The dipole is not a sum of bonds

The standard method for deciding whether a molecule is polar: give each bond a dipole vector pointing toward the more electronegative atom, add them up, and see whether they cancel.

It gets carbon dioxide right, water right, and methane right. It also has an adjustable input, an unaccounted-for contribution, and no notion of exactness.

What the method assumes #

That the molecular dipole moment is a sum of contributions localised on bonds.

A dipole moment is an integral over the whole electron density weighted by position. Decomposing that into per-bond contributions requires deciding which part of the density belongs to which bond, and there is no unique way to do it — the density is a single function and does not come partitioned.

So bond dipoles are a model, not a decomposition. They can be defined within a scheme and they are scheme-dependent, and different partitioning methods give different values for the same molecule.

The assumption also has a second, quieter part. Adding vectors requires the contributions to be independent of one another, and a bond’s share of the density is not independent of what is attached to the other end of the atom it starts from. The C–H bonds in chloroform and in methane are given the same dipole in the usual tables and are not the same bond, which is why the method’s failures grow with the number of different substituents rather than with the size of the molecule.

Electronegativity, which is not one quantity #

The input to the method is worse than it looks.

Pauling’s scale (1932) is derived from bond dissociation energies, comparing a heteronuclear bond with the mean of the two homonuclear ones. It is dimensionless and arbitrary in origin.

Mulliken’s scale is the average of the ionisation energy and the electron affinity, which has units of energy and is a property of the free atom.

Allred–Rochow uses the effective nuclear charge felt at the covalent radius.

Allen’s uses the average valence-electron energy from spectroscopy.

There is no single number called the electronegativity of an atom, because electronegativity is not an observable — it is a useful summary of a tendency, defined differently by different people for different purposes.

They correlate and they do not agree, measured #

“They correlate and they do not agree” is the sentence every source reaches for, this one included until it was checked, and on its own it settles nothing. The usual defence — that the scales correlate at better than 0.99 — is true, and is an answer to a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on opposite sides of carbon, and that is what decides which way a C–H bond gets drawn.

So the four scales are put in rank columns and every element is joined across them. Ranking rather than plotting one against another is forced by the units: two of these scales are dimensionless numbers on a conventional footing and two are energies in electronvolts, and putting them on a common axis requires a linear fit whose coefficients are themselves a choice. A large part of why the scales look more alike in print than they are is that the fit has already been done before the reader sees them.

The counts are the useful output. Pauling against Allen is nearly monotonic, with three discordant pairs out of a hundred and fifty-three. Pauling against Mulliken has twelve, and Mulliken against Allen fifteen. Mulliken is the outlier throughout, and hydrogen is the element that moves furthest — three rank positions between Pauling and Mulliken, in a list of eighteen.

That single displacement is the whole of the argument.

The consequence is a short list of bonds whose polarity would be drawn in opposite directions depending on which of the four tables was consulted, and it is common for two of the four to appear on the same page of the same book.

The carbon–hydrogen bond is the one that matters. Pauling, Allred–Rochow and Allen all place carbon above hydrogen, so the carbon end is drawn negative. Mulliken’s scale places hydrogen above carbon — the mean of hydrogen’s ionisation energy and electron affinity is 7.18 electronvolts against carbon’s 6.27 — and the arrow reverses. Five more ordinary bonds behave the same way: S–H, C–S, C–I, N–Cl and N–Br.

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

Building a quantitative argument on this is building on a scale-dependent construct. Building a qualitative one is worse, because the qualitative claim — which end is negative — is exactly the one the scales disagree about.

The missing contribution #

The bigger practical problem is that lone pairs are not bonds and contribute substantially.

Ammonia’s moment is 1.47 debye. The three N–H bond dipoles alone, summed, would give considerably less — the lone pair on nitrogen points along the threefold axis, in the same direction, and contributes a large share of the total.

Water is the same. Its moment of 1.85 debye is not the vector sum of two O–H bonds; the two lone pairs contribute in the same direction as the resultant of the bonds.

So the method as usually taught omits a term that is often comparable to the ones it includes. It survives because the omitted term usually points the same way as the included ones, and because most exercises ask only whether the moment is zero.

What the symmetry argument gives #

By contrast, the symmetry statement is exact and needs nothing.

A dipole moment is a vector, and every symmetry operation must leave it unchanged. If no direction is fixed by every operation, the moment is exactly zero. That restricts polar molecules to the groups C₁, Cs, Cₙ and Cₙᵥ.

No electronegativities, no bond dipoles, no lone-pair correction, no partitioning of the density. Just the shape, and an answer that is exact where it applies.

Where the two disagree #

The interesting cases are where the shortcut and the exact argument diverge, and there are three families.

Molecules where lone pairs dominate. Carbon monoxide has a small moment of about 0.11 debye with the carbon end negative — the opposite of what electronegativity predicts, since oxygen is the more electronegative atom. The bond-vector method gets the direction wrong. The reason involves the lone pair on carbon and the details of the orbital occupations, and no simple vector picture recovers it.

Molecules of high symmetry with polar bonds. Carbon tetrachloride, sulfur hexafluoride, benzene. The bond-vector method gets these right, and it gets them right by cancellation — which the symmetry argument gets right by necessity. The distinction matters because cancellation of a sum invites the question “how exactly?”, and the answer is that the question is misconceived.

Molecules where the arithmetic is ambiguous. Anything where the bonds are not obviously localised, including delocalised systems, where per-bond contributions are not well defined at all.

Partial charges have the same problem #

Worth noting, because the same criticism applies to a related and widely used construct.

Atomic partial charges are not observables either. There is no unique way to divide a molecular electron density among atoms, and the schemes that do it — Mulliken, Löwdin, Bader, natural population analysis, charges fitted to the electrostatic potential — give different answers for the same molecule, sometimes differing by a factor of two.

That does not make them useless. It makes them model quantities, comparable within a scheme and not across schemes, and a paper quoting a partial charge without naming the scheme has not said much.

The pattern is the same as hybridisation: a construct within a description, useful, and not a property of the molecule.

What to use when #

A practical position rather than a purely critical one.

For “is it polar at all”, use symmetry. It is exact, fast, and needs only the shape.

For “which end is negative”, use electronegativity carefully. It is right most of the time and it is a heuristic, and carbon monoxide is the standing reminder that it can fail.

For a number, measure it or compute it. Dipole moments are measured accurately by microwave spectroscopy and computed accurately by standard methods. Neither route goes through bond dipoles.

Where the number comes from #

The unit is worth a note. The debye is $3.336\times10^{-30}$ coulomb metres — about the moment of an electron and a proton separated by a fifth of an ångström — and it is named for Peter Debye, who developed the measurement of moments by dielectric constant in the 1910s and 1920s.

The reason the method mattered so much historically is that in the 1920s there were very few ways to learn anything about molecular structure. A dipole measurement gave a single number, and a symmetry argument could turn that number into a structural conclusion — that sulfur dioxide is bent, that carbon dioxide is linear — long before there was any way to see either.

That is symmetry doing real work with almost no information, which is the recurring theme of the field this essay sits in.

The cases the shortcut gets right #

It would be unfair to leave the impression that the bond-vector method is useless, because it is not, and knowing when it works is more useful than knowing that it sometimes fails.

It works when three conditions hold: the bonds are clearly localised, there are no lone pairs on the central atom, and the question is only whether the moment vanishes.

Where it fails is where one of the three conditions goes: ammonia and water have lone pairs, carbon monoxide has an electron distribution the electronegativities do not summarise, and delocalised systems have no well-defined per-bond contributions to add.

The honest summary is that the method is a mnemonic for the symmetry result, dressed as a calculation. Where the two agree, the symmetry argument is doing the work.

What it costs to say it properly #

Almost nothing, which is why the vague version is hard to excuse.

The four scales are quoted. They are measurements, or fits to measurements, and quoting them is right — a site that pretended to compute Pauling’s numbers from first principles would be worse rather than better. What is computed is the comparison: the ranks, Spearman’s coefficient between every pair of scales, the count of discordant pairs, and the list of ordinary bonds whose direction is disputed. That is a page of arithmetic over a table of seventy-two numbers.

Two checks make the arithmetic worth trusting, and both were built to be capable of failing.

A scale compared with itself must show no disagreement. If the discordant-pair count were fabricating inversions out of ties or floating-point noise, a scale against itself would produce some. It produces none, and correlates with itself at exactly one.

Both halves of the claim are checked. “They correlate and do not agree” is two statements, and printing it is only honest if both are checked. So every pair of scales must correlate above 0.9 — otherwise the first half is false and these are not four measurements of anything like the same thing — and at least one pair must be discordant, otherwise the second half is false and this essay should be withdrawn. Both conditions are computed, and either failing would withdraw the claim.

Spearman’s coefficient is computed from the ranks rather than from 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 is the kind of small thing that quietly shifts a coefficient in the third decimal place, which is where the interesting part of this comparison lives.

The test the method passes, and the angle at which it fails #

The bond-vector method survives because it makes a claim the symmetry argument cannot: that a bond’s moment is a transferable quantity, the same in every molecule the bond appears in. That claim is testable on one family of molecules and the result is more interesting than either a success or a failure.

Take chlorobenzene, which has one C–Cl bond and a measured moment of 1.69 debye. If the bond moment is transferable, then two chlorines on the same ring should give the vector sum of two such moments, at whatever angle the substitution positions put between them. Two vectors of magnitude $m$ separated by $\theta$ sum to $2m\cos(\theta/2)$, so:

The para result is not evidence, because symmetry forces it and the vector sum could hardly have got it wrong. The meta result is evidence, and it is very good — 1.69 predicted against 1.72 measured, two per cent, from one number measured on a different molecule.

The ortho result is where the method fails, by seventeen per cent, and it fails in the direction that identifies the missing physics. Two chlorines on adjacent carbons are close enough to polarise each other: each one’s electron withdrawal is opposed by its neighbour’s, so each bond’s actual moment is smaller than the isolated bond’s. The vectors are being added with a magnitude the molecule no longer has.

That gives the method a stated domain rather than a general suspicion. Bond moments are transferable when the bonds are far enough apart to be independent, and the distance at which that stops being true is about one bond. Meta works and ortho does not, on the same ring, with the same two substituents.

It is worth noticing what kind of failure that is. It is not a failure of the electronegativity scale, since the same $m$ was used for all three rows and was taken from measurement rather than from a table. It is a failure of the assumption underneath the whole construction: that a molecule’s charge distribution can be cut into pieces that do not notice one another.

Which is why the symmetry argument keeps its place beside it. Symmetry gets the para row exactly, for a reason no amount of interaction between substituents could disturb; the vector sum gets the meta row to two per cent and the ortho row to seventeen, and nothing in it says in advance which.

Where the model stops #

Three limits.

Symmetry gives necessary conditions only. It can forbid a moment absolutely and can never require one, so it cannot predict a magnitude.

Nothing here computes a moment. The calculations here are one-electron and geometric. The measured moments quoted come from the literature, and the figures establish the point groups the argument turns on.

A rank comparison cannot say which scale is right, and none of them is. Showing that four scales disagree does not identify a fifth that would not. Electronegativity is not an observable, so there is no measurement any of these could be checked against; what the comparison establishes is that the quantity is under-determined, not that somebody has made an error. The remedy is not a better scale. It is to stop treating a scale-dependent construct as though it settled a question about a molecule.

The molecules with lone pairs #

Two cases where the omitted contribution is largest, and where the shortcut is most misleading.

Ammonia, at 1.47 debye, is the case where the two contributions point the same way: the three N–H dipoles resolve along the threefold axis and the lone pair points along it too, so the measured moment is larger than the bond vectors alone would give and the excess is the share the lone pair carries.

Both are cases where the vector sum gets the direction right and the magnitude badly wrong, which is the usual outcome — and both are cases where symmetry gives the exact statement that a moment along the principal axis is permitted, without attempting a value.

The instructive comparison is nitrogen trifluoride: same shape as ammonia, same C₃ᵥ group, and a moment of only 0.23 debye. The N–F bond dipoles point the opposite way to the lone pair and largely cancel it. A shortcut that ignores lone pairs cannot produce that, and it is one of the cleanest demonstrations that the omitted term is not small.

Where to read on #

The exact statement is symmetry forbids a dipole, and the search behind it is point groups from coordinates.

The same critique applied to a different construct is hybridisation does not explain.

The method survives because most exercises ask only whether a moment vanishes, and for that question the symmetry argument is both easier and exact.

What the pictures here cannot show. A dipole moment is a property of an electron distribution, and the figures on this page draw nuclei and symmetry elements. Everything quantitative here is quoted rather than computed.