Two derivatives of the same energy, and only one is tabulated

One of the figures on a basis is not a thing: The same electrons written two ways with the density unchanged, and four electronegativity scales that disagree.

Ten essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The quantity no scale prints

Eighteen common bonding elements, placed by their chemical potential against their hardness. Both axes are built from the same two measurements per element; only the horizontal one is ever printed. Hardness spans a factor of nearly four and does not follow the horizontal axis: potassium and fluorine sit at opposite ends of the electronegativity scale and hydrogen, in the middle of it, is the second hardest atom here.

The reconstruction, as a residual. Fifteen bars are invisible at this scale. The three that are not are beryllium, magnesium and nitrogen, out by 0.498, 0.132 and 0.068 electronvolts — and they are exactly the three elements in the table whose anion is not bound.

The electronegativities held fixed and the answer moving anyway. Every scale prints one number per element; what decides how much charge a difference moves is a second number per element, and sweeping it while the first stays put changes the answer by more than the disagreements between scales do.

The value that only exists in the bond

The two derivatives, element by element: the electronegativity that gets tabulated and the hardness that does not. The weighting above is by the reciprocal of the second, so two elements with the same electronegativity and different hardnesses contribute differently to the same molecule.

A mean that is low rather than right

Four unweighted rules against the answer the hardnesses give, on twelve molecules. The geometric mean — Sanderson’s — is the closest on average, at 0.0706 eV against the arithmetic mean’s 0.1666, and it is the closest on only four of the twelve. Every rule below the arithmetic mean is constrained to sit low, which is the whole of why the best-performing one performs best: it is not more nearly right, it is more nearly low.

The inputs, and the correlation the rule rests on: each element’s electronegativity and hardness, both computed from a measured ionisation energy and a measured electron affinity. The two quantities rise together across a row, which is what makes the softest atom usually the least electronegative one.

Two atoms, their electronegativities held fixed, the ratio of their hardnesses swept. The four horizontal lines are the four rules; the marked curve is the answer. They meet at equal hardnesses, which is the one place any unweighted rule is exact — and there they all agree with each other too, which is why the choice between them has never been forced.

Where a closed form stops being one

Each element’s electronegativity from three points on its energy curve, the coefficient a fourth point adds, and what is left of the electronegativity afterwards. Three of the twelve come out negative.

The chemical potential against charge for carbon and lithium with the cubic in. Each curve turns over, and the turning point is where the two roots meet. Carbon’s is three electrons away; lithium’s is seven hundredths.

The quantity the quadratic model is built on, which behaves: the hardness across the table, from the same two measurements the electronegativity comes from. Nothing in this column ranges over two orders of magnitude.

A capacity that is largest where there is none

Each atom at its electron affinity and the capacity the model gives it. The three at negative affinity are the atoms with no bound anion; they are also the ones the model says can accept most.

The full ordering, largest capacity first, with the affinity beside it.

The cubic coefficient of every atom. A negative one means no capacity at all, and three atoms have one.

A correlation is not an account

The capacity against the second ionisation energy, on a logarithmic axis because the capacity spans nearly three orders of magnitude. The circled pair is the argument.

The rank correlation beside the fractions of variance, with the level a quantity would have to reach before it deserved a simpler name.

The fraction of the capacity’s variation each two-parameter form accounts for, against the level the proposal’s own reasoning names.

The worst of the six was the one we asked about

The largest capacity ratio between atoms adjacent in each candidate input, against the floor the capacity itself sets.

The control run two ways: closeness by rank, and closeness by a tenth of the input’s own range.

The fourteen capacities in rank order, spanning a factor of four hundred and thirty.

Six of fifteen change verdict

Each atom’s chemical capacity from a cubic fitted through four electron counts and from a quartic fitted through five.

The atoms in the order the cubic puts them, largest capacity first, with what the quartic says about each.

The seven atoms with a finite capacity under both fits, in the order the cubic puts them, with both values.

A size the fit was not made from

Each atom’s valence shell at the effective charge Slater’s rules give it, with the mean and root-mean-square radii of a hydrogenic orbital of that shell and charge.

The atoms sorted by mean valence radius, with the ratio of capacities between each neighbouring pair.

The worst adjacent pair and the median adjacent ratio for each way of measuring the atoms, with the control beneath.

Four quantities go and one question stays

Chlorine’s energy against the charge it carries, as the exact theory’s straight segments and as the quadratic curve fitted through their ends.

What each part of the construction becomes when the energy is straight segments rather than a smooth curve.

For the cheapest bonds this collection prices, the energy it costs to move one whole electron in the better of the two directions.