Six of fifteen change verdict

A pair test was run against every candidate input and every one of them failed it, and closed by naming the number the whole construction rests on:

The capacity is finite or infinite according to the sign of the cubic coefficient, and that coefficient is the highest derivative of a fit to four points, so it carries whatever the fit’s error carries. Recomputing the capacity from five electron counts rather than four, and seeing which atoms change sign, would say how much of the ordering here is the quantity and how much is the fit’s resolution.

A third of it is the fit’s resolution.

The fifth point, and where it goes in the fit #

The cubic is exact through four electron counts — the anion, the neutral atom, the cation and the dication — which is what fixes its four coefficients with the energy of the neutral atom at zero. The natural fifth point is the trication, whose energy above the neutral atom is the sum of the first three ionisation energies.

Writing the energy as a quartic in the charge and imposing all five conditions has a shape worth reading. The combination that comes out as the electronegativity is still the mean of the ionisation energy and the affinity, and the combination that comes out as the hardness is still half their difference — those are properties of the anion and the cation, and adding points further up does not touch them.

The cubic’s coefficient is a different matter. What the cubic called its third-order term is, in the quartic, the combination c + 2d — so the quantity treated there as a third derivative is a mixture of the quartic’s third and fourth, and the mixing is what a fifth point resolves. That identity is checked rather than trusted: the quartic must reproduce the cubic’s coefficient exactly from the four points the two fits share, or the second fit is a different fit rather than an extension of one.

The capacity generalises exactly too. In the cubic it is the charge at which the chemical potential stops falling — where the two roots of the fixed-point equation collide — and that condition is a linear equation in the charge. In the quartic it is a quadratic, so the capacity is the root nearest zero on the electron-gaining side, and is unbounded when there is none.

Two of the seventeen atoms are dropped rather than estimated. Bromine and iodine’s third ionisation energies are the least firmly established of the set, and the highest coefficient of a five-point fit is exactly the wrong place to put a doubtful number.

Dropping them costs something worth naming. Both are halogens, both have finite capacities under the cubic — 6.37 and 5.74 — and both sit in the band the fifth point turns out to unbound: chlorine, at 6.28, is between them and goes to unbounded. So the two atoms left out are the two whose verdict this essay would most like to have, and the sensitivity computed below says what to expect of them: bromine flips on 5.6 per cent of one measurement and iodine on 6.2, which puts both in the group that a fifth point reaches. The prediction is on the record and the measurement to test it is not quoted here.

Six change, and in both directions #

Carbon, nitrogen, fluorine, silicon and chlorine go from a finite capacity to an unbounded one. Beryllium goes the other way.

Six of fifteen, and the two directions matter: a shift in one direction would be a systematic correction and could in principle be absorbed, and one in both cannot. What a fifth point does to this construction is not a bias but a reshuffle.

And the five it unbounds are not a random five. They are exactly the five largest finite capacities the cubic reports — nitrogen at 31.5, chlorine at 6.28, silicon at 4.72, fluorine at 3.98 and carbon at 3.20. Every atom the cubic placed at the top of its ordering is placed outside the ordering entirely by one more measurement.

That is the answer to the pair test’s question in its sharpest form. The large end of the capacity ordering is not the quantity; it is the fit running out of resolution, and the number 31.5 was never a number.

Beryllium crosses the whole table #

The single most dramatic movement is worth its own paragraph, because it undoes a headline.

The essay that found the capacity largest where there is none reported that the three atoms whose anions are not bound — beryllium, magnesium and nitrogen — have the largest capacities in the set, two of them unbounded. That was the finding: the atoms least able to hold an extra electron are the ones the model says can take the most, which is the model failing its own test.

Beryllium is one of the two unbounded ones. Under five points its capacity is 0.0122 of an electron, the smallest value in the entire set.

So one of the two unbounded capacities in that finding is, on one more measurement, the most tightly bounded capacity there is. The finding’s shape survives — magnesium stays unbounded, nitrogen becomes so — but the atom that crossed did not cross a little. It crossed from one end of the ordering to the other.

The survivors keep their order and lose their values #

Seven atoms are finite under both. Their two orderings are identical — lithium, sodium, potassium, aluminium, boron, oxygen, sulfur, in that order under the cubic and in that order under the quartic, with no pair swapping.

Five of the seven change value by more than a fifth, and three of them roughly halve.

That combination is the opposite of the usual shape. Normally a quantity’s values are more robust than its ordering, because an ordering is decided by differences and differences are where errors accumulate. Here the small capacities keep their order while losing their sizes, and the reason is that they all move the same way: the fourth-order term is a correction of similar sign and similar relative size across the small end, so it slides them together.

The two that barely move are sodium and potassium, and why is the arithmetic of the whole argument. Their cubic coefficient is among the largest in the set — 6.26 and 3.91, against nitrogen’s 0.077 — because their second electron comes out of a closed shell and the ionisation series has a violent kink there. A large coefficient is a large second difference, and a large second difference is one a fifth point has little to correct.

What the movement does to three essays that used these values #

The capacity was not built to be interesting on its own. It was built because the equalisation model needed something to bound how much charge an atom could take, and the three essays after that were spent finding out what it is a function of.

So the movement has to be read against those. The essay that correlated it against the second ionisation energy found 2.9 per cent of the variance accounted for, and its striking pair was nitrogen against potassium — two electronvolts apart in the second ionisation energy and two hundredfold apart in capacity. Under five points nitrogen has no capacity at all, so the pair is gone and the number it produced with it.

What survives is the conclusion and not the demonstration. The capacity is still not a function of the second ionisation energy, because the seven atoms that keep a finite value are still spread over an order of magnitude with no relation to it. But the 2.9 per cent was computed over a set that included five atoms whose values were unbounded and did not know it, and the two-hundredfold ratio was the largest number in an essay whose point was that the ratio was large.

That is the honest cost of this essay and it is worth saying plainly rather than at the end. Three earlier arguments used the capacity’s values as evidence, and the values at the large end are gone. The arguments’ conclusions were all negative — the capacity is not this, is not that — and a negative conclusion drawn from numbers that turn out to be unbounded is still negative, since an unbounded value is no more a function of anything than a large one is.

And the fifth point was not needed to know this #

The cubic’s coefficient is one sixth of I₂ − 2I₁ + A. Flipping the capacity between bounded and unbounded means putting that at zero, which is an additive shift of six times the coefficient — a definite number, computable from the four points alone.

Expressed as a fraction of the measurement it would have to move, it separates the set cleanly. Nitrogen flips on a change of 1.56 per cent in its second ionisation energy. Beryllium flips on 5.1, magnesium on 4.4, phosphorus on 2.2. Sodium needs 79 per cent and potassium 74.

Line those up against which atoms a fifth point actually reached and they are the same atoms. Every one that flips needs under seven per cent; every one that holds needs more than ten. So the question that could not be answered without a fifth point was answerable without one, and the answer would have predicted the fifth point’s own result.

That is worth stating as a method rather than as a coincidence. The capacity is η/3γ, so its relative error is the relative error of γ — and γ is a second difference divided by six, so a fractional error in any one of three measurements is multiplied by the ratio of that measurement to six times the coefficient. For nitrogen that ratio is sixty-four. A quantity built from a second difference of three numbers of order twenty, coming out at 0.077, is a quantity with two decimal places of cancellation in it, and nothing about the arithmetic hides that.

What was computed, and how #

Every energy is quoted. The first ionisation energies, the electron affinities and the second ionisation energies are the ones the earlier essays use; the third ionisation energies are the new quotation and are the whole of the fifth point.

Both fits are exact through their own points — no least squares anywhere — so a coefficient is a combination of measurements rather than an estimate, and the only thing a fifth point does is change which combination.

Twelve things are checked. That the quartic reproduces the cubic’s coefficient exactly, on every atom, from the four points the two share. That several atoms change verdict, and that they change in both directions. That the ones unbounded are exactly the largest the cubic reported, compared as sets rather than described. That the atoms finite under both keep their order exactly. That most of their values move by more than a fifth, and that the ones that do not are the ones with the largest coefficient. That beryllium goes from unbounded to bounded, and that its bounded value is the smallest in the set. And the sensitivity: that the largest finite capacity flips on a couple of per cent in one measurement while the small ones need tens of per cent.

Where this stops #

A quartic is not the truth either. It is a fit through five points and its own highest coefficient carries the same kind of amplification the cubic’s did — less of it, because the second difference being resolved is now a third difference of five numbers rather than a second of four, but the shape of the problem is unchanged. A sixth point would move things again, and the honest claim is about what a fifth point does rather than about what the capacity is.

The exact energy is not a polynomial at all, which is the deeper version of the same reservation. It is straight segments between integers with a kink at each one, so every derivative above the first is a property of a fit rather than of an atom — and the whole family of smooth models is an approximation whose errors are not obviously smaller for a higher degree.

And the third ionisation energies carry the fifth point’s whole weight. Each is quoted to four or five figures and two are quoted to four; the coefficients they enter are differences, so their relative uncertainty is amplified exactly as the second ionisation energies’ is in the cubic. The finding that six atoms change verdict is robust to that, because the changes are large. The values of the quartic capacities are no more trustworthy than the cubic’s were.

The generalisation #

The habit is to compute, before adding a measurement, how large a change in the measurements already in hand would move the answer.

That is cheaper than the extra measurement and it predicts what the measurement will do. Here it would have said in advance which six atoms a fifth point could reach and which nine it could not, from three numbers per atom and a division — and it would have said it without quoting fifteen new energies or building a second fit.

The corollary is about where cancellation hides. A second difference of three numbers of order twenty coming out at a tenth has lost two orders of magnitude to cancellation, and every relative error in the inputs is amplified by that factor. Nothing about the expression announces it; the coefficient is computed, printed and used, and the amplification is visible only by asking what shift in an input would zero it. A quantity worth that question is any quantity defined as a difference of differences, which is most of what a higher derivative is.

Who found it, and when #

The quadratic model of an atom’s energy against electron count is Iczkowski and Margrave, 1961, and Parr and Pearson’s identification of the hardness is 1983. The cubic and the capacity are the earlier essays’ own construction. The ionisation energies and affinities are quoted. What is computed here is the quartic through five counts, the capacities it gives, and the shift in each measurement that would flip each verdict.

The number worth carrying is not 0.0122. It is 1.56 per cent: the change in one measured energy that moves nitrogen’s capacity from the largest finite value in the set to unbounded, computable from the four points already in hand before it.

Still open: an input the cubic cannot contain #

The obvious open question is the one the pair test named and this essay has not touched. Every input tested against the capacity so far is inside the cubic — the coefficients, and the three energies they are built from — so the search was constrained to fail from the start, and what is wanted is a quantity the fit was not made from. The pair test proposed the atomic radius and named the difficulty: radii are tabulated for other purposes and not on a scale all these atoms share. A size computed rather than quoted removes the difficulty entirely — a hydrogenic orbital at the effective charge Slater’s rules give an atom’s valence shell has an exact mean radius, on one scale for every atom by construction, and the cubic knows three energies and no length at all.

The nearer question is what the sensitivity analysis says about the other quantities built the same way. The capacity is the most amplified because it is a second difference divided by six, but the hardness is a first difference and the electronegativity a first sum, and neither has been priced this way. Running the same solve on them — what shift in a measurement would move the hardness by ten per cent, or reverse two atoms’ electronegativities — would say which of the three quantities has been computed robustly enough to carry an argument. It is one division per atom per quantity and the numbers are already in hand.