Four tables and one molecule to disagree about

The table that could not have mattered settled a worry by proving it inapplicable. A molecule made of two elements has partial charges that are one number times a pattern fixed by its formula, so changing the electronegativity table changes only that number — and a ranking cannot notice a rescaling. The five molecules the worry had been raised about were all two-element molecules, and their rank correlations came out identical on all four tables to machine precision.

It then said where the worry does apply, and named the experiment. A series of molecules with the same shape and different central atoms is not one molecule; the multiplier is a property of the pair of elements and is different for every member, so the ratio between two members is a different number on every table. And the interesting output, it said, is not the mechanism but whether the four tables agree about which molecules are the exceptions.

They do agree about which. What they do not agree about is what that molecule is, and the disagreement is larger than the molecule.

What the anchoring does and does not do #

The four scales are in four unit systems: Pauling’s and Allred–Rochow’s are dimensionless numbers on a scale fixed by convention, Mulliken’s and Allen’s are energies in electronvolts. Comparing raw values would compare units as much as chemistry, so each table is multiplied by whatever makes its hydrogen–fluorine separation match Pauling’s — the same convention used for the two-element case, kept unchanged so that both sets of numbers are the same numbers.

That is exactly the structure the two-element theorem is about. The multiplier is one number per table, so inside any single two-element molecule it is a common factor and cancels from every ratio. It does not cancel between two molecules, because the electronegativity difference it multiplies is different in each — and the last column of the table above runs from a fifth on Pauling’s to below zero on Mulliken’s.

The sign is disputed for a quarter of these bonds #

Before any trend, there is the direction of the arrow.

Silicon, phosphorus, lithium and sodium are less electronegative than hydrogen on all four tables, so silane, phosphine, lithium hydride and sodium hydride are hydridic on every account — which is a real chemical fact and is agreed on.

Carbon, sulfur and iodine are less electronegative than hydrogen on Mulliken’s alone. Mulliken’s scale is the mean of an atom’s ionisation energy and its electron affinity, and hydrogen does unusually well by that measure: its ionisation energy is 13.6 eV and its electron affinity is 0.75, giving 7.18, above carbon’s 6.27, sulfur’s 6.22 and iodine’s 6.76. So on that table a C–H bond, an S–H bond and an H–I bond are all drawn the other way round from the way the other three tables draw them.

Three bonds out of twelve, and one of the three is the most common bond in organic chemistry.

This is the same disagreement the rank comparison that measured it directly reported as twelve discordant pairs out of a hundred and fifty-three, seen from the other end. There it was a count of inversions between two rankings; here it is three arrows drawn backwards. The count is the honest summary and the arrows are what a reader of a textbook sees.

It is also worth separating from a different, better-known objection. An electronegativity difference is not a charge transfer — the amount of charge that actually moves depends on how hard each atom is as well as on how much it pulls — and that is a criticism of the model, true on every table at once. What is being measured here is narrower and is not about the model at all: given that the model is being used, the tables it is used with do not agree.

The halide series, and the measurement that settles it #

Group 17 is the only series here with four members whose elements are all in all four tables, and it is the only one with four measured gas-phase dipole moments to be judged against: 1.826, 1.109, 0.827 and 0.448 debye.

Scaling each table to hydrogen fluoride is the fairest treatment available, because none of the four claims to produce a dipole in debye. It also makes the comparison a comparison of shape: every table starts on the measured value and the question is where it goes.

Three of the four have the series falling in magnitude at every step. Mulliken’s does not: it falls to 0.269 at hydrogen bromide and then goes to −0.313, so the series turns round and grows again on the wrong side of zero. That is a table saying hydrogen iodide is polarised the other way from hydrogen fluoride, and it is the only one of the four that says so.

They agree about which molecule and disagree about what it is #

Every one of the four tables has its largest residual at hydrogen iodide: 0.380 debye on Pauling’s, 0.865 on Mulliken’s, 0.431 on Allred–Rochow’s and 0.348 on Allen’s.

So the answer is the first of the two possibilities. If the four tables agree about which molecules are the exceptions, the mechanism is about the molecules; if they do not, it is about a table. They agree. All four find the same molecule hardest, and the mechanism — whatever makes an H–I bond less polar than the trend from H–F, H–Cl and H–Br suggests — is a fact about iodine rather than a fact about anybody’s fit.

That is the reassuring half. The other half is what they say about it.

The four predictions are 0.828, −0.417, 0.017 and 0.100 debye. They span 1.245 debye, which is 2.8 times the 0.448 that was measured. One of them is high by nearly a factor of two, one is on the wrong side of zero, and two are within a tenth of a debye of nothing at all.

A quantity computed from a table, where the tables span three times the quantity, is not a quantity computed. It is a choice of table reported as a number.

No table is right, which is the part that is easy to miss #

The natural next move is to find the best table and use it. The root-mean-square residuals across the halide series are 0.254 debye on Allred–Rochow’s, 0.276 on Allen’s, 0.297 on Pauling’s and 0.505 on Mulliken’s.

The best of the four is better than the worst by a factor of two, and it still misses a four-member series by a quarter of a debye. A verdict computed against a number is only as sharp as the number, and here the numbers being compared are further apart than any of them is from the truth. The disagreement between the tables and the common error of all of them are the same size, which means that choosing between them on this evidence would be fitting to noise.

That is not a complaint about electronegativity. It is a statement about what a charge model of this shape can do: a partial charge proportional to an electronegativity difference, times a bond length, is a two-parameter caricature of a bond dipole, and a caricature that reproduces a four-member series to a quarter of a debye is behaving about as well as it can. The mistake is not using it; the mistake is quoting a number from it without saying which table produced it.

The other groups, and where the agreement is #

Group 15 is the case where all four tables agree completely and agree on something surprising: ammonia’s N–H bond is polarised with hydrogen positive on every table, and phosphine’s P–H bond is polarised with hydrogen negative on every table. Phosphorus and hydrogen sit within a hundredth of one another on Pauling’s — 2.19 against 2.20 — and the sign that falls out of that hundredth is confirmed by the other three.

Group 14 is where all four break, and they break differently. Methane’s C–H is hydrogen-positive on three tables and hydrogen-negative on Mulliken’s; silane’s Si–H is hydrogen-negative on all four. So three tables have the sign flipping down the group and one has it constant, and none of them has the magnitude falling.

Group 1 is where every table has the dipole growing down the group — lithium hydride to sodium hydride — and where the measurements agree with them: 5.88 debye against 6.40. That is the one series here where all four tables and the experiment tell the same story, and it is the series in which the bond is most nearly ionic, which is the regime an electronegativity difference was fitted to describe.

There is one more thing to say about why Mulliken’s is the odd one out, because it is not carelessness. Mulliken’s scale is the only one of the four defined from properties of the free atom alone — an ionisation energy and an electron affinity, both measured on an isolated atom — while the other three are fitted to or derived from something about atoms in molecules. Hydrogen’s ionisation energy is enormous for its position because it has no core to screen its electron, so the free atom is a much stronger attractor than the bonded one, and a scale built from the free atom inherits that.

That is a defensible scale and a defensible reason for it to disagree, which makes the disagreement worse rather than better: there is no argument from carelessness that would let a chemist discard it. The same atom’s capacity to hold charge turns out to be largely a statement about its second ionisation energy, and free-atom quantities behaving unlike bonded-atom quantities is the recurring shape of it.

What was computed, and how #

Every electronegativity value here is quoted: Pauling’s from his thermochemical fit as revised by Allred, Mulliken’s as the mean of the ionisation energy and the electron affinity in electronvolts, Allred and Rochow’s from the effective nuclear charge at a covalent radius, and Allen’s configuration energies. Every bond length is quoted, as an equilibrium internuclear separation in ångströms, and every dipole moment is a quoted gas-phase measurement in debye.

What is computed is the disagreement. Each table is multiplied by the factor that matches its hydrogen–fluorine separation to Pauling’s, and that anchoring is checked rather than assumed — all four must give the same H–F charge separation to a part in $10^{12}$. The charge on hydrogen in a two-atom bond is then half the anchored difference, the bond dipole is that charge times the separation, and the conversion to debye is 4.80320471 debye per electron-ångström.

The refusal is a bond of an element with itself. Hydrogen against hydrogen has no electronegativity difference on any table, so the anchoring must give it exactly no charge separation — a scale that produced a dipole there would be producing one out of its own arithmetic.

Where the model stops #

Nothing here is a calculation of a dipole moment. Each prediction is an electronegativity difference put through one linear rule, anchored on one molecule, and the whole comparison is of four tables against one another rather than of any of them against quantum mechanics. A computed dipole would settle which table is closest to something real; it would not touch the finding, which is that the four disagree by more than the quantity they are predicting.

This is a charge model and not a density. Nothing here computes where the electrons are; it takes a quoted correlation, turns it into two point charges and multiplies by a length. A real dipole is not a sum of bond dipoles, lone pairs contribute and are absent from the model entirely, and phosphine is the standard demonstration of that — its measured dipole of 0.574 debye is mostly its lone pair, which this model cannot see at all.

The intensities the worry was originally about are untouched by any of this, and it is worth saying so plainly: an infrared intensity ranked against an amplitude is a ranking inside one molecule, and the theorem still protects it. What has been shown here is that the caution rightly not raised about those molecules is entirely justified about any comparison between them.

The diatomics are the cases where the model is least unfair, because there is no lone-pair geometry to get wrong and no bond-angle projection to argue about. That is why the halides are the series judged here and the polyatomics are quoted as bond dipoles rather than as molecular ones.

And a bond dipole computed as $q r$ has the wrong shape at large separation. The real quantity saturates and then falls as the bond breaks; this one grows without limit, which is why lithium hydride and sodium hydride come out with the trend right and the magnitude a third of what it should be.

The generalisation #

One theorem made a worry inapplicable, and this finds the boundary of it. Stated together, they are one rule:

A convention that cancels within a system does not cancel between two of them. Inside a two-element molecule the choice of table is a common factor and every ratio is safe. Across a series the factor is different in each member, and every comparison is exposed.

That is the same sentence a self-consistent calculation of bond orders arrives at, where a reference bond order common to one molecule failed to cancel between two of them and put a floor under a decay. Two different conventions, two different fields, one arithmetic.

The practical form is short. A number computed from a table, quoted without the table, is safe only if the quantity is invariant under rescaling that table — and a difference between two systems almost never is.

Who found it, and when #

Pauling’s scale is from 1932 and the halide dipoles have been measured since the nineteen-thirties, so the comparison made here has been available for ninety years and is not new. What is usually done with it is a fit: the standard treatment relates ionic character to the electronegativity difference through an empirical function, Pauling’s own $1 - \exp(-\tfrac14 \Delta\chi^2)$ or Hannay and Smyth’s linear form from 1946, both fitted to the hydrogen halides among others. Those functions absorb most of what is measured here into their parameters.

What is not usually done is to run the same arithmetic on four tables at once and read the spread. The spread is the quantity worth asking for, and it is 1.245 debye at hydrogen iodide.

Still open: a non-linear fit, and a series with no hydrogen #

The obvious open question is the fitted function. Every table above is used linearly, and none of the four standard treatments does that — so the comparison could be made again with Pauling’s own ionic-character expression in place of the linear one, and the interesting question is whether the four tables come closer together or further apart when each is put through it. A non-linear function of a rescaled variable is not a rescaled function, so there is no reason for the spread to shrink, and if it grows the fit is doing more work than the table.

The nearer question is the other diatomic series. Group 1 is where all four tables agree and the trend is confirmed; group 17 is where they disagree and one of them fails. Between those sit the interhalogens — ClF, BrF, BrCl, ICl, IBr — every element of which is in all four tables, every one of which has a measured dipole, and none of which contains hydrogen. Since every disputed sign here is a dispute about hydrogen, a series with no hydrogen in it would say whether the tables’ disagreement is about one awkward element or about the whole idea.