The sum of the exponents, not the softer ion

The error was the row, not the charge took an ionic model that places pairs of closed-shell ions by balancing a Madelung attraction against a four-electron overlap cost, found its errors on six measured separations badly uneven, and sorted them completely by one label: how many of the pair’s two ions have a third-row outermost shell. With none the model is fourteen per cent short, with one it is within seven per cent long, with two it is eighteen per cent long. The charge on the pair sorted nothing.

It reported that as a count on purpose. A count with three values has a failure mode — the groups either overlap or they do not — and a line through six points does not. But it said plainly what the count was standing in for: how far an ion’s outermost p function reaches, which Slater exponents quantify. It named the two candidates, the sum and the ratio of the two ions’ exponents, and said the question was whether a trend was there at all.

There is one, and it is not in the variable the mechanism was describing.

What a continuous measure can add #

A continuous version of the count cannot be tested the way the count was, because it has no groups. And the obvious test, a correlation across six points, is the one the count was chosen to avoid: nearly any quantity that increases from second-row ions to third-row ions will correlate with an error that does the same — which is how, on the same pairs, a control that cannot be a mechanism outranked the mechanism.

So the new measure has to be asked about the one thing the count could not see. Three of the six pairs — Na⁺Cl⁻, K⁺F⁻ and Ca²⁺O²⁻ — each have one third-row ion, and the count gives all three the same value. Their errors are +0.6, +3.2 and +6.7 per cent. A continuous measure either puts them in that order or it does not, and that is the only prediction it makes that the count did not already make.

That prediction is weak on its own. Three points have six possible orders, so a measure carrying no information whatever lands on the right one a sixth of the time. The test therefore has three parts, and a measure has to pass all three: a rank correlation across all six pairs, judged against every one of the 720 orders six points can be put in, so that the chance level is counted rather than assumed; the order of the middle three; and a line fitted through the outer three pairs alone, asked to predict the middle three it never saw.

The softer ion carries nothing #

The mechanism the count was read as pointing to is about the softer ion. A third-row p shell is more diffuse, the argument went, so its overlap with a partner falls off more slowly and the repulsion reaches further — pushing diffuse pairs too far apart and compact pairs not far enough. The continuous version of that is the exponent of whichever ion in the pair is more diffuse.

It carries no information. The rank correlation is −0.36, and 384 of the 720 orderings of six points do at least as well.

The reason is visible in the exponents themselves. The oxide ion’s 2p function has an exponent of 1.925 and the chloride ion’s 3p has 1.917 — the same to under a hundredth. The oxide has two electrons more than its eight protons, and on Slater’s rules that much screening on so small a nucleus makes a second-row shell as soft, by exponent, as a third-row one. So four of the six pairs have a softer ion of nearly identical exponent, and their errors run from −14.9 per cent for magnesium oxide to +17.9 for potassium chloride.

Measuring the softer ion by its mean radius instead of its exponent does not rescue it. The mean radius carries the principal quantum number a second time, so a chloride’s 3p is considerably larger than an oxide’s 2p by that measure, and still the larger mean radius in a pair correlates with the error at only 0.53. The idea that one ion’s reach decides the error does not survive being made into a number, in either of the two different senses of size that apply.

The sum carries it #

The sum of the two exponents is the other thing entirely. Its rank correlation with the error is −0.986. Four of the 720 orderings of six points match it, an exact probability of 0.0056; the count of third-row ions, on the same test, is matched by twenty-four. A line through all six fits with r² = 0.974, falling 22.4 percentage points of error for each unit of exponent.

And it orders the middle three. Sodium chloride’s exponents sum to 5.342, potassium fluoride’s to 5.008 and calcium oxide’s to 4.842, and their errors rise in exactly that order, from +0.6 to +3.2 to +6.7 per cent.

Why the sum should be the variable is not settled here, and the one thing known about it points somewhere specific. Far enough out, the overlap of two Slater functions decays at the rate set by the softer exponent alone — and the softer exponent has just been shown to carry nothing. So whatever the sum is picking up does not live in the overlap’s tail. It lives at the separations the model actually uses, a few bohr, where both functions still contribute and no single exponent sets the decay. Computing the overlap’s local rate of decay at each pair’s own separation, and asking whether that rate orders the six as the sum does, would turn a correlation into a mechanism; nothing here does that.

Seven measures and one chance in six #

The sum was one of two measures named before anything was computed. The other, the ratio of the two exponents, fails on both counts: its rank correlation is −0.60, matched by 174 of 720 orderings, and it puts the middle three in the wrong order.

Five more measures are obvious enough that not trying them would be its own choice: the harder ion’s exponent, the sum and the larger of the two mean radii, the ratio of the mean radii, and the softer ion’s exponent already ruled out. None of them passes the test on all six at the level the sum does. The harder ion’s exponent comes nearest, at p = 0.044, and gets the middle three wrong.

The middle three are where chance has to be counted most carefully.

Two of the seven measures put the middle three in the right order: the exponent sum, and the ratio of the two mean radii. Seven measures each with a one-in-six chance would be expected to produce 1.17 successes by luck, so two is no evidence of anything, and the ratio of the mean radii shows why — it gets the three right while failing the six-pair test at p = 0.175. The same arithmetic runs through the chance calculation behind two electronegativity tables: a classification that succeeds as often as coin-flips would is a classification that has not yet been tested.

What separates the sum is that it passes both tests, and that it was named first. Those are different kinds of support and both matter. Among seven tries, the probability that at least one orders the middle three by chance is 72 per cent; among the two named in advance it is 31 per cent. The advance naming did not make the sum right, but it is most of the reason its success in the middle group counts for anything.

A line that has not seen the middle three #

The strongest test is the one fitted without the points it is judged on.

A line through magnesium oxide, sodium fluoride and potassium chloride — the pairs with no third-row ions and with two — falls 23.8 points of error per unit of exponent. Carried across the middle, it predicts −2.1 per cent for sodium chloride, +5.8 for potassium fluoride and +9.8 for calcium oxide. The errors are +0.6, +3.2 and +6.7.

The order is right. The spread is not: the line predicts 11.9 points between the first and third of the middle pairs, and they are 6.1 apart. Inside the middle group the error falls by 11.5 points per unit of exponent — half the slope the outer pairs set.

So the sum is the right direction and not the whole variable. Something inside the middle group halves its effect, and nothing here says what. The three middle pairs differ in which ion is the third-row one — the anion in sodium chloride, the cation in the other two — and in charge; either could be the missing term, and three points are not enough to separate them. What the out-of-sample line establishes is that the slope a correction would use cannot be read off a fit to all six, because that fit averages two slopes a factor of two apart.

The fit to all six still earns its keep, and how it does so is a caution in itself. Used as a correction — subtracting from each pair’s error what the line predicts — it takes the root-mean-square error over the six separations from 11.4 per cent to 1.8. The worst pair left afterwards is sodium chloride, at 3.1 per cent, and sodium chloride is the pair the uncorrected model had right to 0.6. A correction that improves the set sixfold makes its best member five times worse. That is the same shape as the next order of an expansion making particular cells worse while improving the whole, and as the property that gets worse when a basis improves an energy: a line fitted to a set answers for the set, and a member whose error was already near zero has nowhere to go but away from it.

The tie Slater’s rules build in #

There is one more thing the sum cannot do, and it is exact.

Magnesium oxide and sodium fluoride have exponent sums of exactly 5.850. Every ion in the two pairs has ten electrons, so every 2p shell is screened by the same 4.15 — two inner electrons at 0.85 each and seven neighbours in the shell at 0.35. Each exponent is then the nuclear charge less 4.15, halved, and a pair’s sum is its two nuclear charges added, less 8.3, halved. Twelve and eight make twenty; so do eleven and nine.

So for isoelectronic pairs the sum knows how many protons there are and not how they are divided between the two ions — which is precisely what distinguishes a doubly charged salt from a singly charged one. The model’s errors on the two separations are −14.9 and −13.6 per cent. That 1.3-point difference is below anything the sum can resolve, however well it does elsewhere, and it is a residual built into the choice of variable rather than a failure of the fit. It is also the only place the charge on a pair could still be hiding, having separated nothing when it was tested as a label.

How it was computed #

Nothing in the model was recomputed. The six separations and their errors are the ones the model’s check against measurement produced, from a Madelung attraction balanced against computed overlap repulsions with the quadrature’s own refusals respected. Each ion’s exponent is its Slater effective charge for the valence p shell divided by the principal quantum number, the same charges the model used; a mean radius is (2n + 1)/(2ζ).

Rank correlations use average ranks, so that the tie between magnesium oxide and sodium fluoride counts as a tie and not as whatever order the pairs happened to be listed in. Each p-value is the share of all 720 orderings of the six errors whose correlation with the measure is at least as large in size — exact, with no distribution assumed. Lines are least squares.

The checks are these. The sum’s correlation is below −0.95 with p under 0.01 and r² above 0.95, and its p is below the count’s. It orders the middle three. The ratio does neither. The softer ion’s exponent correlates at under 0.4 in size with p above 0.5, and an oxide’s and a chloride’s exponents are within a hundredth while their pairs span more than thirty points. Exactly two of the seven order the middle three, and the other one fails on all six. The line through the outer three predicts the middle order and a slope more than 1.8 times theirs. And magnesium oxide and sodium fluoride have the same sum to a part in 10¹², with nuclear charges adding to twenty in both.

Where six points and a rule of thumb stop #

Slater’s rules are rules. The exponents are a screening scheme from 1930, not optimised orbitals, and a fitted ionic exponent for an oxide would not be 1.925. The enclosure a tabulated ionic radius corresponds to already varies with the ion’s charge, peaking at the neutral atom, which is a sign that one set of screening constants describes an ion’s reach less faithfully than an atom’s. Everything here is a statement about the model as built, which uses the same rules, so the comparison is internally consistent — but the numbers are not a claim about real ions’ diffuseness.

Six points are six points. The exact test makes the chance level honest and does not make six into sixty; the out-of-sample line is fitted to three and judged on three. And the two separations the quadrature refuses, Mg²⁺S²⁻ and Ca²⁺S²⁻, are exactly the doubly charged third-row pairs that would test the middle-group slope and the division of charge at once. They are unavailable for a reason connected to the question — a compact 2p against a diffuse 3p at short range is where the overlap rule fails — which is worth stating as the specific shape of the limitation.

And any measure that ranks the pairs as the sum does would pass the same rank test. What the sum has that its monotone relatives do not is a reason: it is what an overlap at contact distances depends on.

Naming the candidates first #

The transferable point is about what advance naming buys when the sample is small.

Seven measures were tried, and one of them passing a test worth one chance in six is what seven tries would produce by luck nearly three times in four. Had the sum been found by trying all seven and reporting the best, its success in the middle group would be worth very little. It was not found that way. It was one of two measures named in print before any exponent was computed, and that turns a best-of-seven into a one-of-two, where luck alone succeeds about three times in ten.

That is still not proof, which is why the out-of-sample line and the six-pair test are there. But it shows why one habit is worth keeping: a calculation that ends by naming what the next one should measure has made the next one a test rather than a search.

Where the pieces come from #

Slater’s screening rules are from 1930. The exact permutation test for a correlation is Fisher’s and Pitman’s, from the same decade. The ionic model and its six checkable pairs come from the calculation that found its error uneven; the exponent measures, the tests and the tie are new here.

The shell-count essay deserves the credit for naming the sum and the ratio before either was computed — and for insisting that the count, which has a way to fail, be reported instead of a line that did not.

Still open: a test by stiffening, and the other half of the slope #

The obvious open question is the one the shell-count essay called the nearer one: whether the mechanism can be tested by changing the model rather than inferred from a pattern. If the model’s overlap reaches too far, shortening its range should improve the diffuse pairs and worsen the compact ones — a predicted sign pattern rather than a fitted improvement. That test has to be built with care. A stiffening defined by steepening the repulsion about a common separation produces exactly that sign pattern for any pair above and below the pivot, whatever the mechanism, and so cannot fail. The version worth running changes the range the same way for every pair and lets the pattern come out or not.

The nearer question is the half of the slope the out-of-sample line could not account for. The middle three differ in which ion is the third-row one and in charge, and the doubly charged pair has the largest error of the three. A second doubly charged pair with one third-row ion would separate the two, and the only candidate in these tables, Mg²⁺S²⁻, is one of the two separations the quadrature refuses.