A ranking is not a difference

The four common electronegativity scales are four different quantities: a thermochemical one, an ionisation-energy-and-affinity average, an electrostatic one and a configuration-energy one. Electronegativity is not one quantity measured how far they disagree about order — twelve discordant pairs between Pauling and Mulliken out of a hundred and fifty-three, and six ordinary bonds whose polarity direction is disputed, C–H among them.

This essay measures the other disagreement, which is larger and less often noticed.

The scales run over different ranges #

The first thing to fix is that a difference in Pauling units and a difference in Mulliken units are not the same kind of number. Mulliken’s scale is in electronvolts and the others are dimensionless; the four run from potassium to fluorine over

$$3.160, \quad 7.990, \quad 3.190, \quad 3.459$$

units respectively. So a Mulliken difference of $1.0$ is a quarter of the range and a Pauling difference of $1.0$ is a third — and quoting the two side by side, as tables routinely do, invites a comparison that is not available.

Dividing each difference by its own scale’s range makes them comparable. It is a crude normalisation and it is the least that has to be done.

What the normalised differences say #

The first three rows are unremarkable: the scales agree to within a factor of two, which for a fitted empirical quantity is fine.

The last three are not.

The O–H bond is $0.39$ of the full range on three scales and $0.045$ on the fourth. Water’s bonds are strongly polar on Pauling, Allred–Rochow and Allen, and very nearly non-polar on Mulliken. A factor of nine.

The N–H bond is worse, at a factor of eighteen, for the same reason: Mulliken’s scale puts hydrogen at $7.18$ eV, above carbon at $6.27$ and only just below nitrogen at $7.30$, while the other three put hydrogen well below both.

The P–H bond is the extreme case, at a factor of sixty-two. Pauling has phosphorus at $2.19$ and hydrogen at $2.20$ — a difference of $0.01$, which is nothing — while Mulliken has $5.62$ and $7.18$, a difference of $1.56$, which is a fifth of its range and in the opposite direction.

Where the disagreement lives #

The pattern is not random. Every one of the badly disputed bonds has a hydrogen in it, and the reason is that hydrogen is the atom the four scales treat most differently.

Mulliken’s scale is half the sum of the ionisation energy and the electron affinity, and hydrogen has an ionisation energy of $13.6$ eV — very high, because there is no screening at all. That pushes hydrogen up the Mulliken scale, above carbon and sulfur and iodine.

Pauling’s scale is fitted to bond energies: the extra stability of A–B over the geometric mean of A–A and B–B. On that measure hydrogen sits at $2.20$, below carbon at $2.55$.

Allred–Rochow computes an electrostatic force at the covalent radius, and Allen uses a configuration energy averaged over valence electrons. Both put hydrogen low.

So the disagreement about C–H, N–H, O–H and P–H is a disagreement about what hydrogen is, and each scale is internally consistent about it. A quantity defined four ways will differ where the four definitions pull hardest, and a bare proton with one electron is where they do. That the awkward atom is the one with no core is not a coincidence either. Every one of these scales is, in one way or another, a statement about how tightly an atom holds an electron in the presence of the rest of its own electrons — and hydrogen has none of its own to speak of, so each definition extrapolates rather than measures.

What the scales do agree about #

Rank correlation across the six pairs of scales runs $0.944$, $0.949$, $0.957$, $0.987$, $0.992$ and $0.995$. Those are high, and the two lowest both involve Mulliken.

So the scales are in near-perfect agreement about ordering and in poor agreement about spacing, and that separation is what makes electronegativity usable at all — provided the use is a ranking.

Uses that are rankings, and are safe:

• Bent’s rule — s character concentrates toward the less electronegative substituent — needs only the order, and Bent’s rule, against the substituent series tests it as an ordering across a substituent series.
• The direction of a bond’s polarity, when the two atoms are not adjacent on the scale.
• Which of two ligands is the stronger σ donor, as a first guess — though the spectrochemical series is not electrostatics shows the ordering that actually matters there running on something else entirely.

Uses that need a magnitude, and are not safe:

• Estimating per cent ionic character from $\Delta\chi$.
• Estimating a bond dipole moment.
• Comparing the polarity of two bonds whose $\Delta\chi$ values are close.
• Anything summed over several bonds, where the errors compound rather than cancel.

That division is sharper than the usual advice — the scales correlate well, so pick one and be consistent — and it is better, because being consistent about a quantity whose magnitude is undefined does not make the magnitude defined.

Normalising is a choice too, and here is what it costs #

Dividing by each scale’s own range is the crudest possible way of making four scales comparable, and it deserves the same scepticism the scales get.

The range chosen here is potassium to fluorine, the two extremes among the elements compared here. That choice matters: a range defined by caesium to fluorine would be wider, and the normalised differences would all shrink by a common factor — which changes no ratio in the table, since every entry in a row is divided by the same number.

Ratios within a row are therefore robust and absolute values are not. The factor of sixty-two between the largest and smallest normalised P–H difference survives any choice of range, because the choice divides both. The statement the O–H bond is thirty-nine per cent of the way across the scale does not survive it, and should be read as a way of putting four numbers on one axis rather than as a measurement.

A more careful normalisation would map each scale monotonically onto another, and it exists: it is the rank transformation, which is exactly what the Spearman coefficient uses. Doing that discards the spacings entirely and returns correlations of $0.94$ to $0.99$ — which is the point. Every treatment that makes the scales agree is a treatment that throws away the magnitudes.

Pauling’s own formula, and what normalisation does to it #

The commonest magnitude use is Pauling’s ionic-character expression, $1 - \exp(-\Delta\chi^2/4)$, which turns a difference into a percentage.

It is worth being explicit that this formula is Pauling-specific. Its constant is fitted to Pauling units, so feeding it a Mulliken difference is a unit error, and feeding it a normalised difference is meaningless because the constant was never fitted to normalised units.

There is no version of it that works across scales, and that is the honest end of the matter. A magnitude that only exists in one scale’s units is a property of that scale, and its agreement with measured dipole moments is a property of Pauling’s fit rather than a discovery about bonds.

The dipole is not a sum of bonds makes the related point from the other side: even given perfect bond moments, adding them as vectors is not how a molecular dipole works, because a lone pair contributes and there is no bond to attach it to.

Where the s-character budget comes in #

The hybridisation arithmetic in the angle does not fix the hybridisation supplies a small illustration of the ranking-against-magnitude distinction, from a completely different direction.

That essay computes, from measured bond angles alone, that water’s bond hybrids carry $0.2002$ of an s function each and its lone pairs $0.2998$; that hydrogen sulfide’s bonds carry $0.0353$ and its lone pairs $0.4647$. The lone pair takes more s character than the bonds in every case, which is Bent’s rule with a lone pair as the limiting substituent.

That is a ranking statement and it holds on all four scales, since all four agree that a lone pair — having no substituent at all — is at the extreme.

The corresponding magnitude statement would be that the s character allocated is proportional to the electronegativity difference, and no such relation survives a change of scale: the constant of proportionality would be in units of whichever scale was used, and the ratio between two molecules’ allocations would change by up to the factors in the table above.

A partial charge that can be computed #

One place in this collection does produce a number for how unevenly electrons are shared, and it is worth putting beside the scales because it has the same character.

Six electrons in a ring that is not all carbon computes π charge densities for pyrrole and furan: the nitrogen keeps $1.720$ of the two π electrons it brings and the oxygen keeps $1.791$, so one donates $0.280$ of an electron to the ring and the other $0.209$.

Those are computed rather than tabulated, and they are still not measurements. They come from a Hückel matrix with one fitted diagonal entry per heteroatom, and that entry is an electronegativity parameter under another name. Sweeping it shows what it is worth: moving the nitrogen’s entry from $1.0$ to $2.5$ takes the donation from $0.385$ to $0.160$, smoothly and monotonically.

So the ordering — nitrogen donates more than oxygen — survives any plausible re-fitting, and the value $0.280$ does not. That is the same conclusion as the one above, reached inside a calculation rather than across four published tables, and it is the reason this collection quotes such numbers as comparisons and never on their own.

What would settle it #

A quantity defined four ways, with four sets of units, has no true value. What can be defined without ambiguity is something measurable, and two candidates are in reach of a calculation not done here.

The electron density. Partitioning it between two atoms gives a partial charge, and the partition is itself a choice — Mulliken, Löwdin, Bader, natural population analysis all give different answers, and the spread among them is comparable to the spread among electronegativity scales. Changing the definition does not remove the arbitrariness; it moves it.

The dipole moment. This is genuinely measurable and unambiguous, and it is a property of the whole molecule rather than of a bond. Extracting bond dipoles from it requires a partition, and the partition is the same problem again.

So the honest conclusion is not that a better scale is needed. It is that bond polarity is not a well-defined quantity, and that electronegativity is a useful ordering imposed on a situation that has no natural magnitude. That is a legitimate thing for a concept to be, and the failure mode is quoting a number from it.

Taking three at once rather than two is where the argument stops being about a marginal pair and starts being about a quantity nobody has.

It is worth ending on the molecule everybody would have said was settled, because its bonds are on that list.

The one difference that needs no normalisation #

Normalising is a choice, and the reason it has to be made is that none of the four scales has a natural zero and a natural one. There is a quantity that does, it is measured rather than tabulated, and for diatomic molecules it settles what a difference ought to look like.

A diatomic’s dipole moment and its bond length together give the charge that has actually moved. If a whole electron had been transferred the moment would be $e$ times the bond length; the measured moment divided by that is a pure number between zero and one, with both ends meaning something — zero is a covalent bond and one is an ion pair. No convention enters and nothing is fitted.

For the hydrogen halides:

That is a difference on an absolute scale, and it is the yardstick the four normalised scales can be laid against.

Pauling’s own formula for converting a difference into an ionic character predicts 55, 21, 13 and 5 per cent for the same four bonds. Three of the four are good — within a couple of points, which is better than the construction deserves. The first is 30 per cent high, and it is the bond with the largest difference, which is the region where a difference is supposed to be most reliable.

Two things follow for the essay’s argument.

A difference is checkable after all, for the small class of molecules where a dipole and a length between two atoms are both measured. That is a stronger position than a comparison of orderings alone, which has nothing to compare a difference against at all.

And what the check finds is a failure at the top of the range. The scales’ disagreements were located above in the region of small differences; this comparison locates a separate failure at large ones, where a formula fitted to give something between zero and one overshoots because it was built to saturate and the real charge transfer saturates sooner.

So the honest summary is narrower than differences are meaningless and wider than differences are fine. A difference is a checkable quantity for a diatomic, it is right to within a couple of per cent in the middle of the range, it is unreliable at both ends, and for anything with more than two atoms there is no measurement to check it against at all.

Why this matters more than it looks #

An objection to all of this is that nobody serious quotes electronegativity to three figures, so measuring the disagreement is beating a straw man.

Two answers.

The uses that need a magnitude are everywhere in teaching. Per cent ionic character from $\Delta\chi$ is in every introductory course; so is the classification of a bond as non-polar covalent, polar covalent or ionic at thresholds of $0.4$ and $1.7$ Pauling units. Those thresholds are magnitudes, they are quoted without a scale attached in most places that quote them, and on the Mulliken scale they classify water’s bonds as non-polar.

A quantity with a well-defined ordering and no well-defined magnitude is an unusual thing, and it is worth being able to recognise. It behaves like a rank rather than like a measurement: differences between ranks are not meaningful, the mean of several ranks is not a rank, and multiplying a rank by anything produces nothing. Every rule above follows from that one observation.

The same shape recurs elsewhere. A Hückel β is a fitted parameter whose ratios are meaningful and whose absolute value is a fit; a ligand-field $e_\sigma$ is the same; a lone-pair repulsion weight is the same. In each case the model is useful, its orderings are robust, and multiplying its number by anything is where the trouble starts.

Who found it, and when #

Pauling’s scale is from 1932 and is the oldest; Mulliken’s followed in 1934, Allred and Rochow’s in 1958, and Allen’s in 1989. Each was proposed as an improvement on its predecessors and each is internally coherent.

Mulliken himself was explicit that his scale measured something different from Pauling’s, and the near-agreement between them was treated as evidence that both were measuring a real property. That reading persisted, and the correlation coefficients above are usually quoted in support of it.

What the normalisation shows is that the agreement is entirely in the ordering. Two quantities can have a rank correlation of $0.96$ while differing by a factor of sixty on a particular pair, and the pairs where they differ most are among the commonest bonds in chemistry.

What follows for every use of electronegativity #

This may be as far as the argument goes, because the honest conclusion is a negative one. The scales are different quantities, and the difference is in exactly the half of the concept most uses depend on.

What follows from that conclusion is visible elsewhere. Every place electronegativity appears in these essays, it appears as an ordering: as the direction of Bent’s rule, as the sign of a Hückel diagonal parameter, as the reason a heteroatom donates less than a carbon would. Nowhere does a number from any of the four scales get multiplied by anything, and that is deliberate.