A capacity that is largest where there is none

A cubic fitted through four charge states of an atom — the dication, the cation, the neutral and the anion — breaks down in a specific way: lithium comes out with an electronegativity of −7.907 eV, and two alkali metals have no equalisation solution at all.

The reason is a capacity. A cubic’s chemical potential is a parabola in the charge, so it turns round: beyond some amount of accepted electron the model says taking more would raise the potential, and there is no solution past that point. The idea is easy to notice and easy to set aside:

A largest acceptable charge is a real idea even if this model’s version of it is an artefact, and there is an independent route to one: the anion is unbound for beryllium, magnesium and nitrogen, which is the statement that their capacity at the integer is zero. Comparing the cubic’s capacities against which anions are actually bound would say whether the discriminant is measuring anything at all, and it needs no new data.

It needs no new data, and the comparison is not close.

The three atoms with no anion #

Beryllium, magnesium and nitrogen have negative electron affinities. That is not a small affinity or an uncertain one — it is the statement that the extra electron is not bound, so the true capacity at the integer is zero.

The top of the list is where the model puts the atoms that can accept most. Three of the four are atoms that can accept nothing.

The bottom of the list is where it puts the atoms that can accept least. All three are atoms whose anions exist.

It is worth pausing on nitrogen, because it is the one case where the model has a finite number to be wrong with. Its capacity is 31.5 electrons — the largest finite value in the set by a factor of five over bromine’s 6.4, on an atom that will not hold one. The number is large for a specific reason: nitrogen’s cubic coefficient is 0.077, the smallest positive one, and the capacity is the hardness divided by three times that. A quantity defined as one number over a small number is a quantity whose value is decided by whatever makes the small number small — and here that is a near-cancellation in a second difference of four measured energies, which is the least reliable thing in the calculation.

So the largest number in the table is the one with the least behind it, which is the ordinary fate of a ratio with a difference in its denominator. It is worth saying because the same shape appears wherever this collection divides by a curvature: a susceptibility curve’s fourth parameter is undetermined for the same arithmetic reason.

And the sign does not save it #

The obvious repair is to look at the cubic coefficient rather than at the capacity it implies. A negative coefficient means the chemical potential never turns round, which is the model saying there is no capacity here — and perhaps that is the model’s way of failing loudly.

Three atoms have a negative coefficient: beryllium, magnesium and phosphorus. Two of them have no bound anion. The third binds its own perfectly well, at 0.746 eV.

So the sign does not separate the cases either, and the whole quantity — capacity, coefficient and discriminant alike — is silent about the thing it looks like a statement about.

What it is a statement about instead #

The cubic coefficient is one sixth of a second difference of the energy against charge, built from the second ionisation energy, the first, the affinity and zero. A second difference is dominated by whichever of its four points is furthest from the others, and for these atoms that is the dication.

So the capacity runs against the second ionisation energy, at a rank correlation of −0.503. Lithium’s second ionisation energy is 75.6 eV, because the second electron comes out of the core; that bends the curve sharply; and the model concludes lithium can accept 0.073 of an electron.

The capacity is a statement about removing a second electron, extrapolated four units of charge in the other direction. It is the same extrapolation that gives lithium a negative electronegativity, and it fails for the same reason: the four points are not on one smooth curve, and no polynomial through them is a model of anything.

Why this is worse than a quantity that says nothing #

It would be tidier if the capacity were noise. It is not.

Among the fourteen atoms with a finite capacity, the ordering agrees with the electron affinity at +0.433 — the halogens have large capacities and large affinities, the alkali metals have small capacities and small affinities. A reader who looked only at those fourteen would conclude the quantity is a serviceable if noisy measure of how much charge an atom accepts.

The quantity behaves sensibly across every case that cannot test it, and is at the wrong end of its own ordering for the three that can. That is a shape of failure met three times from different directions: a rank correlation matched by a control that cannot be a mechanism, a formula right for every molecule anybody would check it on, and here a capacity right for every atom that has one.

The common structure is that the cases which would falsify a claim are also the cases hardest to include: an ion pair of similar size and different hardness, a molecule with two centres carrying angles, an atom whose anion does not exist. A test assembled from what is easy to assemble is a test that omits the falsifiers, and the omission is not random — it is correlated with the failure.

What a capacity would have to be #

Since the model’s version is unrelated to the thing, it is worth writing down what the thing is, because the definition turns out to be where the difficulty lives.

At the integers there is no ambiguity. An atom accepts one electron if the anion is bound and does not if it is not, and that is known for eighteen atoms as a measured sign. Beryllium’s capacity is zero, chlorine’s is at least one, and nothing has to be extrapolated to say so.

Between the integers there is nothing to measure. A capacity of 0.4 electrons is a statement about a fractional charge, and a fractional charge is not observable — it is a partition of a density, and every partition is a convention. So the quantity the cubic is computing has no measurement behind it anywhere, and the comparison made here is the only kind available: against the integers, where the answer is known.

That reframes the failure. The model is not getting a hard quantity wrong; it is producing a number for a quantity that exists only inside the model, and the one place where the model’s number can be checked against something outside it is the place it fails. A quantity that can only be checked at the integers should be quoted at the integers, and the capacity’s whole appeal was that it was not an integer.

What survives #

The negative result about the cubic survives untouched and is if anything strengthened. A cubic through four charge states is not a model of an atom’s energy; it is an interpolation through four numbers that happen to be available, and every quantity read off its shape — the electronegativity, the equalisation, the capacity, the discriminant — inherits whatever the interpolation does between the points.

What also survives is the idea worth keeping. A largest acceptable charge is real: beryllium’s is zero and chlorine’s is at least one. The idea is fine and this model’s version of it is unrelated to it, which is a more useful conclusion than either the capacity works or the capacity is meaningless.

And there is a positive statement to take out of it. The quantity that does say whether an atom will accept an electron is the electron affinity, which is one measured number and needs no curve through anything. The reason a curve was wanted at all was to get a fractional charge, and the whole difficulty of electronegativity is that a partial charge is not observable — so the extrapolation is being asked to supply what no measurement can.

The three that could have been the test, and were not #

One more thing is worth extracting, because it is the practical advice this produces.

Beryllium, magnesium and nitrogen are not obscure atoms. They are in every table of electronegativities, their affinities are measured, and the check needs no new measurement and no new calculation. The comparison was available from the first attempt to draw a curve through charge states, and it is rarely made.

That is not a criticism so much as an observation about how a quantity acquires standing. The capacity arrived as a by-product: it was noticed because it explained why two alkali metals had no equalisation solution, which is a defect of the model, and a quantity that explains a defect gets treated as part of the diagnosis rather than as a claim of its own. Nothing about it was ever claimed, so nothing about it was ever tested.

The general form is worth stating, because any body of modelling produces more of them. A model that fails somewhere invites an explanation; the explanation names a quantity; the quantity is then carried forward as though it had been established, when all that was established is that it accounts for the failure. A quantity introduced to explain a failure is a quantity nobody checks, and the habit of giving every claim a test it could fail is exactly what the by-product escapes. The predictors audited elsewhere were all audited because they were offered as predictors; the capacity was never offered as anything.

What is quoted, and what is computed #

The measurements are quoted: first and second ionisation energies and electron affinities for eighteen atoms, from the standard tables. Nothing new is quoted.

Everything else is computed from them: the cubic coefficient as a second difference, the capacity as the charge at which the chemical potential turns round, and both rank correlations.

The two comparisons are made on different sets and the difference is stated rather than hidden. The capacity-against-affinity correlation is over the fourteen atoms with a finite capacity, because a rank correlation cannot use an infinity; the claim about the unbound three is a statement about their position in the whole ordering, where an infinity is simply first.

What this cannot say #

Eighteen atoms is the set with four tabulated charge states, not a sample. They are the main-group elements of the first three rows plus the heavier halogens, chosen because four charge states are tabulated for them. A wider set would add more atoms with bound anions and few with unbound ones, so it would raise the +0.433 and leave the failure exactly where it is.

A negative electron affinity is a measurement with a sign rather than a value. Nitrogen’s −0.07 eV is a number extracted from calculations and extrapolations rather than a spectrum, since an unbound state has nothing to measure. What is solid is the sign, and the sign is all that is used here.

The comparison is against a threshold rather than a size. Bound and not bound is one bit per atom, and the capacity is a real number, so what is being tested is whether a continuous quantity gets three signs right. A stronger test would compare the capacity against the affinity across atoms where both are positive, which is the +0.433 above — and that is the comparison the model passes. Both halves are reported here because reporting either alone would be a choice about which answer to give.

And the cubic is not the only extrapolation available. The piecewise-linear treatment has no capacity at all, because a straight segment’s chemical potential is constant and never turns round — so on that model the question asked here does not arise. That is not an answer to it; it is a different model in which the concept is absent.

What was checked #

The unbound three are among the four largest capacities, checked atom by atom rather than as a group, and by their position in the ordering rather than by their values — which is the only form that survives two of them being infinite.

The three smallest capacities all bind their anions.

The capacity orders with the affinity among the finite cases, checked as a positive correlation. That is the half that could have gone the other way: a quantity that came back at zero there would make this a report that a number is meaningless, and the finding is that it is not.

And it runs against the second ionisation energy, which is the positive half — the quantity is about something, and the something is on the other side of the curve.

The model that gets the alkali metals right by having no capacity at all #

The capacity turns out to be a property of a fit rather than of an atom, and the alternative that keeps coming up disposes of it entirely — by disposing of everything else as well.

On the exact theory, an atom’s energy against electron count is not a curve. It is a sequence of straight segments between integers, with a kink at each one: below the neutral integer the slope is minus the ionisation energy, above it minus the electron affinity, and the two differ by exactly the quantity the quadratic model calls twice the hardness.

That changes the character of every quantity derived from a curve through charge states.

There is no curvature, so there is no hardness in the usual sense. What the quadratic represents as a smooth second derivative is a jump in the first derivative, concentrated at one point rather than spread over the range.

There is no chemical potential at a neutral atom. The left and right slopes differ, so the derivative is undefined exactly where every table evaluates it — and the two one-sided values are the ionisation energy and the electron affinity, which is why the Mulliken average was ever a sensible thing to write down.

And there is no capacity, because the segments are straight and a straight line has no discriminant to go complex. The question asked here does not arise.

What it buys in exchange is the case the smooth models keep getting wrong. Charge transfers between two atoms only if it lowers the total energy, and with straight segments that condition is a single comparison: it transfers a whole electron if one atom’s ionisation energy is below the other’s electron affinity, and nothing at all otherwise. Sodium and chlorine at infinite separation fail that test — 5.14 against 3.61 — so the transfer is exactly zero, which is the right answer and the one every quadratic model gets wrong by four tenths of an electron.

And the alkali metals stop being anomalous. The enormous second ionisation energy that wrecked the cubic is a kink at the closed shell, which a piecewise model represents natively and a polynomial cannot represent at all.

The price is severe and it is worth naming rather than glossing. A model with only whole-electron transfers has no partial charges, no bond polarity and no equalisation in any recognisable form — so it gets the dissociation limit exactly right and has nothing whatever to say about the region every molecule actually occupies. Which is the reverse of the smooth models’ failure, and is why the practical schemes are all attempts to have both.

Still open: how much survives a piecewise-linear energy #

The obvious continuation is the piecewise treatment, which is now the only route left untried. An atom’s energy against electron count is, on the exact theory, straight segments between integers with derivative discontinuities at each one, and that model gets the alkali metals right for free. What it costs is every quantity computed from a smooth curve: it has no hardness, no capacity and no equalisation in the usual sense, and finding out how much of electronegativity theory survives being written in it is a question in itself.

The nearer question is the one the failure above makes available for nothing. If the capacity is essentially a statement about the second ionisation energy, then it should be predictable from it — and a two-parameter fit of the capacity to the second ionisation energy, checked against the eighteen atoms, would say how much of this quantity is the dication and how much is anything else. A quantity that turns out to be ninety per cent of one input is a quantity with a simpler name than the one it has.