A size a confound cannot supply

A closed-shell repulsion ranked against the failure of radius additivity over eight ion pairs gives 0.857, and a formal charge — an integer that cannot be a mechanism — ranked against the same shortfall gives 0.9524. The set is confounded, and a rank correlation over eight points was not going to separate anything.

It named what would: a size.

A rank says the two order together; a size would say whether the repulsion is big enough to matter, and the repulsion’s constant is not free. Evaluating that expression for the eight pairs, in energy rather than in rank, would say whether a repulsion of the computed size displaces two ions by the tenths of an ångström the shortfall amounts to. That is a quantitative test, and unlike a ranking it cannot be passed by a confound.

It cannot be passed by a confound and it is not passed. The computed balance predicts six of the eight separations with a mean error of 0.242 ångström. Adding two tabulated radii predicts the same six with a mean error of 0.183.

The other two are absent for a reason given below, and it is a numerical one rather than a chemical one.

What is being computed #

Three things go into the energy and only one of them is a number somebody chose.

The repulsion is computed. Two filled shells facing each other put four electrons into a bonding and an antibonding combination; the antibonding one rises by more than the bonding one falls, and twice that difference is what the pair costs. It is summed over the three p orbitals of each ion’s valence shell, at the effective charges Slater’s rules give them, and every overlap in it is an integral rather than a parameter.

The attraction is arithmetic. These eight are all rock salt, so each ion has six nearest neighbours and the electrostatic sum over the lattice is the Madelung constant — 1.7476 — times the nearest-neighbour term. Both numbers are structural facts about a structure that is known, and neither is fitted.

The one carried-over number is the Wolfsberg–Helmholz constant, K = 1.75, and it is the constant left out of the ranking precisely so that the ranking would not become a fit. It is the only fitted quantity in this corner: it was fixed on the noble gases, where it produces three contact distances within a tenth of an ångström of the tabulated ones, and it is carried here without adjustment. That is the only way a number taken from a fit can be used in a prediction — and it is the same discipline the caution about that constant already asks for.

No length is fitted anywhere. The equilibrium separation is an output, in the same sense that a contour level here is an output rather than a caption: it is where a computed function has its minimum, and if the function is wrong the number is wrong in a way that shows.

The answer #

The last two columns are the test. The shortfall is how far the sum of two radii is from the measurement; what the repulsion buys is how far the computed balance moves the pair away from that sum. A mechanism would have the two columns equal. They are the same order of magnitude — tenths of an ångström, which is what the question needed — and they have opposite signs on three of the six pairs and a rank correlation of 0.14.

Read as a fraction of the shortfall, the answer scatters across the value a mechanism would have — 0.08, 1.23, 1.96, −1.70, 5.72 and 0.49:

The comparison that settles it #

The strongest form of the result is not about the shortfall at all. It is that the computed balance predicts the eight measured separations no better than the tabulated radii do.

Mean error 0.242 ångström for a calculation with one carried-over constant and no fitted length, against 0.183 for adding two numbers out of a table. The computed model is a third worse than the arithmetic it was meant to explain.

That is a different conclusion from the repulsion is unimportant. The repulsion is worth 0.106 eV for sodium fluoride and 6.72 eV for calcium oxide at their own measured separations, a factor of sixty across a set a table treats alike — and without it there is no minimum at all.

Why the constant cannot be blamed #

A negative result from a model containing a fitted number is worth very little until the number has been moved, so it was moved.

It cannot be moved everywhere, and the bound on it is algebra rather than evidence. For a two-by-two secular problem with unequal diagonals the two roots sum to (α_c + α_a)(1 − K S²) ÷ (1 − S²), so the cost of putting four electrons where two pairs of them started — the sum of the two levels, twice, measured from the two atomic energies — comes out as

2 (α_c + α_a) · S² (1 − K) ÷ (1 − S²).

Both orbital energies are negative and S² ÷ (1 − S²) is positive, so the interaction is repulsive if and only if K exceeds one — for every pair of atoms, at every separation, whatever the overlap happens to be. At exactly one the antibonding level rises by what the bonding level falls, four closed-shell electrons cost nothing, and there is no minimum anywhere to find; below one a filled shell against a filled shell is net bonding. That is a bound on the constant that owes nothing to any fit, and it is why the sweep below starts at 1.20 rather than at 1.00. A minimiser handed the K = 1 energy returns the near edge of its bracket and reports it as an equilibrium, which is the shape of error this collection’s caches are checked against: the number is not wrong to any tolerance, it is an answer to a different question.

Two things fall out of it and the second is the sharper.

The carried-over value is already the best of those tried. A refit would not move it: 1.75 came from three noble-gas contacts and it is where this curve has its minimum, which is a small piece of evidence that the model is not wrong about the shape of a closed-shell repulsion.

And the best it achieves is 0.242 against the radii’s 0.183. So the failure is not a badly chosen constant, and it cannot be argued away by promising a better one — from 1.40 upward the rank correlation between the displacement the model produces and the shortfall it is supposed to explain is 0.14 at every value tried, unchanged across a threefold move in the constant, and at 1.20 it is 0.26 for a reason that is not encouraging: one of the six pairs has been pushed to the near edge of the grid the repulsion is tabulated on, so its displacement is set by where the tabulation stops rather than by the model.

The failure that was fixed on the way #

The first version of this calculation used the σ-facing orbital of each ion and nothing else — the same orbital the ranking used, which is legitimate for a ranking because a ranking has no scale in it.

With a scale in it, that model puts magnesium oxide’s equilibrium at 0.79 ångström, which is inside the cation, and magnesium sulfide’s at 0.94. A closed shell is not one orbital, and the same lesson is recorded where this collection prices a noble-gas contact: doing that calculation with the σ-facing orbital alone puts argon’s contact at a quarter of the observed distance.

Summing over the shell is not a refinement; it is the difference between an answer and a number an ångström out.

The interpolation needed the same care. Reading the repulsion off a straight line between two grid points in the logarithm is two per cent out at the close end of the grid, which moves an equilibrium by a hundredth of an ångström; a parabola through three points is a part in ten thousand. Both statements are checked against a direct evaluation at separations the grid does not contain, because a grid too coarse for an exponential moves every answer and nothing else notices.

The two pairs that are not here #

The s orbitals are missing from the sum, and two whole pairs are missing from the table, and both absences have the same cause.

The overlap is computed by a ninety-point product quadrature over a mapped domain, and checks it against an eighty-point rule over a wider map — two rules sharing neither their point count nor their map, so a disagreement is evidence about the integrand. The tolerance is a part in ten thousand.

A compact cation orbital against a diffuse anion one is where the two rules part company. Asked for every one of the seventy-eight separations each pair needs, the rule agrees with itself everywhere for six of the eight pairs — worst disagreement 6 × 10⁻⁵ — and disagrees at five separations for magnesium sulfide and three for calcium sulfide, at 1.5 × 10⁻⁴ and 1.3 × 10⁻⁴. Those are the two pairs with the most compact cation and the most diffuse anion, and they are the two that are absent.

They are absent rather than approximated, which is the point. A search that wanders across a hundred and twenty separations nobody chose will find those arguments; a fixed grid of twenty-six makes the set of integrals finite, and every one of them can be asked whether the two rules agree before any of them is used. That is why the repulsion here is evaluated on a grid and interpolated between the points rather than evaluated wherever a minimiser happens to look.

The s orbitals go for the same reason one step earlier: a compact 2s against a diffuse 3s fails the check at most separations rather than a few. The missing term is the smaller one, because an s orbital sits inside the p at these charges — but smaller is an argument and not a measurement, and it is the one place here where something has been left out on a judgement.

What a null result of this shape is worth #

The first finding was that a strong rank correlation over a small set can be about a confound. This one is narrower and sharper: the same quantity, priced rather than ranked, does not do the job.

The two are not the same statement and the second is much harder to explain away. A ranking can fail to identify a mechanism while the mechanism is real; a size that comes out uncorrelated with what it is supposed to explain is a mechanism that is not doing the explaining. What survives is the negative half of the ranking — polarisation does not account for the shortfall, at −0.9524 — with a second candidate now removed on stronger grounds than the first.

It is worth being clear about how much of the calculation survives with it. The repulsion is real, it is computed from orbitals rather than fitted, its size is right to the order of magnitude a lattice energy needs, and without it these pairs have no equilibrium at all. A quantity can be correctly computed, physically necessary and still not the explanation of the particular thing it was proposed for, and that has now happened twice in one argument: the polarisation correction is also real, also computed, and also not it.

So the failure of radius additivity has no explanation in any computed mechanism tried here, and that is worth stating plainly rather than leaving as an open question with a promising lead in it. The radius that was tabulated is a fitted quantity, and the most likely remaining answer is the least interesting one: a set of numbers fitted to reproduce distances reproduces them to about a tenth of an ångström, and the residue is what fitting could not absorb rather than a physical effect anybody has isolated.

What is quoted, and what is computed #

Quoted: the eight interionic distances, the Shannon radii, the Madelung constant of rock salt, the coordination number six, and the Wolfsberg–Helmholz constant carried from a calculation on the noble gases.

Computed: every effective charge, every orbital, every overlap integral, every repulsion energy, every equilibrium separation, and every displacement.

The equilibrium is found by golden section on the sum of the two terms, and the cached result is verified on every read by a check independent of the search — the derivative of the total energy must vanish there, evaluated by a finite difference that does not repeat the minimisation.

What this cannot say #

There is no dispersion in it. The attraction is electrostatic and nothing else, which is right for an ionic pair and is why the same calculation needed a quoted C₆ coefficient when it was pointed at two neutral atoms. What that leaves out is not negligible for the larger ions, and it acts in the direction of shorter separations.

The ions are spherical and rigid. How far a neighbour moves an ion’s surface has been measured, so that assumption is known to be wrong by a per cent or two; nothing here corrects for it, and the correction was already shown not to track the shortfall.

And a Madelung constant is a lattice. The measured distances are crystal distances and the balance is written for a crystal, so this has one foot on the other side of a boundary drawn deliberately. What is used is two structural integers and nothing about reciprocal space, which is the least a comparison with crystal data can get away with.

What was checked #

Every equilibrium is inside the range searched, which is what a golden section cannot check for itself: a minimiser handed a monotone function returns an endpoint and reports it as a minimum.

The closed-shell term is a repulsion at every measured separation, checked pair by pair. A sign error there would produce a plausible curve and a nonsense conclusion.

And the refusal is the constant set to zero: a bare Coulomb attraction falls all the way in, so the minimum belongs to the repulsion. A calculation that still found an equilibrium with no repulsion in it would be finding the edge of its own search window.

What a size buys that a rank cannot #

The move made here is worth naming as a method rather than leaving inside its result, because it is the general escape from the trouble a ranking runs into.

A rank correlation asks whether two quantities order together. Any monotone relation between them satisfies it, so a candidate passes on a very weak commitment — and a confound passes on the same weak commitment, which is why the formal charge outranked the mechanism.

A magnitude commits to more. Asking whether a computed repulsion displaces two ions by tenths of an ångström requires the candidate to be right about a scale as well as an order, and there is no free constant left to absorb a mismatch. A confound has no reason to have the right size, because being correlated with the answer costs nothing and being equal to it costs everything.

That is why the negative result here is worth more than the positive one it replaces. The ranking said the repulsion might be the answer; the size says it is smaller than the effect by enough that it cannot be, and no rearrangement of the sample recovers the missing distance.

Still open: what the residue is #

The obvious open question is the measurement everything keeps pointing at. Every candidate has now been tested on a set assembled by somebody else for another purpose, and the set is confounded in a way no amount of care can undo — the eight pairs split four and four by charge, and charge, size and hardness all rise together. A pair of ions of similar size and different hardness would separate them in one comparison, and Zn²⁺ against Mg²⁺ is two rows of data and no new calculation.

The nearer question is what the residue actually is. If the shortfall is what a fit could not absorb rather than a physical effect, that is a statement about the fitting, and it can be tested: refitting a set of radii to these eight distances alone and asking what residue is left would say whether the shortfall is a property of the pairs or of the compromise a universal table makes. Solving for the enclosed fraction that reproduces a distance exactly is already possible, which is the same question asked one step earlier.