Molecular Geometry & Bonding
what the shape decides
What is taught wrongly

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.
15 min read 6 figures Taught confidently, and wrongCounted, not quoted

Every textbook prints a table of electronegativities. Many print two, on facing pages, without remarking on it.

The two are not two measurements of one quantity. They are two different quantities that happen to correlate, and the difference decides which way arrows get drawn on bonds.

Four scales, four orderingsEach column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.PaulingdimensionlessMullikeneVAllred–RochowdimensionlessAllendimensionlessHHLiLiBeBeBBCCNNOOFFNaNaMgMgAlAlSiSiPPSSClClKKBrBrII12 inversions14 inversions2 inversionsSpearman between adjacent columns: 0.957 · 0.949 · 0.995 — high, and not oneranked, because the units do not comparefour definitions, four orderings
Fig. 1 Four scales, four rank ladders, with a line joining each element across them. Lines that stay parallel are agreement; every crossing is a pair of elements two adjacent scales order the opposite way round. Spearman’s coefficient between neighbouring columns is printed at the foot.

Four definitions, none of them equivalent

Pauling’s (1932) comes from thermochemistry. The bond energy of A–B is compared with the geometric mean of A–A and B–B, and the excess is attributed to ionic character. The scale is dimensionless and its zero and its unit are conventions.

Mulliken’s (1934) is the mean of the ionisation energy and the electron affinity of the free atom. It has units of energy, it is a property of an isolated atom rather than of a bond, and it has an appealing physical interpretation as the atom’s resistance to both losing and gaining an electron.

Allred and Rochow’s (1958) is the electrostatic force a valence electron feels at the covalent radius: an effective nuclear charge divided by a radius squared. It is a property of the atom’s size and charge.

Allen’s (1989) is the average one-electron energy of the valence shell, taken from atomic spectra. It is called the configuration energy and is the most directly measurable of the four.

Four constructions, from bond energies, from ionisation, from electrostatics and from spectra. There is no reason for them to agree, and the surprise is that they agree as well as they do.

The defence, and what it evades

The standard response is that the scales correlate at better than 0.99, so the disagreement is negligible.

That is true and it answers a question nobody asked. A correlation coefficient measures whether two lists rise together. It does not measure whether they order any particular pair the same way, and the pairs are what a chemist uses.

Two lists can correlate at 0.99 and still disagree about a dozen specific comparisons, and the disagreements need not be at the ends where nobody cares. They can be anywhere.

So the useful statistic is not the correlation. It is the count of pairs the two scales order differently, and the list of which pairs those are.

The counts

Ranking eighteen main-group elements gives a hundred and fifty-three pairs to compare.

Pauling against Allen: three discordant pairs. Allred–Rochow against Allen: two. Pauling against Allred–Rochow: five.

Pauling against Mulliken: twelve. Mulliken against Allred–Rochow: fourteen. Mulliken against Allen: fifteen.

Mulliken’s scale is the outlier throughout, and the reason is structural rather than accidental. It is the only one of the four that is a property of a free atom in isolation rather than of an atom in a molecule, and free-atom properties do not track bonding behaviour as closely as the other three constructions do.

Hydrogen is the element that moves furthest — three rank positions out of eighteen between Pauling and Mulliken — which is unfortunate, since hydrogen is in more bonds than anything else.

The bonds it reaches

Rank inversions in the abstract are a curiosity. Restricted to bonds that occur in ordinary chemistry, they are a problem.

Bonds the scales disagree aboutFor each bond, which atom each scale calls the more electronegative. Every row is a bond whose polarity would be drawn in opposite directions depending on which of four tables in common use was consulted.bondPaulingMullikenAllred–RochowAllenC–Hδ− on Cgap 0.35δ− on Hgap 0.91δ− on Cgap 0.30δ− on Cgap 0.24S–Hδ− on Sgap 0.38δ− on Hgap 0.96δ− on Sgap 0.24δ− on Sgap 0.29C–Sδ− on Sgap 0.03δ− on Cgap 0.05δ− on Cgap 0.06δ− on Sgap 0.04C–Iδ− on Igap 0.11δ− on Igap 0.49δ− on Cgap 0.29δ− on Cgap 0.19N–Clδ− on Clgap 0.12δ− on Clgap 1.00δ− on Ngap 0.24δ− on Ngap 0.20N–Brδ− on Ngap 0.08δ− on Brgap 0.29δ− on Ngap 0.33δ− on Ngap 0.386 of the ordinary bonds tested have their direction disputedthe gaps are small in every case, which is the point: a small gap still has a signthe consequence of four definitionscompared by rank
Fig. 2 Every ordinary bond tested whose polarity direction is disputed between the four scales. Each row is a bond that would be drawn with the arrow pointing in opposite directions depending on which of four tables in common use was consulted.

C–H is the one that matters. Pauling, Allred–Rochow and Allen all put carbon above hydrogen, so the carbon end is drawn negative. Mulliken puts hydrogen above carbon — 7.18 electronvolts against carbon’s 6.27 — and the arrow reverses.

Every C–H bond in organic chemistry, drawn the other way round, on a scale printed in the same books as the one that gives the familiar answer.

C–S flips between Pauling and Allen on one side and Mulliken and Allred–Rochow on the other. C–I splits two against two. N–Cl and N–Br likewise.

The gaps involved are small in every case, and that is the point rather than a mitigation. A polarity arrow encodes a sign, and a small gap still has one.

Where it does not matter, which is most places

The scales are used constantly and the ambiguity almost never surfaces, and understanding why is what makes the criticism proportionate.

Four scales, four orderingsEach column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.PaulingdimensionlessMullikeneVAllred–RochowdimensionlessAllendimensionlessFFOONNClClCCSSHHPPSiSiAlAlMgMgNaNaLiLiKK5 inversions6 inversions2 inversionsSpearman between adjacent columns: 0.974 · 0.965 · 0.991 — high, and not oneranked, because the units do not comparefour definitions, four orderings
Fig. 3 The same four scales restricted to a set of elements spread widely apart. Across a range this large the lines are nearly parallel and the orderings agree: fluorine is at the top on every scale, potassium at the bottom on every scale, and nothing in between crosses.

That is the case a working chemist meets. Comparing oxygen with carbon, or fluorine with hydrogen, or chlorine with sodium, every scale gives the same answer and the choice is immaterial.

The inversions cluster among elements that are close together on every scale, and close-together comparisons are the ones a chemist has least reason to make. Carbon against sulfur, nitrogen against bromine, carbon against iodine: real bonds, and bonds whose polarity nobody argues about because the effect is small either way.

So the honest statement of the problem is narrow. The scales agree wherever the gap is large, disagree wherever it is small, and the disagreement is therefore confined to cases where the quantity being disputed barely matters. Except for C–H, which is not a rare bond.

Where a symmetry argument replaces it entirely

The alternative worth setting beside all this is a class of question about polarity that needs no electronegativity at all.

water — C2vThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 4 Water, C₂ᵥ, with the group recovered from the coordinates and searched for again at every step of the slider. Its group leaves a direction fixed, so it may be polar — a conclusion reached without any electronegativity whatever.
carbon dioxide — D∞hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.OCOD∞hprincipal axis C83 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 5 Carbon dioxide, D∞h. Its group contains an inversion centre, so the dipole moment is exactly zero — not small, not approximately cancelling, but forbidden for every conceivable electron distribution the molecule could have. The bond-vector method reaches the same answer by adding two vectors that happen to cancel, which is a much weaker statement.
What the group settlesFor each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds
Fig. 6 Both properties for a set of molecules, derived from the point group alone. Nothing in either column required knowing what any bond is made of, and where the answer is no, it is exact.

That is the contrast this site keeps returning to. The symmetry statement is exact, needs no fitted quantity, and answers only a yes-or-no question. The electronegativity route gives a magnitude and gives it with a scale choice buried inside.

Where symmetry can answer, it should. Where it cannot — and it cannot supply any magnitude at all — the electronegativity route is what remains, and its input should be named.

What is computed here, and what is quoted

The distinction matters and is worth being explicit about.

The scale values are quoted. They are measurements, or fits to measurements, and a site that pretended to derive Pauling’s numbers from first principles would be worse rather than better. Every number in the four tables comes from the literature.

The comparison is computed. Ranks, Spearman’s coefficient for every pair of scales, the count of discordant pairs, which elements move furthest, and which ordinary bonds have a disputed direction.

Comparing by rank rather than by value is forced by the units. Two of the scales are dimensionless numbers on a conventional footing and two are energies in electronvolts. Putting them on one axis requires a linear fit whose coefficients are themselves a choice — and a great deal of why the scales look more alike in print than they are is that this fit has already been applied before a reader sees them.

Spearman’s coefficient is computed from the ranks rather than by the usual shortcut formula, because the shortcut assumes no ties and hydrogen ties with itself across Pauling and Allred–Rochow at exactly 2.20. That is a small thing and it shifts a coefficient in the third decimal place, which is where the interesting part of this comparison lives.

The two checks that can fail

The claim being made is a conjunction — they correlate and they do not agree — and printing a conjunction is only honest if both halves are tested.

Every pair must correlate above 0.9. If it did not, the first half would be false and these would not be four measurements of anything like the same thing. All six pairs clear it.

At least one pair must be discordant. If none were, the second half would be false and this essay should be withdrawn. Six pairs are discordant.

Both run in the site’s gate and either would stop the build.

There is a third check, and it exists because a comparison that manufactured disagreements out of ties or arithmetic noise would pass the second test spuriously. A scale compared with itself must show no discordant pairs and must correlate with itself at exactly one. All four do.

That third one is the tripwire, and it is the pattern this site applies to every assertion: an assertion that has never rejected anything proves nothing.

The surprise: it is not an observable, and nothing can fix that

The instinct on reading the above is to ask which scale is right.

None of them is, and the question has no answer, because electronegativity is not an observable. There is no measurement of “the electronegativity of carbon” that the four scales could be compared against. Each is a construction from quantities that are observable — bond energies, ionisation energies, radii, spectral terms — and each construction is a different function of different measurements.

So a fifth scale would not settle it. It would be a fifth construction, correlating with the others at about 0.99 and ordering a dozen pairs differently.

That reframes what the disagreement means. It is not evidence that somebody has made an error; it is evidence that the quantity is under-determined by its definition. Electronegativity names a tendency that several different measurements partly capture, and asking for the number is asking for a precision the concept does not have.

Which is a perfectly respectable position for a chemical concept to be in. What is not respectable is quoting a value to two decimal places without saying which construction produced it.

What it costs

The comparison costs a page of arithmetic over seventy-two quoted numbers.

What the ambiguity costs in use is larger and is paid in three places.

Bond polarity. The bond-dipole method takes an electronegativity difference as its input, so its output inherits the scale choice — and the qualitative conclusion, which end is negative, is exactly what the scales disagree about.

Bent’s rule. The rule says s character follows electronegativity, which means the rule as used is “s character follows whichever scale was chosen”, and the choice is never stated. In practice it is always Pauling’s, which makes the rule a narrower claim than it appears.

Ionic character. The percentage ionic character of a bond is computed from an electronegativity difference by a fitted formula, so it carries two arbitrary choices rather than one.

In each case the ambiguity is invisible in the result. A number comes out, it looks like a measurement, and the scale it depends on has vanished from the record.

Where the model stops

Three limits.

Eighteen elements. The comparison here covers the main-group elements from hydrogen to iodine. Extending it to the transition metals would make the disagreement worse, not better — electronegativity depends strongly on oxidation state there, and most tables quote a single value regardless.

Rank statistics cannot say which scale is right, because there is no reference. What the comparison establishes is that the quantity is under-determined; it does not identify a best construction, and it cannot.

The comparison is over one fixed set of elements. Changing the set changes every rank and therefore every count, so “twelve discordant pairs” is a statement about these eighteen elements and not a universal constant. What is robust across any reasonable set is the ordering of the six pairwise comparisons — Mulliken is always the outlier — and the identity of the elements that move most.

An atom’s electronegativity is not a constant anyway. Every scale here assigns one number per element. The property being summarised depends on hybridisation, oxidation state and what else is attached — an sp carbon is meaningfully more electronegative than an sp³ one, which is exactly the observation Bent’s rule is built on. So the tables are averages over a range that the tables themselves have no way to express.

The clean summary is one sentence. Electronegativity is a genuinely useful idea, defined four ways, quoted to two decimal places, and not an observable — and only the fourth of those four facts is ever printed alongside the tables.

Who found it, and when

Pauling introduced the concept in 1932 in a paper on the nature of the chemical bond, and it was an immediate success because it gave chemists a number where they had previously had an intuition.

Mulliken’s alternative arrived two years later and was, on its face, better founded: it is a property of an atom that follows from two measurable quantities, with no arbitrary constants. It did not displace Pauling’s, and the reason is instructive — Pauling’s is derived from bonds, and chemists were asking about bonds.

Allred and Rochow’s (1958) and Allen’s (1989) each arrived with an argument that it was more fundamental than the others. Both are, in their own terms.

The proliferation is the interesting part of the history. Sixty years produced four scales rather than a consensus, which is what happens when a concept is useful, is intuitively clear, and is not defined by any single measurement. Pauling himself was clear-eyed about it, describing electronegativity as “the power of an atom in a molecule to attract electrons to itself” — a definition that names a tendency and quantifies nothing.

The tables then acquired an authority the definition never supported. That is the failure this essay is about, and it is the same failure hybridisation and the d-orbital account of hypervalency suffer from: a useful construction, repeated until it is treated as a fact about nature.

Where the ladder goes next

The argument this ambiguity damages most directly is the dipole is not a sum of bonds.

The rule that depends on it quantitatively is Bent’s rule.

The exact alternative, where a property follows from symmetry rather than from a fitted scale, is symmetry forbids a dipole.

A reader wanting one rule to take away can have this: an electronegativity difference is a fine input to a qualitative argument when the two elements sit far apart on any table, and is no input at all when they sit close. The threshold is not sharp and it does not need to be, because the cases that matter are the ones where the gap is under about half a Pauling unit — which is exactly where every disagreement in the table above lives, and exactly where a chemist has least business drawing a confident arrow.

What the pictures here cannot show. The rank ladders on this page contain no values, only positions, and that is deliberate — the values cannot be plotted on a common axis without a fit, and the fit is the thing being questioned. A reader wanting to see how far apart two elements are on a scale is wanting a comparison that the four sets of units do not permit.

Where this goes next

moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds The dipole is not a sum of bonds Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero. sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron Bent's rule, against the substituent series More electronegative substituents get more p character, so the angle between them shrinks. The rule predicts a trend, the trend is observed, and the quantity it depends on turns out not to be one quantity. HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms Why water is bent The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.