The residue is below its own noise

An additive model of ionic separations says that d(AX) − d(AY) − d(BX) + d(BY) is exactly zero, for any four ions and any assignment of radii whatever. The combination is a subtraction with no free parameters, so whatever it comes to on real separations is what no set of radii can reproduce — and the residue that is two numbers computes it on the two complete blocks available.

It got −0.031 Å for the alkali halides and −0.056 Å for the alkaline-earth chalcogenides. Both negative, and it said so plainly: with two of them, a systematic sign is a coin toss. A larger block would settle it, and a systematic sign would mean something — it would say that a large cation with a large anion sits closer than additivity predicts, which is a statement about polarisability rather than about size.

There is a way to get a larger block without quoting a single new measurement. A model exists that computes an equilibrium separation for a pair of ions from their charges and the overlap of their outer p shells, with no length fitted anywhere. Four cations and four anions give sixteen separations, and sixteen separations give thirty-six two-by-two blocks against the measurements’ two.

Building them is easy. Reading them is the essay.

How well the model does on one distance #

Six of the sixteen pairs have a measured separation the model can be checked against. It gets sodium chloride right to 0.017 Å — six parts in a thousand — and potassium fluoride to 0.085. It gets sodium fluoride wrong by 0.314 Å, magnesium oxide by 0.313, and potassium chloride by 0.563, which is eighteen per cent.

The mean is 0.242 Å. For a model with no fitted length in it — the radial functions from Slater’s rules, the charges formal, the repulsion from an overlap — that is a respectable result and it has been treated as one. It gets the ordering of the six right, it gets the difference between a first-row and a second-row anion right, and it does all that from an atomic number and a charge.

It is also the number every other number here has to be read against.

The residue is smaller than the error #

The residue is a difference of four separations, each carrying that error, so at best it inherits it and at worst it accumulates it. The measured residues are 0.031 and 0.056 Å.

A quarter of an ångström of error on each of four terms, differenced to find three hundredths of an ångström. That arithmetic is available before anything is computed, and it says the answer will be noise.

It is worth being careful about how much cancellation to hope for, because that is the only thing that could rescue it. A systematic error common to all four separations does cancel: the alternating sum kills any term that depends on one ion alone, which is precisely why the residue is a residue. What does not cancel is the part of the error that depends on the pair — and that is most of it here, since the four terms involve four different cation–anion combinations with different shells and different overlaps. Sodium fluoride is wrong by −0.314 and potassium chloride by +0.563; those two appear in the same block, with opposite signs in the alternating sum, and they do not cancel, they add.

It is noise. The twenty-one residues run from −0.246 to +0.150 with a root-mean-square of 0.1118 Å — twice the larger of the two measured values — and they split nine negative to twelve positive. That split is not a finding about the sign of the non-additivity. It is the model’s own error, sorted and plotted.

The size of that scatter says something about how the error is built. If the four separations in a block each carried an independent error of about a quarter of an ångström, the alternating sum would scatter by about twice that, near half an ångström. The survey scatters by 0.11 Å, a quarter as much, so a large share of the error does cancel: the part that belongs to one ion and follows it from pair to pair. What is left is the part that belongs to the pair, and it is still twice the larger measured residue.

The one check, and it fails #

The decisive test is not the scatter but the comparison, and there is exactly one available.

The alkali-halide block — sodium and potassium against fluoride and chloride — is measured at −0.031 Å. The model returns +0.115 Å. Wrong sign, and nearly four times too large.

The other measured block is not reachable at all, for reasons in the next section, so that is the whole of the check. One comparison, and it fails in exactly the way the error arithmetic predicts. Had the model happened to get the sign right, that would have been one agreement out of one — worth almost nothing on its own, and the twenty other blocks would still have carried no information. Getting it wrong is more informative than getting it right would have been, which is the shape a good control has: it can fail, and failing is informative.

That is also the arithmetic above, made concrete. The alkali-halide block is NaF − NaCl − KF + KCl, and the model’s errors on those four are −0.314, +0.017, +0.085 and +0.563. Combined with the block’s signs that is −0.314 − 0.017 − 0.085 + 0.563 = +0.147 Å of error, against a measured residue of −0.031. The wrong sign is not bad luck; it is those four numbers.

That settles what the twenty-one-block survey is worth: nothing, on this question. A model that gets the sign wrong on the case it can be checked against carries no information about the sign of the cases it cannot.

What a useful survey would have looked like #

It is worth saying what the survey would have needed, because the requirement is quantitative and short.

To decide a sign that is 0.031 Å, the combined error on the alternating sum has to be well under that — call it 0.01 Å, which on four terms with independent errors means about 0.005 Å on each separation. That is two per cent of a bond length for the shortest pair here and a fifth of a per cent for the longest, and it is fifty times better than this model manages.

Nothing about the model is close to that, and no adjustment of it would be. The four-electron cost is a two-level estimate with one borrowed constant in it; the Coulomb term uses formal integer charges on point ions; the radial functions are single Slater exponents. Each of those is worth several per cent on a distance on its own.

So the finding is not “this model is not quite good enough”. It is that the gap between what the question needs and what the method delivers is a factor of fifty, which is a difference of kind. That is a more useful thing to know than a marginal result would have been, and it is why the confound that outranked its mechanism is the right comparison: both are cases where the instrument was measured before its answer was believed.

What the quadrature refuses #

The overlap rule carries an independent check on itself, and it refuses magnesium against sulfide and calcium against sulfide: a compact second- or third-row p against a diffuse sulfide at three bohr is outside what the quadrature can do to its own tolerance. It says so rather than returning a number, which is the calculation working as intended.

Those two separations appear in fifteen of the thirty-six blocks, so twenty-one survive. And one of the fifteen lost is the alkaline-earth chalcogenide block — one of the two the measurements supply, and the one with the larger residue.

That is worth stating rather than absorbing quietly. A survey that dropped two awkward pairs without saying so would look like a survey of everything, and it would have had two measured blocks to check against instead of one.

Why fifteen blocks is most of them #

Losing fifteen of thirty-six to two refused separations looks disproportionate until the combinatorics is written down, and it is worth writing down because it is a general feature of surveys built out of blocks.

A block needs four separations. A separation appears in every block whose cation pair contains its cation and whose anion pair contains its anion — which with four of each is three cation pairs times three anion pairs, or nine blocks. Two separations sharing an anion between them therefore account for nine plus nine less the blocks they share, and here that comes to fifteen.

The lesson generalises to any such survey: a single unusable cell in an n-by-n table removes about a quarter of the two-by-two blocks, and two of them remove nearly half. So the robustness of a block survey to missing data is much worse than the fraction of missing cells suggests, and the fraction to quote is the fraction of blocks rather than the fraction of separations. Here it is two of sixteen separations and fifteen of thirty-six blocks — an eighth against five twelfths.

The tripwire that passes #

There is one thing here that is exact, and it is the reason the residue is worth computing at all.

Take four numbers for the cations and four for the anions, invented for the purpose, and set every separation to the sum of the two. Every one of the twenty-one residues comes out zero to machine precision. The alternating sum has no free parameters, so it is not a fit’s leftover and no better fitting improves it — the departure from zero is what an additive description cannot carry, whatever radii are chosen.

So the arithmetic is right and the inputs are not good enough for it. Those are different failures and it is worth keeping them apart: nothing above suggests the residue is not a real quantity, or that it is not worth having its sign. It says that this particular route to more of them supplies numbers too coarse to be differenced.

What was computed, and how #

Four cations — sodium, potassium, magnesium, calcium — and four anions — fluoride, chloride, oxide, sulfide — from a standard tabulated set. For each pair, an energy of a Coulomb term between the formal charges and a four-electron cost from the overlap of the two outer p shells, minimised by golden section over the same bracket the equilibrium calculation uses. Every minimum is checked to be interior and to have a vanishing derivative.

The check requires eleven things, and the ones that matter are these: that the quadrature refuses exactly two separations and reports them; that this costs exactly fifteen blocks; that the model reproduces the measured separations to more than a tenth of an ångström and not better; that this exceeds the largest measured residue by more than a factor of three; that the model’s own residues have a root-mean-square larger than the measured ones; that they split in sign with both signs well represented; that exactly one measured block survives; that the model’s residue on it has the opposite sign to the measurement; and that an additive set of radii gives zero on every block to machine precision.

The last is the tripwire and the one before it is the finding.

Where the model stops #

The model is a caricature and the essay does not depend on its being better than one. What it depends on is the size of its error, which is measured against six real separations rather than assumed, and which is what makes the conclusion robust: any model with a comparable error on a single distance is equally unable to resolve a residue of this size, whatever its physical content.

A model an order of magnitude better would change that. What would be needed is roughly a hundredth of an ångström on each separation, which is what a modern electronic-structure calculation on an ionic crystal reaches — so the question is answerable, and not here. Tabulated radii are also not the route, because they are fitted to the separations and reproduce the residue by construction as a quarter of itself on each bond.

And the sixteen pairs include eight that no crystal supplies at one-to-one stoichiometry — sodium against oxide, magnesium against fluoride, and the rest — so most of the survey is of model diatomic ion pairs rather than of anything measured — a size a confound cannot supply makes the same point about a different set of pairs. That would have been a caveat worth arguing about if the survey had produced a result. It did not, and the caveat is moot.

The generalisation #

The transferable thing is a calculation to do before a survey rather than after it.

A quantity formed as an alternating difference of n terms cannot be resolved by a method whose error on one term exceeds it. Here the residue is 0.03 to 0.06 Å and the model’s error is 0.24 Å, so the survey was decided before it was run — and the arithmetic takes one line and needs only the model’s accuracy, which was already measured.

The habit that prevents the wasted work is to ask what a derived quantity’s error is, not just what the primary quantity’s is. It is the same discipline as pricing a claim by the error that would overturn it, turned round: there the question was how large an error would have to be, here it is how small. Differencing amplifies: a fourth difference of four numbers with independent errors σ has an error of 2σ, and even a perfectly correlated model error survives if it is not identical across all four terms — and it is not, since the four terms involve four different pairs of ions.

The second transferable thing is what to do when the resolution test comes back negative. The model gets the one available sign wrong, and the temptation is to report the twenty-one anyway with a caveat, because twenty-one numbers look like more evidence than two. They are not evidence at all, and the caveat would not have made them so.

Who found it, and when #

Non-additivity of ionic separations is old and well known; that a set of radii cannot reproduce a complete two-by-two block is a two-line identity. The model, the sweep, the accuracy measurement and everything above are new arithmetic, done to answer the sign question.

This is a question whose answer needs data not in hand, and the honest response is to compute the bound rather than the quantity — the same move a claim’s error bar made from the other direction. What it establishes is a negative and a boundary: the two measured negatives are still two negatives, and a model at this accuracy cannot make them three.

The one thing worth carrying forward is the number 0.005 — the accuracy on a single separation that would make the question answerable — because it converts “better numbers are needed” into a specification anybody can check a source against.

Still open: a measured larger block, and where the model’s error comes from #

The obvious open question is the measured route, deliberately not taken here: the alkali halides supply a five-by-four block from standard tables, which is twelve independent residues from measurements rather than from a model. It needs twenty quoted separations, and quoting numbers from memory is not good enough — so it needs a stated source, and with one it is arithmetic rather than computation and it answers the question properly.

The nearer question is the model’s error itself, which was measured on six pairs and then used as a bound. Two of the six are wrong by fourteen per cent and one by eighteen, while two are right to under one — so the error is not uniform and something distinguishes the pairs it does well from the pairs it does badly. Sodium chloride and potassium fluoride are the good ones and sodium fluoride, magnesium oxide and potassium chloride the bad; the pattern is not obviously size or charge, and finding what it is would say whether the error can be reduced by fixing something identifiable rather than by changing the model.