The shortfall was a length, not a strength

An ionic model built from Slater p functions, a Madelung attraction and a four-electron cost for two closed shells overlapping gets six measured rock-salt separations wrong in a pattern that follows the row of the periodic table. Contracting the third-row shells removed the pattern, and contracting them one ion at a time — chloride by 1.302, potassium by 1.387, calcium by 1.441 — put the middle group in its own order and landed potassium chloride, fitted on nothing, among the second-row pairs.

What it left is an offset. At the per-ion contractions every pair is short, by −13.56 per cent for sodium fluoride to −14.86 for magnesium oxide, and the six agree with one another to 1.30 points. A model whose errors are all the same size is a model with one thing wrong, and both essays before this one named the obvious thing: the strength of the repulsion. A four-electron cost is proportional to an overlap squared times a constant that the model sets rather than measures, and a common shortfall is what refitting that constant is for.

Refitting it is one number, fitted once, with the three contractions held where they were. The question it answers is the one both essays asked: whether range and strength together put every separation within the 1.3 points the pairs already agree to, or whether something is still missing.

A strength fans the pairs out #

The fit is regula falsi on the mean error, with the six equilibria found by golden-section search on each pair’s energy curve at each trial strength. The mean reaches zero at a strength of 6.15 times the model’s own.

At that strength the six are no longer together. Sodium fluoride sits at −0.03 per cent, the three pairs with one third-row ion between −2.54 and +2.77, potassium chloride at −3.33, and magnesium oxide at +5.47. The spread is 8.81 points — nearly seven times what it was.

And it did not arrive there by accident at the end. The spread grows from the first step: 3.08 points at a factor of two, 6.72 at four, 10.06 at eight, and on to 21.8 at 128. There is no strength that closes the mean and keeps the pairs together, because the pairs start moving apart the moment the strength moves at all.

So strength and range together do not account for the six. At the strength the offset asks for, the model is right on average and nearly seven times worse in the one respect it had got right: the pairs’ agreement with each other.

A strength moves an equilibrium by a length #

Why it fans them out has a short answer. At an equilibrium the Madelung attraction’s slope balances the repulsion’s. If the repulsion falls off as a decaying exponential with a decay length ρ, multiplying its strength by κ is the same as shifting it outward by ρ ln κ — the separation at which the stronger repulsion has fallen to the old one’s value. The attraction’s slope changes slowly by comparison, so to first order every equilibrium moves out by ρ ln κ.

That is a length. A common shortfall is a fraction. The two agree only if every pair’s decay length is the same fraction of its separation.

They are not. Measured from each repulsion’s own logarithmic slope at the per-ion equilibrium, the decay lengths run from 0.282 bohr for sodium fluoride to 0.365 for magnesium oxide, and over the separations that is 0.060 for potassium chloride to 0.108 for magnesium oxide. Times ln 6.15, those predict moves of 10.8 and 19.6 per cent. The moves measured are 10.5 and 20.3, and every pair lies within six per cent of its prediction.

The oxides move furthest because their repulsion is longest compared with their separation. Magnesium oxide and calcium oxide are the most compact pairs relative to the reach of their closed shells — doubly charged ions pulled close by a Madelung attraction four times as strong — so the same strength factor pushes them out by the largest fraction. Potassium chloride, the largest pair, moves least.

This is the whole of the failure, and it is structural. Any constant that multiplies a short-ranged repulsion moves every equilibrium by roughly the same length, and six equilibria at separations from 3.4 to 5.1 bohr cannot all move by the same fraction that way.

A range scale carries them together #

The thing that moves every separation by one fraction is a change of length scale. The model has one: the orbital exponents, which set how far each closed shell reaches. The contractions already fitted change them ion by ion; the question is what one more factor, multiplying every ion’s exponents alike, does to the offset.

It closes it, and it closes it with the pairs together. At a factor of 0.870 on every exponent — every orbital made about fifteen per cent more diffuse, with the per-ion contractions kept in proportion — the mean error is zero and the six sit within 0.98 points of one another: sodium fluoride at +0.63 per cent, the others between −0.35 and +0.08. That is tighter than the 1.30 they started at. Every overlap on the grid passes the same quadrature check the per-ion factors passed.

Set on one plane, the two refits are two paths from the same starting point. The strength path runs towards zero mean while climbing in spread; the range path runs to zero mean along the floor. They answer the same question — close the offset — and only one of them leaves the model better than it found it.

A range is a strength and a stretch #

It works for a reason that can be written down exactly, and the exactness is worth more than the fit.

A Slater function’s shape depends on its exponent ζ and the distance r only through ζr. So the overlap between two of them, with every exponent multiplied by λ and the centres a distance R apart, equals the overlap of the unscaled pair at a distance λR. The four-electron cost depends on nothing but those overlaps, so scaling every exponent by λ turns the repulsion V® into V(λR). The Madelung attraction is −A/R and has no length of its own.

Write R′ for λR. The scaled model’s energy is then −λA/R′ + V(R′), which is λ times −A/R′ + V(R′)/λ — the unscaled model with its repulsion’s strength multiplied by 1/λ. Its minimum is the strength-refitted model’s at κ = 1/λ = 1.149, and the actual separation is that one stretched by 1/λ.

So a common range scale is a strength change of 1.149 followed by a uniform stretch of 14.9 per cent. The stretch moves every separation by the same fraction, which is what the offset needed. The strength part is small and moves the pairs unevenly, by ρ/R times ln 1.149 — between 0.8 and 1.5 per cent — and that unevenness happens to run against the per-ion residue, which is why the spread tightens rather than merely surviving.

The identity is checked rather than assumed: the scaled model’s six separations are computed directly, and computed again as the strength-refitted model at 1/λ stretched by 1/λ, and the two agree to within two tenths of a per cent on every pair — the residue being the interpolation of the repulsion on a fixed grid.

Three fits, side by side #

Read pair by pair, the difference is plain. The per-ion contractions put all six in a tight cluster fourteen per cent short. The strength refit centres the cluster by stretching it: magnesium oxide overshoots by more than five per cent while potassium chloride is still three per cent short. The range scale moves the whole cluster to zero intact.

What this settles about the model is a question this line of essays has carried since it opened: whether the residue of a sum of radii belongs to the pairs or to the compromise a table makes. Here, once each ion’s range is right relative to the others, what is left belongs to a single length — how far every closed shell reaches in absolute terms — and nothing else. The model’s exponents come from Slater’s screening rules, which assign each shell one number from a count of the electrons inside and beside it and were fitted as a compromise across atoms; that one factor corrects them all together is a statement about those rules, not about the six pairs.

What fifteen per cent more diffuse does to a contour #

The line of essays this one belongs to began with drawn surfaces: the contour a table of radii implies, which for three noble gases encloses between 99.38 and 99.75 per cent of the density and collapses once the atoms are charged. The same scaling that makes the identity exact says what the common factor does to any such surface.

A contour drawn at a fixed enclosed fraction is a sphere of some radius r, and the enclosed fraction depends on the exponent and the radius only through their product. Multiply every exponent by 0.870 and every contour, at every enclosed fraction, moves out by 1/0.870 — 14.9 per cent — for every ion alike. The separations moved out by the same 14.9 per cent, less the one per cent or so the small strength part takes back.

So in this model a common change of every ion’s reach and a common change of every contact distance are the same change, which is why one factor could close a common offset and a strength could not: a strength moves contact distances without moving any ion’s contours at all, so it can only close the offset by changing how far into each other the shells are pressed, and that depth differs from pair to pair. Which enclosed fraction the fitted contact distances correspond to is a separate calculation, and whether it is the same for anions as for cations is the question the first essays in this line found a table could not answer.

What the fits cost, and what they tested #

Four numbers have now been fitted to six separations: three per-ion contractions, each chosen to put one pair on the second-row mean, and one common scale, chosen to zero the mean. Counted that way the fit has two separations’ worth of room to fail in, and it used both. Potassium chloride was fitted on nothing, and after the common scale it sits at +0.03 per cent. The two second-row pairs were never fitted against each other, and their disagreement, 1.30 points at the start, is 0.75 after the scale.

In lengths, every one of the six is now within about two hundredths of an ångström of its measured value. That is the comparison the size a confound cannot supply set up when this model was first balanced against its own repulsion: it predicted the six to 0.242 ångström where adding two tabulated radii did so to 0.183, and the model lost. With four fitted numbers it now wins by an order of magnitude, which is less than it sounds — a table of radii is fitted too, one number per ion less one that only trades cations against anions — and more than nothing, because the table spends six numbers on these seven ions and the model four.

What the model does not do, and was never going to do, is settle the residue a table leaves: a fourfold alternating difference of distances that is below the noise of any table built from them. That residue is a statement about additivity, and a model with one length per ion is additive in lengths by construction.

The earlier essays’ reading of the pattern also changes shape. The sum of the exponents carried the row pattern because the third-row exponents from Slater’s rules were too diffuse relative to the second-row ones; the per-ion contractions corrected that ratio; and the common scale now says the absolute value was too compact for all of them together. The two corrections point in opposite directions and are the same kind of statement — about how far a closed shell reaches — at two levels: relative between rows, absolute across the table.

What was computed, and how #

The model is the one every essay in this line has used: Slater p functions for the outer shell of each ion, at the exponents Slater’s rules give and the per-ion contractions multiply; the four-electron cost of two closed p shells overlapping, summed over the three p components along the internuclear axis and interpolated on a fixed grid of separations; the Madelung attraction of the rock-salt lattice; and six pairs whose measured separations are tabulated. Every equilibrium is a golden-section minimum over the grid, and every one is checked to lie inside it.

The strength refit multiplies the four-electron cost by κ after it is built, so the overlaps are exactly the per-ion model’s. The range scale multiplies every ion’s exponent, contraction included, by λ and rebuilds every overlap, each checked against the quadrature on every grid point. Each fit is regula falsi on the mean relative error.

The checks, run wherever these figures are drawn: at unit strength the second-row pairs are the unscaled model’s and potassium chloride is the per-ion prediction; one strength zeroes the mean; each pair moves in the order of its decay length over its separation and within six per cent of that times ln κ; the spread grows with the strength at every step and the refit opens it more than five-fold; one range scale zeroes the mean with a spread below the starting one, every overlap passing the quadrature; and the range scale agrees with a strength of 1/λ stretched by 1/λ to two tenths of a per cent. The refusal is the strength of one, which must reproduce the per-ion contraction exactly.

Where the model stops #

One exponent per ion. The identity holds for any orbital scaled as a whole, however many exponents it has; what one exponent per shell limits is the fit. A single Slater function has one decay length, a real ion’s density falls off with more than one, and a common scale on one exponent stands in for a change in the whole tail that a better orbital might distribute differently between its inner and outer parts.

A fitted scale, not a derived one. The factor 0.870 is one number fitted to six pairs, and its success is that the spread fell, which it was not fitted to do. It is not a prediction of what a better orbital would give, and a Hartree–Fock ion’s outer-shell tail would be the test of whether fifteen per cent is the right amount of diffuseness or only the amount this model needs.

And the pairs share ions. Six pairs made of seven ions do not give six independent checks of a common scale; they give the check that one scale serves every ion in every partnership, which is weaker than six and stronger than one.

Who found it, and when #

Slater’s rules, the Madelung constant and the measured separations are quoted, and the scaling of a Slater function’s shape with its exponent is textbook. The strength refit, the reading of each move as a decay length times the logarithm of the factor, the common range scale, and the identity that makes a range scale a strength and a stretch are computed here.

Still open: whether the scale belongs to the ions, and a seventh pair #

The obvious open question is whether the common factor is really common. It was fitted as one number for seven ions, and the spread it leaves is under a point, but the per-ion factors were fitted before it with the second-row ions held at one — so a scale that differs between anions and cations is not excluded, it is absorbed. Fitting two scales, one for each charge sign, with the six pairs’ spread as the test would say whether the diffuseness the model lacks belongs to the anions, as the neutral-atom origin of the exponents would suggest, or to every ion alike.

The nearer question is a pair the fit has not seen. Everything here is six separations and seven ions; a seventh measured pair built from ions already fixed — lithium fluoride, or sodium bromide once bromide’s shell is in the model — would be predicted by the per-ion contractions and the common scale with nothing refitted, which is the test this line of essays has used each time a construction has been fitted.