The diffuseness was not the anions'

An ionic model’s closed shells repel through the overlap of their outer orbitals, and the reach of those orbitals is set by their Slater exponents. Three things have been asked of those exponents so far. Three third-row shells needed their own contraction, one per ion, to remove the row pattern from the errors. Then every separation fell about fourteen per cent short, all six to 1.3 points, and one factor on every exponent closed that shortfall at 0.870, leaving a spread of about a point, where a strength constant had made things far worse.

That factor was fitted as one number for seven ions. The argument that found it noted what that concealed: a scale differing between anions and cations was not excluded, it was absorbed. And there was a specific reason to suspect one. Anions are where a free-ion description is weakest, and a model whose diffuseness is short might well be short of it there.

The question takes two numbers, $\lambda_+$ on every cation’s exponents and $\lambda_-$ on every anion’s. By the scaling identity already established, a change of both together is a strength plus a stretch and has been understood, so the only new thing is the ratio $r=\lambda_-/\lambda_+$.

A sharp minimum, at first sight #

Hold the three per-ion contractions where the common scale left them. At each ratio, solve for the size that brings the mean error of the six pairs back to zero, then read how far apart the six still are.

The held curve is close to decisive. It is 5.2 points at $r=0.8$, 2.3 at 0.9, 1.44 at 0.95, and then 0.98 at exactly one, with a corner in it, up through 1.17, 1.70 and 2.32 to 3.31 at 1.25. Put another way, anything but one common factor makes the six pairs worse, and it looks as though the scale belongs to every ion alike.

The reason to distrust that is in how the contractions were obtained. Each third-row factor was chosen to put one pair on the second-row mean, at equal scales. Move the ratio and hold those factors, and the pairs carrying a third-row ion are moved away from where they were set. So some part of the corner at one is the contractions’ own, showing where they were fitted. The earlier argument’s warning about absorption applies just as much in this direction.

The test with nothing absorbed #

The way to see that is to notice that the pairs are not all alike. Two of the six, sodium fluoride and magnesium oxide, contain no third-row ion, and no contraction acts on them. What happens to them depends on the ratio and on nothing else that has been fitted. That suggests a stricter procedure, run at each ratio separately:

Set the size so that the mean of those two is zero. Refit chloride’s contraction to put sodium chloride on that line, potassium’s for potassium fluoride, calcium’s for calcium oxide. Three pairs are then right by construction and three numbers are left over: sodium fluoride and magnesium oxide, whose errors are equal and opposite, and potassium chloride, to which nothing has been fitted at all.

The lower curve in the first figure is what results, and it is a different shape. From a ratio of 0.95 to 1.1 the spread is 1.44, 1.20, 1.14 and 1.35 points. The corner has gone, and so has any claim that one ratio stands out. Refitted, the data do not pick out equal scales, or indeed a value within that band. The sharp minimum was a statement about the order in which things were fitted.

What remains decisive is the outside of the band. Leave the cations at their own exponents, which needs the anions at 0.738, and the two untouched pairs stand at +3.15 and −3.15 per cent, a spread of 6.3 points. Leave the anions alone with the cations at 0.771 and the spread is 2.36. So neither charge alone can carry the scale. And the two are not symmetric: the suspicion that started this, that it was the anions’ diffuseness that was short, is the worse of the two by nearly a factor of three.

Reading the two pairs that cannot hide, they are equal at $r=1.072$, with the cations at 0.845 and the anions at 0.906. That is the opposite of the suggestion: the cations are made slightly the more diffuse. It would be unwise to make too much of that seven per cent, since each pair still carries the model’s other approximations. But its sign does at least mean that the data do not ask for extra anion diffuseness.

Why magnesium oxide and sodium fluoride differ on this #

The two pairs’ different response to the ratio can be measured directly, by moving each scale alone at the common size. In sodium fluoride a one per cent change in either scale moves the separation by about half a per cent, 0.53 for the cation and 0.55 for the anion. In magnesium oxide the two come apart, and not in the expected direction: 0.68 for the cation and 0.45 for the anion. The tighter partner is the one with the larger lever.

That runs against a natural picture in which the soft outer anion is all that matters, and the sum of the exponents gave the first warning that it would. A plausible reading is that the two ions do different jobs in an overlap. The diffuse function decides how fast the overlap falls with distance, and the tight one decides how much amplitude there is to sample that tail with, through its normalisation, which depends strongly on its own exponent. Magnesium oxide has the tightest cation and the most diffuse anion of the set, with exponents of 7.85 and 3.85 before scaling, so it shows the separation of those two jobs most clearly. That reading is an interpretation of two measured derivatives and was not separately calculated. The derivatives are enough to account for the crossing: as the cations are made more diffuse relative to the anions, magnesium oxide lengthens faster than sodium fluoride, and at some ratio they meet.

The pair nothing was fitted to #

Potassium chloride was fitted on nothing, and it is the honest test of the set. At equal scales it stands at +0.82 per cent, which is a fair prediction given that the published version, at slightly different contractions, read +0.03. It rises steadily with the ratio, reaching +1.66 per cent at 1.3, and falls through zero near 0.88.

So the two tests disagree. Sodium fluoride against magnesium oxide asks for a ratio of 1.07, and potassium chloride for 0.88. No value satisfies both, and there is no reason one should, in a model with this little in it. That leaves a floor of about a point in the three residues, whatever is done with the two scales: 1.14 points at the best ratio, and the published 0.98 at equal scales was reached by a different recipe for the same contractions, not by a better one. That number, about a point, has been the model’s limit for several essays, and it is now a reasonable thing to call a limit.

The contractions themselves respond unevenly.

Calcium’s barely moves, and potassium’s settles quickly on the upper side of the band. Chloride’s falls steadily, from 1.32 to 1.24 across the useful range: it is the third-row anion that absorbs the change in the anions’ scale, which is why its one-pair fit and its effect on potassium chloride are bound together. That is the reason one contraction was not enough for the shell in the first place, and it shows the same trade from another angle.

A seventh pair, and what it tested #

The essay before this one proposed a pair the fit had not seen, built from ions already in hand, as the test. Sodium bromide needs a shell not yet in the model, and lithium’s single closed shell has none of the p-type overlap the model’s cost is built from, so neither is quite what the lead imagined. The model does include sulfide, with two measured rock-salt partners, so a slightly weaker form of the test can be run: fit sulfide’s contraction on calcium sulfide at each ratio, then predict magnesium sulfide with nothing else changed.

It fails, and by more than anything fitted has. At equal scales the prediction is 2.85 per cent short, and it does not come within a point and a half of right at any ratio tried. The natural reaction is that sulfide’s contraction was tuned to calcium’s particular pair. But the reason it fails is visible in the crystal, and the failure has a very old name.

In a rock-salt lattice, two anions on neighbouring corners of a face are $\sqrt2$ times the cation–anion distance apart. In magnesium sulfide that is $\sqrt2\times2.602=3.68$ Å. Twice sulfide’s usual radius is 3.68 Å. The anions are touching. Magnesium is too small to hold them apart, so the lattice’s size is set by sulfide against sulfide, and the distance is longer than a cation–anion balance alone would make it. The model counts only cation–anion repulsion, so it has no way of producing that, and it comes out short in exactly the direction this predicts. Calcium sulfide, the fitted pair, has its sulfides 4.02 Å apart and well clear of each other. Magnesium oxide, the pair that gave no such trouble here, has oxides 2.98 Å apart against twice oxide’s radius of 1.40, also clear.

This is the observation that started ionic radii. Landé argued in 1920 that sulfides and selenides of small cations, where changing the cation hardly changes the spacing, had their anions in contact, which gave an absolute size for the anion without any assumption about the cation. So the radius of 1.84 quoted here is not independent of this argument, since radii of this family were calibrated on contact like this: the coincidence is partly circular. But the prediction’s direction and the existence of the mechanism are not, and they turn a failed test into a statement about what kind of test it was. It tests whether the lattice is held apart by what the model says holds it apart, and on magnesium sulfide it is not.

The whole set #

Counted as a fit, the model now has two scales and three contractions for five pairs, and it reproduces them exactly at the crossing ratio; the numbers that test it are potassium chloride, 1.12 per cent, and magnesium sulfide, for which sulfide’s own contraction is a sixth parameter and a seventh pair is then predicted. Given that, a single figure of merit would be misleading. The fair summary is that a band of ratios is equally good, that no charge can be left out, and that no out-of-sample pair was predicted to better than about a per cent.

Where this leaves the question of what the residue of a sum of radii belongs to is a modest but useful place. It is no longer true that the remaining error is a choice between ions. The reach of a closed shell is short by about the same amount in both charges, to the resolution the six pairs allow. That sits well with what the contours of noble gases and charged ions already suggested, which is that a drawn surface is not a property that scales with the charge alone.

What was computed, and how #

Every separation is the minimum of the model’s energy curve, through the same route that produced the published numbers: the Madelung attraction, the four-electron repulsion between shells built from the overlap of Slater p functions at each ion’s scaled exponent, with each overlap on the grid checked against numerical quadrature. On the held line, the size at each ratio is found by secant iteration on the mean of all six errors. On the refitted line it is found on the mean of the two pairs with no contraction, and each contraction by secant iteration on one pair’s error alone, starting from the same values at every ratio. Sodium chloride is fitted to get chloride’s factor, potassium fluoride potassium’s and calcium oxide calcium’s. Sulfide’s is fitted on calcium sulfide with the others fixed.

The claims hold where they are stated: the held spread’s minimum at one, the refitted spread within 1.5 points over the band, one crossing of the untouched pairs between 1.05 and 1.1, and each extreme more than half as far apart again. The refusal is the model’s identity with itself: a refit at equal scales must recover the published size of the scale to a per cent, and it comes to 0.872 against 0.870.

Where the model stops #

A few overlaps fail the check at the far left. At ratios of 0.75 and 0.9, where the contractions on potassium, chloride and calcium become strong against a diffuse partner, two and one of the checked overlaps disagree with quadrature, respectively. The 6.3-point spread at the far left is well-determined for the two pairs without third-row ions, whose overlaps pass, and it is those that carry the conclusion.

Six pairs do not resolve ratios to better than about a tenth. The band is a limit of this data set, and adding pairs is the way to narrow it. Those available are mostly of the wrong structure for a rock-salt model, or carry a new parameter.

The model’s exponents are a convention. Calling one a scale of an ion’s “diffuseness” is a statement about Slater p functions at a stated exponent, not about a density. A real ion’s density in a crystal is shaped by its neighbours in a way no single exponent can express, and that has to be part of why a floor exists at all.

Nothing here touches the Madelung part. Every pair is rock-salt with one Madelung constant and full ionic charges. The error scale of about a point is small against what the covalency in magnesium oxide or calcium oxide might contribute in a fuller treatment.

Anion contact has not been modelled. The argument about magnesium sulfide is geometric and direct, and no energy for sulfide against sulfide has been computed.

Who found it, and when #

Landé’s 1920 calibration from anion contact began the absolute scale. Wasastjerna and Goldschmidt continued it, and Pauling’s radius-ratio rules made the stability limit a familiar part of textbook chemistry, with sulfides and halides of small cations as the standard examples. Slater’s rules, 1930, supplied the exponents. The Born–Mayer form dates from 1932, with Gordon and Kim’s 1972 electron-gas pair potentials a later step toward computing such repulsions from densities. That the scale of an ion’s density is in need of adjustment, and differently for anion and cation in a crystal, has long been recognised in those approaches. What was computed here is the two-scale test on this model, the distinction between its held and refitted forms, and the anion contact as the reason for its one failed prediction.

A minimum is a statement about what was held #

The habit worth carrying is to ask, when a scan finds a sharp minimum exactly at the value where the rest was fitted, how much of it was left behind by that fit. The same was true in a different form of the earlier ratio scans. A contraction fitted at one setting makes that setting look special, and the honest comparison is to refit at every setting and see how much of the preference is left. Here it was enough to turn a verdict into a band.

Still open: the anion’s own repulsion, and a bromide #

The obvious open question is the term magnesium sulfide asks for. An anion–anion repulsion across the face diagonal can be computed from the same overlaps, summed over the twelve second neighbours, and it should matter little for the pairs that have been fitted and a good deal on the contact side. The prediction is specific: magnesium sulfide should lengthen and calcium sulfide scarcely move. If so, the fit’s floor may fall too, since oxides and fluorides with small cations lie nearest that condition.

The nearer question is a pair with no new number. Sodium bromide needs bromide’s fourth-row shell, and the first such shell has no fitted contraction, but potassium bromide then becomes the same sort of test potassium chloride has been, with one new parameter fitted on one pair and another predicted. Unlike magnesium sulfide, that pair has no anion contact to complicate its reading.