One contraction for two conditions

The error was the row, not the charge found that an ionic model — a Madelung attraction balanced against the repulsion two closed shells produce through their overlap, checked against six measured separations — puts two compact second-row ions about fourteen per cent too close together and two diffuse third-row ions eighteen per cent too far apart, with the mixed pairs in between. It proposed a mechanism: the model’s overlap reaches too far for a diffuse shell. And it proposed a test, which is the part worth quoting exactly, because the whole of this essay turns on it:

If the overlap’s range is too long, then stiffening it — using a shorter-ranged repulsion at the same strength — should move the two-third-row pair down and the two-second-row pairs up, and should move the mixed pairs least. That is a one-parameter change to the model with a predicted sign pattern rather than a fitted improvement, which makes it a test rather than a tuning.

The sum of the exponents, not the softer ion then found that the shell count was standing in for a continuous quantity, and that the obvious continuous version of the mechanism — how diffuse the softer ion is — carried nothing. The range explanation had lost its most natural form. It still had the test.

The test as proposed cannot fail. A version that can was built and run, and the range explanation passes it.

Stiffening about a separation #

“A shorter-ranged repulsion at the same strength” has to be made precise before it can be run, and the natural way to make it precise is to hold the repulsion’s value fixed at some separation and make it fall off faster on either side. Choose a separation — five bohr, between where the compact pairs sit and where potassium chloride sits — keep each pair’s repulsion unchanged there, and raise the steepness of its logarithm by a quarter.

The model’s own repulsion does exactly what the prediction said. Magnesium oxide moves out by 0.50 bohr and sodium fluoride by 0.35 — the second-row pairs, whose errors are short, go up. Potassium chloride moves in by 0.33 — the pair of two third-row ions, whose error is long, goes down. The three mixed pairs move least: calcium oxide by 0.12, potassium fluoride by 0.02, and sodium chloride by essentially nothing.

Then the same construction was applied to a repulsion with no shells in it. Every pair gets one exponential of one range, 0.6 bohr, with its size set only so that the pair’s own equilibrium is unchanged; nothing about 2p or 3p, compact or diffuse, enters it anywhere. Steepened about the same five bohr, it moves magnesium oxide out by 0.58 and sodium fluoride by 0.46, potassium chloride in by 0.34, and the mixed pairs least.

The pattern is the pivot’s. A repulsion held fixed at five bohr and made to fall faster is larger inside five bohr and smaller outside, so a pair whose equilibrium is inside is pushed out and a pair whose equilibrium is outside is let in — and under both repulsions the shift simply falls with the pair’s own separation, from the most compact to the most diffuse. The second-row pairs sit inside the pivot because they are compact; potassium chloride sits outside because it is not. The sign pattern the test predicted is a statement about where the pairs sit, which the shells decide, and the construction would have produced it from any repulsion whatever. The two repulsions differ in exactly one place, where they cross zero: sodium chloride, the pair nearest the pivot, moves in by a thousandth of a bohr under one and out by four hundredths under the other.

It is the same trap the control that outranked the mechanism sprang on these same pairs from the other side. There a label that could not be a mechanism sorted the errors as well as the proposed one; here a repulsion that cannot carry the mechanism reproduces the predicted response. In both cases the check that exposes it is the same: run the argument on something that lacks the ingredient, and see whether the result survives.

Contracting every shell #

The obvious repair is to stop choosing a separation. The range of an overlap repulsion is set by how far the orbitals reach, and a Slater function’s reach is its exponent, so contract every p function by the same factor and let each pair find its own new equilibrium. Nothing is pivoted and nothing is chosen but the factor.

It has to be done carefully, because the model builds each ion’s orbital and its orbital energy from one effective charge, and contracting the orbital through that charge would deepen its energy too and change the repulsion’s strength along with its range. The four-electron cost of two closed shells has a closed form in their overlap — twice the sum of the two orbital energies, times S²(1 − K)/(1 − S²), which is the antibonding level rising further than the bonding one falls summed over four electrons — so the range can be changed on its own: contracted orbitals for the overlap, the model’s own energies and constant for everything else. With nothing contracted, the result is the model’s repulsion again, to within the model’s own rounding.

At a factor of 1.1 every separation shrinks, by between 9.7 and 10.2 per cent. The six errors all move down together. The spread between the most negative and the most positive narrows from 32.7 percentage points to 30.1, which is less narrowing than rescaling every distance by the average change would give, and the root-mean-square error rises from 11.4 per cent to 14.4.

That is not a test either, for the opposite reason. A change that shortens every separation by nearly the same fraction cannot move some pairs up and others down, so it cannot produce the predicted pattern whether the explanation is right or wrong. It asks a different question — whether the model’s overall length scale is off — and answers that one clearly: making every shell more compact makes the model worse.

The version that can fail #

What the explanation actually claims is narrower than either construction. It says the third-row shells reach too far compared with the second-row ones. So contract the third-row p exponents alone and leave the second-row ones exactly as they are.

That has a property neither earlier construction had. The two second-row pairs contain no third-row ion, so their errors do not move at all: they stay at −14.9 and −13.6 per cent, mean −14.2. Everything else moves, and there are two things it has to do. The three pairs with one third-row ion have to come down onto the second-row pairs, and potassium chloride, with two, has to come down onto them too. If a third-row range defect is what separates the groups, one factor will do both. If it is not, the factor that aligns one group will leave the other stranded, and nothing in the construction makes the two factors agree.

The pairs with one third-row ion come onto the second-row pairs at a factor of 1.3765. Potassium chloride comes onto them at 1.3463. The two factors are 2.2 per cent apart.

Whether that is close enough needs a criterion stated before the answer, and the one that makes sense is the model’s own scatter within a group. At the factor that aligns the middle group, potassium chloride is left 2.0 points below the second-row mean; at the factor that aligns potassium chloride, the middle group is left 1.1 points above it. The three middle pairs themselves span 5.2 to 5.3 points at the two factors. So each group is misaligned by less than the spread inside the middle group, and by about as much as the 1.3 points separating the two second-row pairs from each other. One contraction meets both conditions to within the scatter the model has without any contraction at all.

Nothing in the construction guaranteed that. The two conditions involve different pairs, one with one contracted ion and one with two, at different separations and charges, and nothing about contracting a shell ties the factor at which three mixed pairs arrive to the factor at which potassium chloride does; the two curves in the figure were free to cross the second-row line far apart. The explanation made a prediction that could have failed, and it did not.

What is left #

At the aligning contraction the shell pattern is gone. The six errors, which spanned 32.7 points as built, span 5.2. What is left is two things, and neither is the pattern.

One is an offset. Every pair now sits near the second-row pairs’ −14 per cent rather than near zero, so the root-mean-square error is 14.7 per cent against 11.4 as built. The model was never going to be made accurate by fixing a range: the second-row pairs, which the contraction does not touch, were already fourteen per cent short, and a uniform shortfall is a statement about the repulsion’s strength — the constant K, which was fitted on noble-gas contacts and carried across to these ions unchanged, or the Madelung balance — rather than about its range.

The other is the middle group’s own order. Sodium chloride, potassium fluoride and calcium oxide keep exactly the order they had as built at every contraction from 1.0 to 1.5, and their spread falls only from 6.1 points to 5.2. A change that acts on every third-row shell at once moves all three together and cannot reorder them. That is the half of the slope the exponent sum could not account for, met again from the other side, and it is now clearly a separate question from the pattern between groups.

Read together, the spread and the root-mean-square error say what kind of result this is. The spread falls from 32.7 to just over 5 at the two aligning factors, stays there to a factor of 1.4 and rises again beyond it; the root-mean-square error first improves slightly, to 8.9 per cent at a factor of 1.1, and then rises steadily. The first measures the pattern and the second measures the model, and the contraction repairs the first while spending the second.

How large the defect is #

A factor of 1.35 to 1.38 on an exponent divides a Slater function’s mean radius by the same amount: the third-row p functions would need to be 26 to 27 per cent smaller for this model’s errors to lose their dependence on the row. That is a large correction to a set of screening constants, and it is worth saying what it is a correction to.

It is not a claim that real chloride or potassium ions are a quarter smaller than Slater’s rules make them. The test holds the second-row functions fixed by construction and moves only the third-row ones, so what it measures is how the third-row range must change relative to the second-row one; it cannot tell a third-row shell that reaches too far from a second-row shell that does not reach far enough, and nothing here has run the second-row version. And Slater’s rules describe an orbital’s size in one sense among several, while the repulsion depends on the overlap at contact distances, which is a different sense again — and even the tabulated radii of these ions enclose different fractions of their density depending on their charge. What the test establishes is that, within this model, the shell pattern is fully accounted for by one relative range and nothing else about the shells.

How it was computed #

Every separation is the minimum of a Madelung attraction for a rock-salt lattice against six neighbours’ closed-shell repulsion, the same balance the shell-count essays measured against, found by golden-section search over a grid of separations on which the repulsion is interpolated. The repulsion is the four-electron cost summed over three pairs of p orbitals, from overlaps integrated on a quadrature grid whose own agreement check is run at every contracted exponent; no contracted overlap was refused and every equilibrium is an interior minimum.

The two aligning factors are found by regula falsi from the bracket a scan over 1.0 to 1.5 supplies, so each is located in a handful of equilibria rather than a bisection’s thirty.

The checks are these. With nothing contracted, the decoupled repulsion reproduces the model’s at every separation the model resolves, to within the model’s own rounding, and reproduces the six errors the earlier essays reported. Steepened about five bohr, the model’s repulsion moves the second-row pairs out and potassium chloride in, a shell-blind exponential does the same, both move the mixed pairs least, both shifts fall steadily with separation, and the two cross zero within one pair of each other. Contracting every exponent shrinks every separation by nearly the same fraction, narrows the spread by less than a rescaling would and worsens the model. Contracting the third-row exponents alone gives two aligning factors within three per cent of each other; at either, the misalignment left is smaller than the middle group’s spread; the pattern falls from over thirty points to under six; and the middle group keeps its order and nearly its spread.

One thing turned up along the way that belongs to the model rather than to the test. The model evaluates its four-electron cost as a difference of two nearly equal energies, which loses precision where the overlap is tiny, and beyond about eleven bohr it cancels to nothing for three of the six pairs. Every equilibrium here lies between 3.4 and 7.0 bohr, where the two evaluations agree, so none of the equilibrium separations quoted for these pairs is affected.

What one contraction factor cannot settle #

The test contracts one shell against another. It cannot say which shell is wrong, only that their ranges are wrong relative to each other by about a third; expanding the second-row functions instead is a different construction, since the second-row pairs would then move too, and it has not been run.

Six pairs make three groups, and the two conditions are each carried by few pairs — potassium chloride alone carries one of them. That is enough for the test to fail if the explanation were wrong, and not enough to say that the factor is known to better than a few per cent.

The offset is left alone. Refitting the strength at the aligning contraction would say whether one constant then brings all six within the middle group’s scatter, and it is a different repair from the one tested here; putting both into one fit would turn a test back into a tuning.

And the two refused separations, magnesium sulfide and calcium sulfide, are again the pairs that would have mattered — doubly charged, with a third-row anion — because they would give the middle group a fourth member and potassium chloride a companion.

A test that could have come out the other way #

The transferable point is about what makes a prediction a test.

The proposed stiffening predicted a sign pattern rather than a fitted improvement, and said explicitly that this made it a test rather than a tuning. That was the right instinct and it was not sufficient. A sign pattern is only evidence if the construction producing it could have produced a different one, and a pivot between the compact and diffuse pairs cannot: it hands the second-row pairs one sign and the third-row pair the other before any physics enters. The prediction was correct and the test was empty.

The repair was to find a construction with two conditions and one free parameter. Two independent requirements met by a single number is a structure that fails unless the explanation is right, and it can be checked for emptiness the same way the pivot was — by asking whether anything in the construction forces the two factors together. Nothing does, and they came out 2.2 per cent apart.

That shape — more conditions than free parameters, and a check that the agreement is not built in — is what a test is, and it is worth asking of any proposed test before it is run rather than after.

Where the pieces come from #

Slater’s rules, the Madelung constant for rock salt and the measured separations are quoted; the four-electron cost of two closed shells is the standard two-orbital result with its overlap kept. The shell pattern and its proposed test come from the shell-count essay, and the exponent-sum analysis from the one after it. The shell-blind control, the three constructions and the two aligning factors are computed here.

The shell-count essay deserves the credit for writing down a test sharp enough to be checked for emptiness — and for saying, in advance, that a uniform improvement would refute the explanation, which is exactly the statement that made the uniform contraction’s result readable.

Still open: the strength, and the middle group’s order #

The obvious open question is the offset. At the aligning contraction every pair is about fourteen per cent short, and that is the kind of error a single strength constant exists to absorb. Refitting K once, with the contraction fixed rather than fitted alongside it, would say whether range and strength together account for all six separations to within the middle group’s scatter — or whether something else is still missing once both are fixed.

The nearer question is the middle group’s order, which neither the exponent sum nor any contraction of a whole shell can reach. Its three pairs differ in which ion carries the third-row shell and in charge, and calcium oxide, the doubly charged one, sits at the top of the group throughout. A per-ion contraction — the cation’s third-row shell and the anion’s separately — is a two-parameter version of the same construction with three conditions on it, and it would say whether the order follows the ion that is diffuse or the charge the pair carries.