Where a closed form stops being one

Equalisation in closed form establishes that the exact answer to electronegativity equalisation is a hardness-weighted mean of the free-atom values, and that Sanderson’s geometric mean is better than it looks because it errs downwards, which is the direction no power mean of order below one can err.

All of that rests on a quadratic. An atom’s energy is expanded in its electron count, three measured points fix a linear and a quadratic term, and the chemical potential is therefore linear in the charge — which is what makes the equalisation condition a linear equation with one solution and a closed form.

The natural next question is what happens past that. A cubic energy makes the fixed-point condition a quadratic, so there are two roots and something has to choose between them: is the second spurious, or is it the ionic solution?

The choice turns out to be easy. What is not easy is everything upstream of it.

The cubic coefficient, in closed form #

The quadratic model fits $E(q)$ through three points: the anion at $q = -1$, the neutral atom at $q = 0$, and the cation at $q = +1$. A cubic needs a fourth, and the natural fourth is the dication, whose energy above the neutral atom is $I_1 + I_2$.

Writing $E(q) = aq + bq^2 + cq^3$ and imposing all four:

$$b = \frac{I_1 - A}{2} = \eta, \qquad a = \chi - c, \qquad c = \frac{I_2 - 2I_1 + A}{6}.$$

The cubic coefficient is one sixth of a second difference of the ionisation series. That is the form worth having, because a second difference is large exactly where the series has a kink — and the series has a kink precisely where the second electron comes out of a closed shell.

The third column is the model’s electronegativity, because the slope of the cubic at zero charge is $a = \chi - \gamma$ and not $\chi$.

So a cubic term does not correct the electronegativity. It replaces it, and for the alkali metals the replacement is negative. Electronegativity is not one quantity is the familiar version; this is the same finding arriving from inside a single scale rather than from a comparison between four. Lithium goes from being the least electronegative element in the table to being fifteen electronvolts below it.

That is a much larger statement than anything about roots, and it arrives before the equalisation is even set up.

Hydrogen has no fourth point at all #

There is a second problem, and unlike the first it is not a matter of degree.

A hydrogen atom has one electron. H⁺ has none. There is no second ionisation energy to measure, there never will be, and hydrogen therefore has no cubic coefficient — not a small one, not an estimated one, none.

Every molecule equalised here except one contains hydrogen. So the cubic model applied to methane is cubic for the carbon and quadratic for the four hydrogens, which is not a model of anything; it is two models sharing a molecule. That is a harder version of the difficulty a cage of one element raises: there the question was whether one number can describe an atom in different environments, and here two atoms in the same molecule are not even being described by the same functional form.

The consequence is visible immediately. Hydrogen’s $\chi$ is 7.176 and it keeps it. Oxygen’s falls from 7.540 to 5.982. So under the cubic, hydrogen is more electronegative than oxygen, and water’s polarity comes out the other way round: the linear model puts $-0.020$ on the oxygen and the cubic puts $+0.063$. Ammonia flips too.

Both linear values were already near zero — the Mulliken numbers put hydrogen above carbon, as a comparison of the scales shows — so the flip is between two small numbers. What is new is the size: the cubic triples the magnitude and reverses it, on the strength of one element having no fourth data point.

The two roots, and the easy part #

With the cubic in, the chemical potential is

$$\mu(q) = -(\chi - \gamma) - 2\eta q - 3\gamma q^2,$$

so setting it equal to a molecule’s $\mu$ gives $3\gamma q^2 + 2\eta q + (\mu + \chi - \gamma) = 0$ — a quadratic, with two roots.

Choosing between them is done by a limit rather than by inspection. As $\gamma \to 0$ the model must return the quadratic answer, and only one root does: the one taken with a plus sign, which goes to $-(\mu + \chi)/2\eta$. The other goes to $-2\eta/3\gamma$, which diverges as the cubic term is switched off, and a solution that runs away when a correction is removed is not a solution of the corrected problem.

So the second root is not the ionic solution. But calling it spurious is also wrong, and the numbers say why:

Carbon’s second root is six electrons away and is plainly nothing. Lithium’s is at fourteen hundredths of an electron, which is well inside the range molecules actually use — and it is not a second solution to argue about, because the two roots collide first.

What the collision is, and why it matters more #

The discriminant $4\eta^2 - 12\gamma(\mu + \chi - \gamma)$ goes negative below a chemical potential, and at that point the two roots meet at

$$q^{*} = -\frac{\eta}{3\gamma}.$$

Past it there is no real root at all: the atom cannot reach that chemical potential, whatever charge it is given. The cubic model has given every atom a capacity — a largest amount of electron it can accept — where the quadratic model had none.

The capacities are the last column above and they are not of one size. Carbon can take 3.198 electrons, which is generous to the point of meaninglessness. Lithium can take 0.073.

Lithium’s capacity is smaller than the charge equalisation routinely assigns an atom. So for lithium the cubic does not correct the linear answer and does not offer a second one; it runs out of solutions before the answer is reached.

A molecule with no answer #

The clean demonstration is a molecule of two alkali metals.

Lithium’s cubic electronegativity is −7.907 and sodium’s is −3.416, so sodium is the more electronegative of the two and must accept charge. Sodium’s capacity is 0.122 of an electron. The bisection on the chemical potential finds no value at which the two charges sum to zero, and the routine returns no solution rather than a number.

Li–Na has no answer. Li–K has no answer. An ordinary molecule still does, so the refusal is about those atoms rather than about the method.

That is the sharpest form the finding takes, and it is not the form the question anticipated. The question was which of two roots to take. The answer is that for the elements where the question is interesting, there are no roots.

What the failure is actually about #

None of this is a defect of cubic expansions. It is a defect of the fourth point.

The first three points — anion, atom, cation — are three charge states of the same valence shell, and a smooth curve through them is a reasonable description of an atom giving or taking a fraction of an electron. The dication of an alkali metal is a different kind of object: its electron has come out of a closed shell, at a cost four to fourteen times the first ionisation, and a polynomial forced through that point is being asked to describe two regimes with one curve.

The size of $\gamma$ is a direct measurement of how different the two regimes are. Lithium: 10.912. Nitrogen, whose second ionisation continues the series smoothly: 0.077. A hundred and forty-fold range in a coefficient that a smooth model would have varying gently across the table is the arithmetic saying that the fourth point does not belong on the same curve. It is the same diagnosis a predictor’s own ties give, reached from the parameters rather than from the data: a number that behaves like this is not measuring what the model says it measures.

Three elements — beryllium, magnesium and phosphorus — have negative $\gamma$, so their curves bend the other way and their capacity is unbounded. That is the same statement with the sign reversed: the second difference is not a small correction of one sign but a quantity whose magnitude and sign both track shell structure.

What this says about the closed form #

The closed form — the hardness-weighted mean — is not threatened by any of this, and the reason is worth stating plainly.

The quadratic model is not an approximation to the cubic one. It is a model fitted to three states an atom actually visits, and the cubic is a model fitted to four, one of which it does not. Going from three points to four is not going from a first approximation to a better one; it is changing which measurements the model is about.

So the honest boundary is: the closed form holds wherever the charges are small enough that a three-point fit describes the energy, and the way to find out whether they are is to compute the charges and look, not to add a term. Every charge computed in these equalisations is well under half an electron, which is the condition, and the cubic does not test it — it replaces the question.

What is quoted, and what is computed #

Three measurements per element are quoted — the first ionisation energy, the electron affinity and now the second ionisation energy — and every one of them is a measurement. Everything else is computed: the cubic coefficient from a closed form in those three, the electronegativity as a slope, the two roots as a quadratic formula, the capacity as a discriminant, and the equalisation by bisection because there is no longer a closed form to use.

The branch is chosen by a limit and the choice is checked: at a chemical potential near neutrality the physical root must agree with the linear model’s charge to a tenth of itself, and the other root must be more than an electron away. Both are checked, so a branch chosen the other way round would fail rather than merely look odd.

What this cannot say #

A polynomial in the electron count is not the only way to add a term. The literature has integer-constrained and piecewise-linear treatments that avoid this entirely by refusing to interpolate between charge states, and nothing here says anything about those; it says what a cubic does.

The second ionisation energies are quoted to three decimals and are worth less than that here, because the conclusion turns on differences of order ten electronvolts. No conclusion here would change if any of them were wrong in the second figure.

And an atom in a molecule is not an atom. The whole equalisation picture assigns a free-atom energy curve to an atom in a compound, and the departure that assumption makes is not measured here and is probably larger than the cubic term.

The capacity is a property of the model rather than a measured quantity. Nothing in an experiment says that an atom cannot accept a tenth of an electron; what the arithmetic says is that this particular curve cannot describe it accepting one, which is a statement about the curve.

What the cubic requires #

The cubic coefficient is largest where the second electron leaves a closed shell — lithium’s more than ten times carbon’s, stated as a ratio rather than as two values.

Hydrogen has none, and the routine returns exactly zero for it rather than an estimate.

The physical branch agrees with the linear charge to a tenth of itself at a chemical potential near neutrality, and the other root is more than an electron away — the pair of checks that makes the branch a choice justified by a limit.

Carbon’s second root is several electrons away and lithium’s two roots collide before a tenth of an electron has moved, with the capacities differing by more than twentyfold — so the capacity is a property of where the kink is and not of atoms in general.

Three elements come out with a negative electronegativity and the elements with no kink do not, checked in both directions.

A molecule of two alkali metals has no solution, and an ordinary molecule still has one — again both directions, because a routine that refused everything would be reporting a bug.

And water’s polarity reverses, which is the one consequence a reader can check against a fact they already have.

What a negative electronegativity would mean if it were true #

Lithium’s $-7.907$ electronvolts is reported as an oddity, and it is worth saying what the number actually claims, because the claim is not subtle and its falsity is the sharpest evidence available that the cubic has failed.

Electronegativity is minus the chemical potential — minus the rate at which the energy rises as electrons are added. A negative electronegativity is therefore a positive chemical potential, which says that adding an electron to a neutral lithium atom raises its energy and removing one lowers it.

An atom of which that were true would ionise spontaneously in vacuum, shedding an electron with no field applied and no photon absorbed, and go on doing so. Lithium does not. Its first ionisation energy is 5.39 electronvolts, measured, positive, and known to five figures.

So the failure is not that a fitted quantity came out with an unexpected value. It is that the fit produces a statement contradicted by the single most elementary measurement available on the element.

The cause is locatable in one comparison. A cubic through four charge states is fitted to the anion, the neutral, the cation and the dication, and the dication is reached by removing a second electron. For carbon that costs 24.4 electronvolts against a first ionisation of 11.3 — a factor of 2.2. For lithium it costs 75.6 against 5.4, a factor of fourteen, because the second electron comes out of a closed shell.

A cubic has three degrees of freedom above its constant, and one data point sitting fourteen times further out than the others will dominate all three. The curve is dragged to pass through the dication, and everything it says about the region between the anion and the neutral — which is the only region a molecule ever occupies — is whatever is left over after that.

That is a general hazard of fitting a smooth function through points of wildly different magnitude, and it has a general remedy: do not include a point the model was not built to describe. What makes it specific here is that the point in question is the closed-shell jump, and a closed-shell jump is a discontinuity — the one feature no polynomial of any order can represent, and the feature the alkali metals have most sharply.

Still open: a piecewise model, the capacity, and the charge itself #

The obvious open question is the piecewise treatment the literature uses instead. An atom’s energy against electron count is, on the exact theory, a sequence of straight segments between integers with derivative discontinuities at each one — which is the opposite of a smooth polynomial and gets the alkali metals right for free, because a discontinuity at the closed shell is exactly what the huge second difference is trying to represent. Computing the equalisation with segments rather than a curve would say how much of the closed-form answer survives, and the interesting part is that a piecewise-linear model has no hardness at all.

The nearer question is the capacity, taken seriously rather than as a symptom. 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 is zero at the integer. 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.

And there is a third direction, which is to stop expanding altogether. The quantity equalisation actually wants is a charge, and a charge is not observable — an integer nobody measured is the same difficulty, reached from a metal complex rather than from a table of scales.