The error was the row, not the charge

The residue is below its own noise asked whether a model built from Slater radial functions and a four-electron overlap cost could decide the sign of a non-additivity residue — a quantity built by differencing four separations, which is three or six hundredths of an ångström. It could not, and the reason was blunt: the model reproduces the separations themselves to a mean of about a quarter of an ångström, so its error on one distance is several times the whole of what four of them are being differenced to find.

Along the way it recorded something it did not chase, as is sensible when a residue turns out to be two numbers and only one of them is the business at hand. The error is not uniform. Two of the six checkable pairs are right to under one per cent and three are wrong by thirteen to eighteen, and it named the good and the bad and said the pattern “is not obviously size or charge”.

It is not size and it is not charge. It is which row of the periodic table the ions come from.

Two labels, and why six points are enough #

Each pair carries two labels that are independent of each other.

The shells: whether each ion’s outermost p shell is the second or the third. Na⁺, Mg²⁺, F⁻ and O²⁻ are all 2p⁶; K⁺, Ca²⁺ and Cl⁻ are all 3p⁶. So a pair contains zero, one or two third-row ions.

The charge: whether the pair is singly or doubly charged.

Six points is a small number and it would be no number at all if the two labels moved together. They do not, and that is checked before anything is tested: both charges appear at more than one shell count, so each label can be varied with the other held. Mg²⁺O²⁻ is a doubly-charged pair with no third-row ion and Ca²⁺O²⁻ is a doubly-charged pair with one, so the shell varies at fixed charge; Na⁺F⁻ and Mg²⁺O²⁻ have the same shells and different charges, so the charge varies at fixed shell.

A set of pairs in which charge and shell were confounded would give exactly as clean a picture and it would mean nothing. Establishing that they cross is not a formality, it is what makes the rest of this essay a measurement.

The shell sorts it completely #

With no third-row ions the model is short by 13.6 and 14.9 per cent. With one it is long by 0.6, 3.2 and 6.7. With two it is long by 17.9.

The three groups do not overlap, they run in order, and the narrowest gap between adjacent groups is 11.2 percentage points against a total range of 32.7. That is separation with a great deal of room in it, not a marginal sort.

And the two doubly-charged pairs land inside their shell groups rather than beside each other. Mg²⁺O²⁻, at −14.9, sits with Na⁺F⁻ at −13.6 and nowhere near Ca²⁺O²⁻ at +6.7, which is the other doubly-charged pair. The charge does not merely fail to predict; it is actively cross-cut by the thing that does.

By charge, the doubly-charged pairs alone span 31.4 of the 32.7 points the whole set covers. A label whose single group nearly reproduces the full range of the data is a label with no information in it.

The sign is the interesting part #

The model does not merely fail worse on some pairs. It fails in opposite directions: two compact second-row ions come out about fourteen per cent too close together, two diffuse third-row ions eighteen per cent too far apart.

That rules out the repair anybody would try first. An error that were a fixed offset, or that scaled with the separation, could be absorbed by adjusting the radii — which is what an additive-radius model is, and which is the thing the residue was defined to measure the failure of. But no set of radii produces an error that reverses sign with the shell, because a radius correction is additive and adding a constant to each ion’s size cannot make a model too short in one place and too long in another by comparable amounts.

So this is a defect in the model’s shape rather than in its parameters, and the shell is the only thing in the model that the shell decides: the range of the overlap repulsion. The overlap is the only place a shell enters this model at all — the Coulomb term knows charges and distances and nothing else — so if any label is going to sort the errors, the shell is the one the model’s own structure predicts. A third-row p shell is more diffuse, its overlap with a partner falls off more slowly, and the repulsion it produces reaches further — the same diffuseness that decides how far a table’s drawn surface sits from a nucleus, and that a neighbour can move. If the model’s overlap is too soft — falling off too slowly — it will push diffuse pairs apart too much and compact pairs not enough, which is the pattern exactly.

What this does to the verdict on the residue #

It is worth asking whether an identified error is a smaller problem than an unidentified one, and the answer here is no — but it is a differently shaped one.

The verdict was that the model cannot decide the sign of the residue, because its mean error on a single separation is 0.24 Å against residues of three to six hundredths. That arithmetic is untouched. Knowing where the error comes from does not make it smaller.

What changes is the prospect. An error with no pattern is noise, and noise is reduced only by a better model in the general sense — more physics, more parameters, more work. An error that runs cleanly with one identifiable property is a bias, and a bias has a chance of being corrected by fixing the one thing it runs with. Two of the six pairs are already right to under one per cent, which says the model’s structure is not hopeless; the other four are wrong in a direction the shell predicts.

So the honest statement is that the verdict stands and its outlook does not. The model is still unable to decide a residue’s sign, and it is now a model with one identified defect rather than a model that is generally too crude — and those two situations call for different next steps. The first calls for repairing the overlap’s range; the second called for abandoning the approach, which is roughly what was concluded and which this suggests was premature.

None of that is a claim that the repair would work. It is a claim about which experiment is worth running, and the last section says what it is.

What was computed, and how #

Nothing was recomputed. The six model separations and the six measured ones are the ones already computed, obtained by minimising a Coulomb term against a four-electron overlap cost over a quadrature grid, with the overlap rule’s own verdict consulted before the minimiser runs — which is why there are six checkable pairs and not eight, and which is the habit of refusing rather than approximating applied to a quadrature.

The whole of this essay is labelling those six and asking whether either label sorts them. The test is deliberately not a correlation coefficient. On six points a correlation is available between the errors and almost any label, including labels invented for the purpose, and it would come back with a number that reads like evidence. The test used here is whether the groups a label makes have overlapping ranges — every error in one group below every error in the next — which cannot be satisfied by accident on data that do not have the structure.

Six checks. One is the confound check: that both charges appear at more than one shell count. Three concern the shell: that its groups separate, that the gap between them is a clear margin rather than a hair, and that the sign reverses between the extreme groups. Two concern the charge: that its groups do not separate, and that one group alone spans most of the range. That last pair is what stops the essay being a story about one label that happened to work — a label is only shown to be the explanation when a plausible rival is shown not to be.

Where the model stops #

Six pairs is six pairs. The shell count takes three values and one of them has a single member, so “two third-row ions gives +17.9 per cent” rests on K⁺Cl⁻ alone. The pattern would be much better supported by the pairs the quadrature refused — Mg²⁺S²⁻ and Ca²⁺S²⁻ are exactly the doubly-charged, third-row cases that would test it, and they are the ones the overlap rule declines to compute.

That is worth stating as the specific shape of the limitation rather than as a general caveat. The refusals are not random: they concentrate where a compact 2p meets a diffuse 3p at short range, which is the regime this explanation is about. So the cases most able to confirm or break the finding are precisely the ones unavailable, and the mechanism that makes them unavailable is related to the mechanism being proposed.

The shells are read from the electron count — ten electrons is 2p, eighteen is 3p — which is the obvious assignment and is exact for these seven ions. Nothing here would survive being applied to a transition-metal ion, where the outermost shell is not a filled p shell at all, and where the radius that was tabulated is doing rather more work than a single number should.

And “third-row” is standing in for diffuseness without measuring it. The proposal is that the overlap’s range is wrong, and the shell is a proxy for range rather than the quantity itself. A better version of this would use the fitted Slater exponents, which are available, and ask whether the error runs with those continuously — six points is thin for that, which is why the categorical version is the one reported.

The generalisation #

The transferable point is about ruling a label out rather than in.

The natural way to report this finding is that the shell explains the error, and on its own that is weak: with six points and enough candidate labels, something will sort them. That is the same hazard a control that outranked the mechanism found from the other direction, where the label that sorted best was the one nobody had proposed as an explanation. What makes it strong is the second half, that the charge — the label anybody would try first, and the one first named — is shown not to sort them, on the same six points with the same test.

That pairing is worth making a habit of. A classification result should carry the rival it beat, and the rival should be the one a reader would have guessed. A finding reported as “X explains it” invites the question “was Y tried”, and a finding reported as “X separates the groups and Y does not” has answered it before it is asked.

There is a stronger version available when the rival is not merely plausible but preferred. Charge is the label first reached for and the label a reader reaches for, because the Coulomb term is the visible half of this model and charge is what it takes. Showing that the visible half is innocent and the half nobody was looking at is responsible is worth more than showing that some label works, and it is the same shape of result whenever a model has an obvious part and an unobtrusive one.

The second point is about what a confound check is for. Establishing that the two labels cross took one line and produced no result, and without it the whole essay would have been unfounded — because on six points it is entirely possible for two labels to coincide, and then the clean one is being credited with the other’s work. The check earns its place by being capable of stopping the essay, which is the only property a check needs.

Why a count and not a continuous measure #

There is an obvious improvement available and it is worth saying why it is not made here.

“How many third-row ions” takes three values, and diffuseness is continuous. The Slater exponents that set how far each ion’s outermost p function reaches are available, so the error could be plotted against those instead of against a count — and that would give a slope rather than three groups, which is what any correction would eventually need.

The reason it is not done is the number of points. A continuous fit through six points would return a slope and an intercept whatever the data did, with no test that could fail; a reader would get a line and a residual and no way to tell either from a line drawn through noise. The categorical version has a test that can fail, and it is the overlap of the groups’ ranges — which is a property of the arrangement of the points rather than of a fitted object, and which would have come out negative if the errors had been scattered.

That is the trade. The categorical statement says less and can be checked; the continuous one says more and cannot be, at this sample size. The right time for the continuous version is when the pairs the quadrature currently refuses become available, since that would take the set from six to eight and, more importantly, would add points in exactly the region the proposed mechanism is about.

There is a general habit in that. When a sample is small enough that a fit cannot be falsified, the thing to report is the coarsest statement that still has a failure mode — and to say what would make the finer statement worth making.

Who found it, and when #

Slater’s rules, the ionic radii and the measured separations are all quoted. The model, the six-pair check and the observation that its error is uneven come from the residue analysis. The classification is new here, and it is arithmetic over numbers that already existed — the entire contribution is two labels and a test for overlap.

The residue analysis deserves the credit for writing down that the pattern was not obvious rather than proposing one. It named the good pairs and the bad pairs, said what it had ruled out by eye, and stopped — which left a question with an answer in it rather than a guess to be inherited.

Still open: the exponents, and a test by stiffening #

The obvious open question is the exponents. The shell is a proxy for how diffuse an ion’s outermost p function is, and the Slater exponents quantify it. Plotting the error against the sum or the ratio of the two exponents, rather than against a count that takes three values, would say whether the relationship is continuous and would give it a slope — and a slope is what a correction needs. Six points will not support much, but the question is whether the trend is there at all.

The nearer question is whether the proposed mechanism can be tested directly rather than inferred from a pattern. If the overlap’s range is too long, then stiffening it — using a shorter-ranged repulsion at the same strength — should move the two-third-row pair down and the two-second-row pairs up, and should move the mixed pairs least. That is a one-parameter change to the model with a predicted sign pattern rather than a fitted improvement, which makes it a test rather than a tuning: a change that improved everything uniformly would refute the explanation as surely as one that made things worse.