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

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.
9 min read 6 figures Taught confidently, and wrongSymmetry settles itCounted, not quoted

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 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. 1 The same question answered from the point group alone. Nothing here required an electronegativity, and where the answer is no, the moment is exactly zero rather than a sum that happens to cancel.

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.

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.

They correlate and they do not agree, and they order some elements differently. 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.

Building a quantitative argument on it is building on a scale-dependent construct.

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.

ammonia — C3vThe 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.HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 2 Ammonia, C₃ᵥ. The symmetry permits a moment along the threefold axis, which is all the group can say. What actually produces most of it is the lone pair, which the bond-vector method does not count.

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 3.336×10−303.336\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.

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. 3 The textbook case. Two identical polar bonds pointing opposite ways, vectors cancelling, moment zero. The method gets it right — and so does the symmetry argument, which additionally says the answer is exact rather than a cancellation that happens to work out.
ethene — D2hThe 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.HHCCHHD2hprincipal axis C23 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms
Fig. 4 Another: ethene, with four polar C–H bonds arranged so that every one is opposed. D₂ₕ, an inversion centre, and a moment of exactly zero. The vector sum and the symmetry argument agree, and only the second explains why no measurement could ever find otherwise.

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.

Where the model stops

Two 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. This site’s machinery is one-electron and geometric. The measured moments quoted come from the literature, and the figures establish the point groups the argument turns on.

The molecules with lone pairs

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

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. 5 Water, 1.85 debye. The two O–H bond dipoles have a resultant along the twofold axis, and the two lone pairs point the same way and add to it. Counting only the bonds understates the moment substantially.
ammonia — C3vThe 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.HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 6 Ammonia, 1.47 debye. The three N–H dipoles resolve along the threefold axis, and the lone pair points along it too. Here the lone pair contributes a large share of the total.

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 the ladder goes next

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

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

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 by this site.

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 Symmetry forbids a dipole Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise. HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms Point groups from coordinates A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not. HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms Hybridisation does not explain Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.