene and naphthalene tied at 0.618 and measured 0.940 apart. 41.0 per cent of the 3.10 eV the six molecules span.

That agreement is the check that this is the same measurement rather than a new one. The two computations are written independently — one specialised to the ionisation set and one generic over rows of predicted-and-measured pairs — and they agree to the last bit.

The control, which had to say nothing #

An instrument that condemned every model would be measuring itself.

So the fourth predictor is a control chosen because it cannot be caught: the angle strain of a planar equilateral ring, which is a different number for every ring size by construction. Six rings, six distinct predicted values, no ties, and the instrument returns nothing about it.

That is the right answer and it is a real constraint. A predictor whose values are all distinct is not thereby a good predictor — the ring-strain account of what a ring’s strain actually is is wrong in ways computed elsewhere — but it is not refutable this way, and an instrument that pretended otherwise would be worthless.

The pattern across the four #

The ordering is worth reading twice, because the obvious guess about it is wrong. The ranking is by how coarse the predictor’s input is, not by how well the model is thought of. Hückel theory has a far worse reputation than VSEPR and comes out with less than half its unexplained share; the spin-only formula, which is taught as a rule of thumb, comes out best of the three. What is being ranked is not the quality of the physics but the width of the channel through which the physics is being asked to speak: VSEPR reads two small integers and is worst; the spin-only moment reads one and is better, because the set it is applied to varies less in the direction it cannot see; the Hückel eigenvalue reads a whole graph and is in between; the angle strain reads a continuous geometry and has no ties at all.

That is not a law — a continuous predictor can be worse than a discrete one, and a model whose residual looks excellent can be the wrong model entirely — but it is the mechanism, and it points at the repair. A predictor with ties is improved by giving it an input that separates the tied pairs, and the input is named by the tie itself: NH₃ against PH₃ is a change of row, Cr³⁺ against Co²⁺ is a change of orbital angular momentum, ethene against benzene is a change of ring count. Each tie is a diagnosis as well as a refutation.

What the instrument is not #

Two things it is worth separating this from, because both are more familiar and neither is the same.

It is not a correlation coefficient. A correlation asks how much of the variation a predictor tracks and answers with a number that depends on the sample and on the form assumed. A tie asks whether there exist two systems the predictor equates and the measurement does not, and answers with a gap. The first can be made to look good by choosing a sample that varies mostly along the axis the model reads; the second cannot be improved by any choice of sample that keeps the tied pair in it.

And it is not cross-validation. Holding data back tests whether a fit generalises, which is a question about the fitting. A tie tests the model’s form, and it does so on data the model was fitted to — which is normally the one place a test is worthless, and here is the one place it costs nothing. The two ask different questions and a model can pass either and fail the other.

What is quoted, and what is computed #

Every measurement here is quoted, and all are standard values: nine magnetic moments, four bond angles, six π ionisation energies and six ring strain energies. Nothing new is claimed about any of them.

Every predictor is computed by the library that owns it — eigenvalues by diagonalising an adjacency matrix, arrangements from the two counts, moments from the unpaired count, strains from the ring’s interior angle and a quoted bending constant.

The ties are found at a tolerance of $10^{-9}$ for the three exact predictors. They are not near-ties made to look exact by a loose threshold: two ions with the same count are assigned the same moment by a formula in an integer, and two molecules with the same counts are assigned the same arrangement by a model that reads nothing else.

What this cannot say #

A within-tie spread is a floor, not the error. It says the model cannot do better than that; it says nothing about how much worse it is elsewhere, and the true unexplained share is larger.

Four predictors on small sets, and the sets were chosen by the essays that built them rather than for this audit. Nine ions, six molecules, four molecules, six rings — these are the sets this collection quotes, and a larger set would find more ties and probably wider ones. The numbers here are what these data support and not a survey.

Nothing here is about whether the models are useful. VSEPR predicts that ammonia is pyramidal and that is worth more than any of this; the audit measures resolution, which is a different property from being right about the thing the model was invented for. A model can have a large unexplained share on one set and be the best available account of something else entirely.

And the tie is only as exact as the claim that the model reads nothing else. VSEPR with a lone-pair weight fitted per molecule has no ties, because it has a continuous input — but that weight does not transfer between molecules, which is the same difficulty arriving in a different form: a model with enough freedom to break its ties is a model with a parameter per system.

What the instrument requires #

Three of the four predictors have exact ties, found at a tolerance of $10^{-9}$, and the separations within them are required to be smaller than $10^{-6}$ — so a near-tie admitted by a loose threshold would fail the check rather than pass it.

The VSEPR angle cannot account for most of the variation in the angles it predicts, stated as an inequality above 80 per cent so that a change to the underlying data cannot quietly turn the finding into a smaller one.

The spin-only moment cannot account for a fifth of the variation in the moments it predicts.

The control has no ties and its unexplained share is exactly zero — required to be zero rather than small, because a procedure reporting a large share for every predictor would be measuring itself.

And the general instrument reproduces the original tie spread exactly, to the last bit, which is what makes this the same measurement rather than a second one that happens to agree.

Whether the audit is asking the predictor’s own question #

An instrument that reports ninety-two per cent of the variation unaccounted for invites an obvious objection, and it deserves an answer rather than a defence.

The objection is that a predictor should be audited on the claim it makes. VSEPR’s primary output is a shape — linear, bent, trigonal pyramidal, see-saw, square planar — and as a classifier of shapes it is extremely good: given a count of bonding and lone pairs it names the arrangement, and it names it correctly across essentially the whole of main-group chemistry. Auditing a classifier by how well it predicts a continuous coordinate will make almost any classifier look useless.

So the audit is fair only if the continuous claim is one the predictor actually makes. It is. The rule as taught does not stop at naming the arrangement: it says that lone pairs occupy more room than bonding pairs and therefore compress the remaining angles, which is a quantitative statement about a number of degrees, and it is used as one — a reader is expected to conclude that ammonia’s angle is smaller than methane’s and water’s smaller again.

That clause is what the audit measures, and the audit says it accounts for eight per cent of the spread in the angles it is applied to.

Two consequences follow and they should be kept separate.

The classification survives. Nothing here touches the shape prediction, which is a different claim with a different success rate, and a reader who uses VSEPR to decide that sulfur tetrafluoride is a see-saw is not affected.

The quantitative clause does not. Predicting that an angle is smaller is a claim about a sign and the predictor gets the sign right; predicting how much smaller is a claim about a magnitude, and on that the predictor is very close to a constant.

The right comparison to have in mind is the null one. A predictor that returned the same angle for everything would account for zero per cent of the variation. Eight per cent is not zero, and it is much nearer zero than it is to a working quantitative model — which is the honest summary of a rule that is excellent at one of the two things it is used for.

Still open: the repair each tie names #

The obvious open question is the repair each tie names. Every tied pair differs in a stated way — a row of the periodic table, an orbital contribution, a ring count — so adding that one thing to the predictor and asking how much of the tie it closes is a computation of the same size as the audit. The interesting output is not whether the fit improves, which it must, but whether one added input closes all the ties of a given predictor or whether each needs its own, which is the difference between a missing term and a missing model. A single input that closes every tie of a predictor is a correction to the model; a different input for each tie says the model was never the right shape.

The nearer question is about ties that are not exact. Every tie here is exact because every predictor reads integers or a graph; a continuous predictor has near-ties instead, and the same argument applies with the spread compared against the separation rather than against zero. Two systems a hundredth apart in the predictor and a whole unit apart in the measurement bound any model whose slope is finite — and the bound is a statement about the derivative rather than about the function. Making that precise would extend the instrument from the discrete predictors, where it is sharp and easy, to the continuous ones, where most of chemistry’s models live.