A dispute is a small difference

Twelve hydrogen bonds run through four published electronegativity tables, anchored so the four can be compared, show that a quarter of them have no agreed direction: three of the twelve have hydrogen at the negative end on Mulliken’s table and at the positive end on the other three.

Every bond in that set contains hydrogen, so every dispute in it is a dispute about hydrogen by construction. The closing question named the way out — the interhalogens, five compounds whose elements are all in all four tables, all with measured dipoles, and none containing hydrogen. A series with no hydrogen in it would say whether the disagreement is about one awkward element or about the whole idea.

All four tables agree about every interhalogen. But that answers a smaller question than it appears to, because the thing that decides a dispute turns out not to be which elements are involved.

What a disputed direction costs #

It is worth being clear what is at stake before measuring where it happens, because the answer is not that a table is wrong.

The table that could not have mattered established the boundary case: for a molecule of two elements, the choice of table cannot change anything a reader would act on, because every table agrees on the ordering and the molecule’s dipole direction follows from that alone. The hydrogen bonds show where that protection runs out — a bond whose direction is not agreed is a bond whose contribution to a molecular dipole has no agreed sign, and a sum of such contributions is not a quantity at all.

That matters because bond dipoles are almost always used as a sum. The dipole is not a sum of bonds is about why the sum fails even when every term is agreed; this is about terms that do not have a value to be summed.

One number, and nothing in between #

Take every bond in the comparison — twenty-five real bonds plus the six interhalogen pairs — and compute, on each of the four tables, the difference in electronegativity divided by that table’s own range over the elements it covers. Dividing by the range is what makes the four comparable: two of the tables are dimensionless numbers on a scale fixed by convention and two are energies in electronvolts, so a raw difference would be a comparison of unit systems.

The six bonds the tables disagree about all have a mean normalised difference below 0.0507. The twenty-five they agree about all have one above 0.0584. Nothing in the collection lies between those two numbers.

That is a complete separation on a single quantity, and it does not mention elements. A bond is disputed when its electronegativity difference is small — small enough that the four tables’ own disagreement about the value is larger than the value itself — and the sign then depends on which table is consulted. The disputed hydrogen bonds are in that group because H–C and H–S happen to have small differences, not because hydrogen is peculiar.

The threshold has a natural reading, and it is worth stating even though nothing here tests it: the four tables are four attempts to measure the same thing, and they agree to within a few hundredths of their own ranges. A difference smaller than that is a difference below the resolution of the concept, not merely below the resolution of one table. On that reading a disputed bond is not a bond the tables get wrong — it is a bond about which electronegativity, as a body of practice, does not have an answer.

It is not the tables being far apart #

The natural alternative explanation is that a disputed bond is one where the tables are unusually inconsistent. It is worth killing because it is more comfortable than the truth — it would make the dispute a property of the tables’ disagreement rather than of the bond.

The spread does not separate the groups. C–S is disputed and has the smallest spread of any bond in the collection, 0.0318; O–H is agreed and has nearly the largest, 0.3625. The contested bonds are scattered through the same range of spreads as the rest. What they have in common is being close to zero on the horizontal axis, and only that.

The relation between the two quantities is the obvious one once stated: a sign flips when the difference is smaller than the disagreement about it. A bond with a large spread and a larger difference is safe; a bond with a small spread and a smaller difference is not. It is the ratio that matters, and since the spreads are all of a similar order, the numerator carries almost all of the variation.

There is a small asymmetry worth noticing in the numbers. The spreads run from 0.032 to 0.363, a factor of eleven, while the differences run from 0.0006 to 0.664, a factor of a thousand. So the denominator of that ratio varies over a much wider range than the numerator does, which is exactly why one of the two quantities separates the groups and the other does not. It is not that the spread is irrelevant in principle; it is that on this collection it barely varies compared with what it is being divided into.

What the interhalogens were #

The proposed test comes out clean, and the clean result is the uninformative one. All four tables agree that fluorine is richer than chlorine, chlorine than bromine, bromine than iodine, and every combination of those — six pairs, four tables, twenty-four verdicts, no disagreement.

Measured against the boundary, five of the six sit between 3.2 and 9.2 times clear of it. I–F, the widest, has a difference nine times what a dispute requires. Those five could not have been contested by any of these tables, and their agreement is not evidence about anything.

The series was chosen for two properties — measured dipoles exist, and no hydrogen is involved — and neither is the property that makes a disagreement possible. That is not a criticism of the choice: the aim was a series that could be checked against measurement, and it is the only one available. It is a statement about what the test could have found.

Which bonds are actually disputed #

The six are worth naming, because the list is not what the hydrogen set would suggest. They are C–S, N–Cl, C–I, S–H, C–H and N–Br, in order of increasing difference. Two contain hydrogen and four do not — a fact the hydrogen set could not have shown, because it was twelve hydrogen bonds and contained no others to compare against.

So a hydrogen-free dispute was available in this collection all along — four of them — and the interhalogens were not the only hydrogen-free series that could have been tried. What distinguishes C–S, N–Cl, C–I and N–Br from Cl–F, Br–F and I–F is not the presence of an element but that they join partners of nearly equal electronegativity across different groups, where the tables have no shared periodic trend to fall back on. Two halogens are always in one group with a monotone ordering every table reproduces, which is the structural reason the series could not have disputed anything.

The one that came close #

One interhalogen is interesting, and it is the one nobody would have picked. Br–Cl has a difference of 0.0584, which is 1.07 times the boundary — and it is the smallest difference of any agreed bond in the entire collection. The series does contain a near miss; it is simply not the one with the measured dipole anybody would have reached for first.

Set it beside N–Br, the most comfortably disputed bond at 0.0507, and the two are separated by less than a hundredth of a table’s range. On one the four tables’ marks straddle zero and on the other they do not, and that is the whole of the difference between a bond whose direction is a fact and one whose direction is a choice of reference book.

So the honest form of the answer to the closing question is: the disagreement is about neither one awkward element nor the whole idea. It is about a threshold, and hydrogen appears on the wrong side of it twice out of twelve because two of its bonds are nearly non-polar.

That framing also settles a question left open about the hydrogen set. What a dipole cannot tell apart is about molecules a measurement cannot distinguish; this is the same shape one level down, at the bond. Two bonds with differences of 0.05 and 0.06 are not meaningfully different in polarity, and the fact that the tables sort one into “disputed” and the other into “agreed” is an artefact of asking for a sign rather than a magnitude. The threshold is sharp because the question is binary, not because the chemistry is.

What was computed, and how #

Four published tables — Pauling, Mulliken, Allred–Rochow and Allen — for the eighteen elements all four cover. For each bond, the difference on each table divided by that table’s range across those eighteen; a bond is disputed when the four normalised differences do not share a sign.

Normalising by the range rather than anchoring to a chosen bond matters here and did not matter for the hydrogen set. Anchoring rescales each table so a nominated bond comes out the same, which is right when the question is about magnitudes; but a rescaling by a positive factor cannot change a sign, so for this question any positive normalisation gives the same verdicts. That independence is worth having explicitly: one coordinate, three point groups is about a conclusion that survives a change of description, and a verdict immune to the normalisation is the same kind of claim — it belongs to the tables rather than to the arithmetic applied to them. The range is used because it needs no bond to be nominated.

The refusal is an element against itself. Its difference is exactly zero on all four tables, and it has to be reported as having no direction rather than as a disagreement — a sign test on zero would otherwise count the arithmetic’s own tie as a dispute, and would report every element in the table as contested with itself.

Zero dipole is not no interaction makes the companion point about a molecule whose dipole vanishes by symmetry: a quantity being zero is not the same as there being nothing there. Here a bond dipole near zero is not nothing either — the charge separation is small but the bond is real, and the ambiguity is in the sign rather than in the existence.

Where the model stops #

The boundary is fitted from six disputed bonds, and six is few. Its position is only known to lie between 0.0507 and 0.0584, and the claim that it is sharp rests on there being nothing in that interval — which is a statement about this collection of thirty-one bonds rather than about chemistry. A wider set would populate the gap and turn a clean separation into a probability.

The elements are also eighteen main-group ones, being those all four tables cover. Transition metals are where electronegativity scales differ most and are absent entirely, so nothing here says whether the same threshold governs them. Nor does anything here involve a lone pair, which the lone pair is not the missing term showed carries its own contribution to a molecular moment — a bond-level threshold says nothing about a quantity that is not a bond.

And a normalised difference is not a physical quantity. It is a way of putting four incommensurable tables on one axis, and its numerical value carries no meaning beyond the ordering it produces — which is why the boundary is quoted as a range and the interhalogens’ distances as multiples of it rather than as differences.

One more limit belongs to the framing rather than the data. A dipole is not what an infrared spectrum sees established that a dipole’s derivative is what a measurement responds to, and nothing here touches derivatives — a bond whose sign is disputed has a disputed contribution to the moment and a separately disputed contribution to its change under a stretch, and the second does not follow from the first.

The generalisation #

The lesson is about designing a test rather than about electronegativity, and it keeps turning up from different directions.

The hydrogen-bond finding is real — some bonds have no agreed direction — and so is the question about its scope. The series proposed to answer that question was chosen for the properties that made it available: the data existed, and it excluded the suspected culprit. Neither property has anything to do with the mechanism that produces the effect, so the series had almost no chance of showing the effect whether or not the effect is general.

The check that would have caught it is cheap and it is the one performed here: before running a test, compute where its cases sit on whatever axis the effect is known to vary along. If they all sit far from the interesting region, the test can only confirm. That is the same arithmetic as choosing sizes so that a scaling variable matches — a test’s cases have to be placed where the answer could go either way, and placement is a calculation rather than a judgement.

Here it would have taken one line: the interhalogens’ differences run from 1.07 to 9.2 times the threshold, so five of the six carry no information and the sixth is worth looking at closely.

The habit generalises past test design. Any binary verdict computed by comparing a quantity against zero — a sign, an ordering, a “which is larger” — has a region near zero where the verdict is decided by the measurement’s own scatter rather than by the thing measured, and the width of that region is the scatter. Reporting the verdict without reporting how far the case sits from the boundary hands a reader a fact where there is a coin toss. Computing rather than quoting makes that distance available for nothing, which is the only reason the boundary above could be located at all.

Who found it, and when #

That electronegativity differences below a few tenths of a Pauling unit make bond polarity ambiguous is old and general knowledge, and the various tables’ disagreements for near-equal partners have been noted since Allred and Rochow’s own comparisons in the 1950s. Nobody discovered the threshold here.

What is new here is the measurement: that the separation is complete on these thirty-one bonds, where the boundary lies, and that the series proposed to test the effect sits almost entirely outside the region where the effect can occur.

Still open: whether the boundary is sharp #

The obvious open question is more bonds. Thirty-one is enough to see a gap and not enough to know it is real, and the four tables cover eighteen elements — a hundred and fifty-three pairs, of which a fifth are used here. Running all of them would either populate the interval between 0.0507 and 0.0584, making the boundary a soft one with a width worth quoting, or leave it empty, which on a hundred and fifty cases would be a much stronger claim than it is on thirty-one.

The nearer question is Br–Cl and the measurement. It is the one interhalogen close enough to the boundary to be worth a real test, all four tables put it on the same side by a small margin, and it has a measured gas-phase dipole moment — and no interhalogen dipole is yet part of the comparison. Adding that one number, from a stated source, would let the four tables’ predicted bond dipole for Br–Cl be compared against the measurement in the same way the hydrogen halides were compared. It is the one case in the proposed series where the comparison could distinguish the tables, and it needs a single quoted value rather than a calculation.