Three contractions for one shell

An ionic model built from Slater p functions, a Madelung attraction and a four-electron cost for two closed shells overlapping gets six measured separations wrong in a pattern that follows the row of the periodic table. Sodium fluoride and magnesium oxide, with no third-row ion, come out short by −13.56 and −14.86 per cent. Sodium chloride, potassium fluoride and calcium oxide, with one, come out long by +0.62, +3.16 and +6.71. Potassium chloride, with two, comes out long by +17.88.

Contracting the third-row p exponents alone was the test built so that it could fail. The second-row pairs are untouched by construction, and one factor has to bring two different things onto them: the mean of the group with one third-row ion, and potassium chloride. The two factors came out at 1.3765 and 1.3463, 2.2 per cent apart, and at either one the row pattern was gone.

It left two things. An offset: every pair sat near the second-row pairs’ fourteen per cent short rather than near zero. And the middle group’s order: sodium chloride below potassium fluoride below calcium oxide, unchanged at every factor from 1.0 to 1.5, because a change that acts on a whole shell moves all three pairs together. The nearer of the two was the order, and the construction proposed for it was a contraction per ion — the cation’s third-row shell and the anion’s separately.

Each pair fixes one ion #

The three pairs in the middle group have exactly one third-row ion each, and that ion is a different ion in each: chloride in sodium chloride, potassium in potassium fluoride, calcium in calcium oxide. So each pair’s error depends on one ion’s contraction and on nothing else that moves, and each pair fixes its ion’s factor on its own.

The target is the one the shared-shell test used, the mean of the two second-row pairs’ errors, −14.21 per cent. Nothing about the second-row pairs changes.

Chloride needs 1.302, potassium 1.387 and calcium 1.441.

The first thing to see is that these are the middle group’s order. Sodium chloride had the smallest excess and needs the least contraction; calcium oxide had the largest and needs the most. A contraction of the whole third-row shell could not reach that order because it gave the three ions one factor, and one ion at a time it is simply the order of what each ion needs.

The curves also say where that order comes from, and it is not where it might have been. Three ions could need different factors because the three pairs respond differently to a contraction — a steeper response reaching the target sooner — or because they start at different distances from it. Between factors of 1.25 and 1.5 the three errors fall by 9.1 points for sodium chloride, 8.7 for potassium fluoride and 10.0 for calcium oxide: close to one slope, about nine points for every quarter of contraction. What differs is the start. As built, the three pairs sit 14.8, 17.4 and 20.9 points above the second-row mean, and at nearly one slope a pair that starts further away needs more contraction to arrive. So the order of the factors is the order of the excesses the pairs began with — which is why a single factor could align their mean and leave their order exactly as it was.

The second thing is the size of the spread. Dividing an exponent by a factor divides a Slater function’s mean radius by the same amount, so chloride’s p functions would be 23 per cent smaller, potassium’s 28 and calcium’s 31. The three span 10.7 per cent in factor — against the 2.2 per cent between the shared-shell test’s two conditions.

That is also the obvious objection. Three factors fitted to three pairs cannot fail to fit them, and the figure on its own shows nothing but a fit. What would make it a test is a condition none of the three factors was fixed on.

The pair nothing was fitted on #

There is one. Potassium chloride contains two of the three ions — potassium and chloride — and neither of their factors was taken from it. The construction has no free parameter left when it reaches potassium chloride, so its error at those two factors is a prediction.

At potassium’s 1.387 and chloride’s 1.302, potassium chloride’s error is −13.81 per cent. The two second-row pairs span −13.56 to −14.86, so the prediction lands inside their band, 0.40 points from its middle. Potassium chloride began 32 points away from them, the largest error in the set, and it arrives among them at factors taken from two other crystals.

The band is the right yardstick and it is worth saying why. The target is the mean of two pairs that themselves differ by 1.30 points, so nothing aligned to that mean can be said to have landed on it more closely than the two pairs that define it agree with each other. A prediction inside the band is as good as the reference allows.

The same figure tests the reading the per-ion construction was set up to rule out. Potassium and calcium carry the same third-row shell as cations, and if the middle group’s order followed only which ion was diffuse, one factor should serve both. It does not. Given potassium’s factor, calcium oxide’s error is −12.21 per cent, above the band. Given calcium’s factor, potassium chloride’s is −15.38 per cent, below it. The shared-cation reading fails on both of the pairs it can be carried to, in opposite directions.

The sizes matter as much as the directions. The band is 1.30 points wide, so a pair aligned to its middle can be 0.65 points off and still sit among the second-row pairs. Potassium chloride at its own ions’ factors is 0.40 points off. Calcium oxide at potassium’s factor is 2.00 points off, three times the half-width, and potassium chloride at calcium’s is 1.17 points off, nearly twice it. The shared-cation reading does not narrowly miss a generous target: its larger miss exceeds the band’s whole width, and its smaller one is nearly three times the per-ion prediction’s.

It also misses in the direction the factors predict. Potassium’s factor is smaller than calcium’s, so giving it to calcium under-contracts calcium oxide and leaves it long, above the band; calcium’s factor is larger than potassium’s, so giving it to potassium over-contracts potassium chloride and leaves it short, below. Neither result is a surprise once the three factors are known, and that is the point: they are what the factors say, carried to pairs the factors were not fixed on.

Two routes to potassium #

Potassium’s factor can be found a second way, and the two ways use different crystals.

Hold chloride at the 1.302 sodium chloride fixed, and ask what potassium factor brings potassium chloride onto the second-row mean. The answer is 1.401. Potassium fluoride’s was 1.387. The two routes agree to 0.95 per cent, and they share nothing but the potassium ion: one passes through fluoride, the other through chloride, and chloride’s factor came from sodium chloride.

Calcium, the other cation with that shell, needs 1.441, which is 3.9 per cent above potassium fluoride’s value and 2.9 per cent above potassium chloride’s. The two determinations of one ion’s factor are four times closer to each other than either is to the factor of the ion next to it in the table.

This is what additivity would look like if a contraction were a property of an ion. It is not guaranteed by the construction. The four-electron cost depends on the overlap between a cation’s functions and an anion’s, and nothing requires a factor that suits potassium against fluoride to suit potassium against chloride, where the overlap is between two diffuse shells rather than one diffuse and one compact. Within this model it does, to one per cent.

The single factor was two averages #

The per-ion factors also say what the shared-shell test’s agreement was.

The shared-shell factor that aligned the group with one third-row ion was 1.3765. The mean of chloride’s, potassium’s and calcium’s factors is 1.3769 — 0.03 per cent away. The factor that aligned potassium chloride was 1.3463. The mean of chloride’s and potassium’s is 1.3448, 0.11 per cent away.

So the two conditions of the shared-shell test were two averages, and the reason they agreed to 2.2 per cent is that they averaged overlapping ions. The group mean contains chloride, potassium and calcium; potassium chloride contains chloride and potassium. They differed by exactly the amount calcium’s extra contraction lifts a three-ion mean above a two-ion one.

That does not make the shared-shell test wrong. It was built to fail if the row pattern were not a range effect, and it did not fail, and the per-ion factors confirm it: every third-row ion needs a contraction, and one factor for each is enough to reach every pair. What it changes is what the agreement measured. Two conditions met by one factor looked like a single property of the third-row shell. They were a property of three ions, averaged twice.

There is a general caution in that, and it has turned up elsewhere in these calculations: a test with more conditions than parameters is only as strong as the independence of its conditions, and two conditions that are means over overlapping members are less independent than they look.

The factor rises with the charge #

The three ions are isoelectronic. Chloride, potassium and calcium each carry eighteen electrons, and what changes along them is the nuclear charge — seventeen, nineteen, twenty — and with it the ionic charge, −1, +1, +2.

The factor rises at every step: 1.302, 1.387, 1.441. The more positive the ion, the more its p functions have to be contracted relative to Slater’s rules for this model to put it at its measured distance.

It is tempting to read that as a property of the ions, and there are three things it could be a property of, which these three ions cannot tell apart.

The ion’s charge. Chloride is negative, potassium singly positive, calcium doubly positive, and the factor follows that order.

The ion’s nuclear charge. Along an isoelectronic series the nuclear charge is the ionic charge plus eighteen, so it follows the same order by definition, and no set of eighteen-electron ions can separate the two.

The pair’s charge. Calcium’s factor was fixed by calcium oxide, the only doubly charged pair in the group, whose Madelung attraction is four times that of a singly charged pair at the same separation. A contraction fitted against a stronger attraction need not be the contraction the same ion would need against a weaker one. Potassium’s factor was checked through two partners and agreed with itself; calcium’s has one partner and no check.

So the order is real within the model and its reading is open. The row, not the charge sorted the six pairs into groups, and within the third row the charge — of the ion or of the pair — orders the contractions, which is as far as three ions and one prediction go.

What survives of the standing claims #

The row pattern is a range effect in the third-row shells, and that stands. The per-ion factors confirm it rather than replace it: every third-row ion needs a contraction, and nothing about the second-row ions had to change.

The shared-shell factor is an average. Its two conditions are means of the per-ion factors to a tenth of a per cent, and the per-ion factors span five times the gap between them.

The middle group’s order is the same defect, resolved per ion. It was unreachable by one factor per shell and it is the order of the factors per ion.

Within the row, the contraction follows the charge, of the ion or of the pair, and the construction cannot say which. The exponent sum caught half of the middle group’s slope and could not account for the rest; the per-ion factors are where the rest was.

The offset is untouched. Every pair still sits about fourteen per cent short, and fixing the range was never going to fix that.

What was computed, and how #

The model is the one the radius calculations built: Slater p functions for each ion’s outer shell, a Madelung attraction for rock salt, and the four-electron cost of two closed shells written in closed form in the overlap, so that an ion’s orbital range can be changed without changing its orbital energy or the strength constant. Each equilibrium is a golden-section minimum over the overlap grid, and every contracted overlap is checked against the quadrature’s own agreement test before it is used.

The contraction is keyed by the ion rather than by its shell, so potassium’s third-row functions and calcium’s can take different factors in the same calculation. Each factor is a root of the pair’s error less the second-row mean, bracketed by a scan from 1 to 1.5 in steps of an eighth and refined by the Illinois form of regula falsi, since every evaluation is a full equilibrium. The scan stops at 1.5 for a reason: contracted to 1.75, calcium’s p functions against oxide’s at three bohr give an overlap that fails the check recomputing it, which is the quadrature saying the arguments are outside what it can do. 1.5 is also the largest factor the shared-shell test used, and every scanned contraction below it passes.

Seven results are checked. Each middle pair lands on the second-row mean at its ion’s factor, with every overlap accepted and every minimum interior. The factors rise along chloride, potassium, calcium. Potassium chloride at its ions’ factors lands inside the second-row band. The potassium factor potassium chloride needs agrees with potassium fluoride’s to under one and a half per cent, with calcium’s more than three times further away. The shared-shell factors are the means of the per-ion factors of their ions to half a per cent. The per-ion factors span more than four times the gap between the shared-shell conditions. And the refusal: one factor for both cations puts calcium oxide and potassium chloride outside the second-row band.

Where the model stops #

Six pairs, three ions and one prediction. Potassium chloride is the only pair the construction was not fitted on. It is enough for the construction to fail, and a single prediction landing inside a 1.3-point band is not a measurement of how additive the contractions are.

Relative, not absolute. The second-row functions are held at Slater’s values by construction, so a factor of 1.302 says how chloride’s range must change relative to fluoride’s and oxide’s, not that a real chloride ion is 23 per cent smaller than Slater’s rules draw it. Slater’s functions describe one of several senses of an ion’s size, and the repulsion depends on the overlap at contact.

Three readings of the order are confounded. The ion’s charge and its nuclear charge move together along an isoelectronic series, and calcium’s factor comes from the only doubly charged pair, so a rise with the ion’s charge and a rise with the pair’s Madelung attraction look the same here.

And the refused pairs are still refused. Magnesium sulfide and calcium sulfide, doubly charged with a third-row anion, would give sulfide a factor to set beside chloride’s and calcium a second pair to be tested on; the overlap quadrature does not accept them at this model’s separations, which has cost these calculations the same two pairs before.

Who found it, and when #

Slater’s rules, the Madelung constant and the measured separations are quoted, and the contraction of an isoelectronic series with nuclear charge is textbook chemistry. The per-ion factors, the prediction for potassium chloride and the reading of the shared-shell conditions as averages are computed here.

Still open: the strength, and an ion with a second pair #

The obvious open question is still the offset. With every pair’s range set by its own ions, the remaining error is a shortfall of about fourteen per cent common to all six, and a single strength constant is what such a shortfall is for. Refitting it once, with the three factors fixed rather than fitted alongside it, would say whether range and strength together bring every separation within the second-row pairs’ own 1.3-point agreement, or whether something else is still missing once both are set.

The nearer question is additivity. The one-per-cent agreement between potassium’s two factors is one ion checked through two partners. The same check for chloride needs a second chloride pair with a cation whose factor is known, and for calcium a second calcium pair — which is exactly what the refused sulfides would have supplied. A partner for calcium that the quadrature accepts would turn one prediction into two and say whether a contraction really belongs to an ion.