A size the fit was not made from

The pair test was put to the chemical capacity against every candidate input the cubic contains and every one failed, and it said why the exercise had been circular:

Everything tested here is inside the cubic, so the search was constrained to fail from the start; the interesting version needs a quantity the cubic was not fitted to. The atomic radius is the obvious candidate, it is uncontroversial, and radii are tabulated for other purposes — but not on a scale all fourteen of these atoms share.

The difficulty is in the word tabulated.

A size that is computed #

A tabulated radius is a compromise between measurements made for other reasons. Ionic radii come from crystal spacings and are defined only for a stated coordination; covalent radii come from halving bond lengths and are defined only for elements that bond that way; van der Waals radii come from contact distances in solids. There is no single table covering lithium, carbon, iodine and potassium on one footing, which is exactly the difficulty that made the pair test worth running.

Computing one removes the difficulty rather than working around it. A hydrogenic orbital with principal quantum number n, angular momentum l and effective nuclear charge Z has a mean radius of (3n² − l(l+1))/2Z in units of the Bohr radius, exactly, and a root-mean-square radius with a closed form of the same kind. Give each atom its valence shell and the effective charge Slater’s rules assign that shell, and every atom in the set has both numbers on one scale by construction.

And neither number is anywhere in the cubic. The cubic is fitted to three measured energies and nothing else; a Slater effective charge is a function of the nuclear charge and the electron configuration, and a mean radius is a function of that and the quantum numbers. The fit could not have produced either, which is precisely what the pair test was asking for.

The sizes it gives are ordinary. Fluorine’s valence 2p has a mean radius of 0.96 bohr and potassium’s 4s has 10.9 — a factor of eleven across the set, with the two rows and the two long-period elements in the right order and the right places. Nothing about them is a finding; they are an input.

Two definitions rather than one, because a single definition would leave the answer resting on a choice. If the test says the same thing about both, the answer is about size rather than about which size.

The test, and the floor it is read against #

The test is the pair test’s, unchanged. Sort the atoms by the candidate input, take every adjacent pair, and report the largest ratio of capacities among them. An input the capacity is a function of would have no adjacent pair with a large ratio — that is what being a function means at the resolution the set allows.

Adjacency is by rank rather than by value, which the pair test established is not a detail. Run on a raw scale the test reports a capacity ratio of 43.9 for the capacity against itself, because the capacities span two orders of magnitude and a tenth of a linear range is enormous in ratio terms. A proximity test on a raw scale is not a proximity test on the quantity.

The control is the capacity sorted against itself, and it is the floor. With fourteen atoms spread over two orders of magnitude, even neighbours in the capacity itself differ — by up to 4.95, between bromine and nitrogen. No input can beat that, so every number here is read against it.

Sixteen times the floor #

The worst adjacent pair in the mean radius is lithium and iodine, whose capacities differ by a factor of 78.7.

That is sixteen times the floor. And it is not a single outlier: the median adjacent ratio is 6.92, itself above the floor’s worst case, so the typical neighbouring pair in size differs in capacity by more than the worst neighbouring pair in the capacity itself.

Lithium and iodine are the pair worth looking at. Lithium’s valence 2s has a mean radius of 4.62 bohr and iodine’s 5p has 4.80 — five per cent apart, adjacent in the sorted list with nothing between them — and their capacities are 0.073 and 5.74. Two atoms of essentially the same computed size, one with the smallest capacity in the set and one near the largest.

So the answer the pair test was put to is no, and it is no by a wide margin. A size is not a quantity the capacity is a function of.

The pair is worth one more sentence because of what the two atoms are. Lithium’s valence electron is a 2s at an effective charge of 1.30 and iodine’s is a 5p at 7.60 — three principal quantum numbers apart, with nearly six times the charge, arriving at the same radius because the two effects cancel. So the coincidence in size is not a near-coincidence in anything else, and an input that cannot tell them apart is an input that has thrown away the whole of what distinguishes them. A size is one number extracted from a configuration, and the extraction is lossy in exactly the direction that matters here.

One thing about the test is worth stating before the second definition, because it explains why the second exists. The worst adjacent pair’s ratio is a ratio of capacities, and the input’s only job is to say which atoms are adjacent. So the number depends on the input through the order it induces and through nothing else: rescale the input by any strictly increasing function — take its logarithm, cube it, divide it by the periodic table’s row number — and every adjacent pair is the same pair and the worst ratio is unchanged to the last bit. The test is a statement about a quantity’s order, which is the property the argument about slope floors on one scale turns on, arriving here from the other side.

That is why two definitions of size are a real check rather than a second look at the same thing. If the root-mean-square radius were a strictly increasing function of the mean radius across these atoms, the two would be the same input to this test and running both would prove nothing. They are not: the two orders disagree — silicon and bromine swap — and the multiset of adjacent ratios differs enough to move the median from 6.92 to 6.20. Two inputs that order the set differently, agreeing on the verdict, is evidence; one input written twice is not.

Both definitions, one verdict #

The root-mean-square radius orders a few atoms differently from the mean — silicon and bromine swap, and so do a couple of others — and it produces the same worst pair at the same ratio of 78.7, with a median of 6.20 against 6.92.

So the verdict is about size and not about which size. That is the whole reason for computing two, and it is the sort of check that is cheap before the fact and impossible after: an argument resting on one definition of a quantity with several is an argument whose next reader will ask about the others.

Where it lands among the six already tested #

The pair test was run against six inputs, all of them inside the cubic, and reported their worst adjacent pairs against the same floor. Placing the size in that list is the comparison this essay exists to make.

The six run: the cubic coefficient at 7.94, the hardness at 10.45, the electronegativity and the first ionisation energy both at 24.25, the electron affinity at 43.86, and the second ionisation energy at 192.57. The floor is 4.95.

Size, at 78.7, is fifth of the seven. It is worse than four of the six quantities the cubic is built from and better than only one of them — the second ionisation energy, which is the one the pair test found worst and is the one its own question had named.

That placement is the useful form of the negative result. An input from outside the fit was supposed to be the interesting case precisely because the inside ones were guaranteed to fail; it turns out to fail more than most of them. So the capacity is not merely a quantity nothing inside the cubic predicts — it is a quantity for which the best predictor available is a coefficient of the cubic itself, at 7.94 against a floor of 4.95, and everything else is further away.

Which leaves the cubic coefficient as the only near miss in seven attempts, and it is not an independent input at all: the capacity is the hardness over three times that coefficient, so a coefficient that nearly predicts the capacity is a ratio nearly predicting its own denominator. The pair test said as much about it. Seven attempts, and the only one that comes close is the one that cannot count.

What the correlation says, and why it is not the question #

Size and capacity have a rank correlation of −0.433. Larger atoms tend to have smaller capacities, and the association is respectable — not strong, and not nothing.

That is not a contradiction of anything above and it is worth being explicit about why. Both quantities track the periodic table. Size falls across a row and grows down a group; the capacity, being built from ionisation energies, does something related. Any two quantities that both follow the table will correlate, and a correlation of that origin is a statement about the table rather than about either quantity.

The pair test asks something the correlation does not. It asks whether two atoms the input cannot distinguish have capacities the input could have predicted, and the answer is that atoms five per cent apart in size are eighty-fold apart in capacity. A quantity can correlate respectably with another and still fail to be a function of it at any useful resolution, and this line of work has met that before: three essays earlier a rank correlation of 0.857 was matched by a control that could not possibly be a mechanism, which is what made the correlation worthless as evidence.

So the negative result here is a stronger negative than a small correlation would have been, because it survives a respectable one.

What was computed, and how #

Each atom’s valence shell is stated rather than derived — the outermost occupied subshell of the neutral atom’s configuration — and the effective charge comes from Slater’s rules applied to that shell. Both radii are closed forms: no quadrature, no grid, no fit.

Six things are checked. That the test runs on the atoms with a finite capacity and a valence shell, which is the same set the pair test used. That both definitions of size have a worst adjacent pair differing by several times what the control allows, checked separately for each so that either could fail alone. That the two definitions agree about how badly it fails, within a factor of three, which is the statement that makes the verdict about size. That the control gives the smallest worst-pair ratio of any ordering, which is what says the test measures the quantity rather than the sort. And that the rank correlation is respectable, so that the correlation is on the record as the thing the test is not asking about.

The control’s role is the one worth understanding. If some input beat the capacity’s own worst adjacent pair, the test would be measuring the sorting rather than the input — an ordering can always be constructed in which neighbours have close capacities, and that ordering is the capacity’s own. Requiring that no input beats it is requiring that the test has not been inverted.

Where this stops #

A Slater radius is a caricature and the caricature is the point. Slater’s rules are a fit to atomic energies from 1930 and they give the 2s and the 2p the same effective charge when the two are separated by several electronvolts. A better radius would be a different number. What it would not be is a differently ordered number to the extent that matters here, because the ordering of atomic sizes is not in dispute and the failure is a factor of eighty between neighbours rather than a close call.

And the test is a test at the resolution of fourteen atoms. With more atoms the adjacent pairs would be closer in the input and the ratios would fall; with a hundred atoms an input that failed here might pass. What the test says is that at this resolution a size cannot predict a capacity, which is the resolution anybody actually has.

The placement among the six is a placement among numbers that moved. Five of the atoms lose their finite capacity when a fifth point is added, and the pair ratios above are computed over the set that still has one — the same set the pair test used, so the comparison is like for like. What it means is that all seven of these numbers are ratios over a quantity whose large end is provisional, and the ordering among them is more trustworthy than any of their values.

The deeper reservation belongs to the capacity itself. Five of the atoms in the set have no finite capacity once a fifth point is added to the fit, so the quantity being tested against a size is a quantity whose large end is the fit running out of resolution. Testing an input against a partly illusory quantity is a limited exercise, and what makes it worth doing is that the answer is negative: an input that failed to predict a quantity’s illusory part and its real part alike has failed either way.

The generalisation #

The habit is to compute an input rather than quote one, when what is needed is one scale across a set nothing was measured on.

A tabulated quantity carries the purpose it was tabulated for, and the purposes rarely line up with the set at hand. The usual response is to splice tables together and note the seam, which imports two conventions and a discontinuity. The alternative is to define the quantity from something every member of the set has — here a nuclear charge and a configuration — and accept that the result is a model’s size rather than a measurement’s. For a test about ordering and proximity that trade is almost always worth taking, because a model’s size is monotone and internally consistent where a spliced table is neither.

The corollary is the two definitions. Having decided to compute a quantity, compute it twice by different reasonable routes and report both. It costs one closed form, it is the only available check that the answer is about the quantity rather than about the definition, and the reader’s first question is what it answers.

Who found it, and when #

Slater’s screening rules are 1930. The hydrogenic radial expectation values are older than quantum chemistry. The capacity and the pair test come from the essays this one follows. What is computed here is the valence size of each atom by two closed forms, the pair test against both, and the rank correlation the test is to be read against.

The number worth carrying is not 78.7. It is five per cent and eighty-fold: two atoms whose computed sizes differ by a twentieth and whose capacities differ by a factor of eighty, adjacent in the sorted list with nothing between them.

Still open: the energy that is not a curve #

The obvious open question is the one route named earlier and not taken. Every quantity in this family — the electronegativity, the hardness, the capacity — is a derivative of a smooth curve fitted through an atom’s energy at integer electron counts. The exact theory says the energy is not smooth: it is straight segments between integers with a derivative discontinuity at each one, and that model gets the alkali metals right for free, which is exactly where the cubic gives lithium a negative electronegativity. What it costs is every quantity computed from a curvature, and finding out how much of this family of quantities survives being written in it is a question in itself rather than a calculation.

The nearer question is the one this essay’s arithmetic makes cheap. A size failed the pair test; so did every quantity inside the cubic. What has not been tried is a pair of inputs — the test asks whether two atoms close in one quantity have close capacities, and two quantities define a plane in which proximity means something different. The two-variable boundary already computed here is the apparatus, a size and a hardness are the obvious pair, and the answer would say whether the capacity is a function of nothing available or of something available in two pieces.