Molecular Geometry & Bonding
what the shape decides
What symmetry decides

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.
9 min read 6 figures Symmetry settles itTaught confidently, and wrong

A molecule’s point group follows from its coordinates. A dipole moment is a vector. A symmetry operation of a molecule must leave every property of that molecule unchanged, including that vector.

Those two sentences settle the question completely.

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 Which molecules may be polar, from the group and nothing else. No bond dipoles were added, no electronegativities consulted, and the answers are exact.

The argument

Suppose a molecule has a dipole moment μ\boldsymbol{\mu}, and suppose it has a symmetry operation RR. Since RR leaves the molecule indistinguishable from itself, it must leave μ\boldsymbol\mu unchanged: Rμ=μR\boldsymbol\mu = \boldsymbol\mu.

So the dipole must lie along a direction that every operation of the group fixes. If no such direction exists, the only vector satisfying the condition is the zero vector, and the moment must vanish.

Which groups leave a direction fixed? A rotation Cₙ fixes its own axis. A mirror fixes every direction in its plane. So a group built from one axis and mirrors containing it — Cₙᵥ — fixes the axis. So does Cₙ alone, and Cs fixes the plane, and C₁ fixes everything.

Anything else kills it. A horizontal mirror reverses the axis. A perpendicular twofold axis reverses it. An inversion centre reverses every vector. So only C₁, Cs, Cₙ and Cₙᵥ may be polar, plus the linear C∞ᵥ.

What that rules out at a glance

Carbon dioxide is D∞ₕ: it has an inversion centre, so its moment is exactly zero.

Methane is Td: no direction is fixed, so zero.

Boron trifluoride is D₃ₕ: the horizontal mirror reverses the threefold axis, so zero.

Benzene, ethene, sulfur hexafluoride: all have inversion centres, all exactly zero.

Water is C₂ᵥ and may be polar. Ammonia is C₃ᵥ and may be polar. Hydrogen cyanide is C∞ᵥ and may be polar.

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. 2 Carbon dioxide, D∞ₕ. The inversion centre at the carbon reverses every vector, so the dipole must be its own negative — which only the zero vector can be. The C–O bonds are strongly polar and the molecule is not.

“May be”, not “is”

The rule gives a necessary condition and not a sufficient one.

A C₂ᵥ molecule may be polar. Whether it actually is, and how large the moment is, depends on the electron distribution, and symmetry says nothing about magnitude. A molecule could in principle be C₂ᵥ with a moment that happens to be very small.

So the useful form is: symmetry can forbid a dipole absolutely, and can never require one. Where it forbids, the moment is exactly zero and no measurement will find otherwise.

Against the bond-vector argument

The account usually taught runs differently: assign each bond a dipole based on the electronegativity difference, add them as vectors, and see whether they cancel.

For carbon dioxide the two C–O vectors point opposite ways and cancel; for water they do not. The method gets both right.

It also has three problems.

It needs the bonds. Electronegativity differences are required, and electronegativity is a derived and scale-dependent quantity with several incompatible definitions.

It ignores lone pairs. A lone pair contributes to the moment, often substantially, and is not a bond. Ammonia’s moment is much larger than the N–H bond dipoles alone would give, because the lone pair points the same way.

It has no exactness. Vector cancellation says a sum comes to zero given the values used. The symmetry argument says the quantity is zero for a reason no values can affect, which is a different kind of claim.

The general theorem behind it

This is one instance of a rule that runs through the whole subject.

An integral over all space vanishes unless the integrand is unchanged by every operation of the symmetry group. A dipole moment is such an integral — the charge density weighted by position — and position is a vector, so the integrand transforms as a vector.

The same theorem gives which orbitals may overlap, which spectroscopic transitions are allowed, and which vibrational modes are infrared active. Four apparently different rules, one argument.

That is worth noticing because it means the dipole rule is not a special trick. It is what happens when a general statement about integrals meets a particular integrand.

What follows for measurement

Two practical consequences.

A measured moment identifies a structure. If a molecule of formula AB₂ has a measurable dipole it cannot be linear, because linear AB₂ is D∞ₕ. Sulfur dioxide is polar, therefore bent — established before any diffraction was done on it.

A zero moment does not identify one. Many structures give zero, so failing to find a moment narrows the possibilities without fixing them.

That asymmetry is typical of symmetry arguments: they exclude cleanly and include weakly.

The isomer case

The sharpest everyday use is distinguishing isomers.

Cis- and trans-1,2-dichloroethene have the same formula and the same bonds. The trans isomer has an inversion centre and is C₂ₕ, so its moment is exactly zero. The cis isomer is C₂ᵥ and has a substantial one.

No calculation of the bonding is involved. Two structures, two groups, two answers, and a dipole measurement tells them apart definitively.

Reading the table

The comparison across molecules is worth walking, because the pattern in which groups qualify is not arbitrary.

methane — TdThe 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.HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms
Fig. 3 Methane, Td. Four threefold axes pointing in four different directions, so no direction is fixed by every operation — and the moment is exactly zero however polar the individual C–H bonds are.

Td, Oₕ, D∞ₕ, D₆ₕ, D₃ₕ, D₂ₕ — all forbidden. Every one either has an inversion centre or has operations pointing in enough directions that none survives.

C₂ᵥ, C₃ᵥ, C∞ᵥ, Cs, C₁ — all permitted. Each has a direction every operation leaves alone: the principal axis, or a direction in the mirror plane, or any direction at all.

boron trifluoride — D3hThe 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.FFBFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 4 Boron trifluoride, D₃ₕ. Adding the horizontal mirror to what would otherwise be C₃ᵥ is what kills the moment: the mirror reverses the threefold axis, so the only vector it can fix is zero.

That last comparison is the useful one. Ammonia and boron trifluoride both have a threefold axis and three polar bonds. Ammonia is pyramidal, has no horizontal mirror, and is strongly polar. Boron trifluoride is planar, has one, and is not. One structural difference, one symmetry operation, and a property that goes from 1.47 debye to exactly zero.

Where the model stops

Three limits.

Equilibrium geometry only. A vibrating molecule is instantaneously less symmetric, so a formally non-polar molecule has a fluctuating instantaneous moment. That is why carbon dioxide, with a permanent moment of exactly zero, absorbs infrared radiation strongly — the asymmetric stretch produces a changing moment even though the average is zero, and that absorption is why it is a greenhouse gas.

One conformation. A flexible molecule averages over conformations, and the observed moment is that average.

Environment, which is one more reason a computed or measured number is needed for a magnitude. A molecule in a solvent or a crystal is perturbed, and an induced moment can exist where the isolated molecule’s permanent one vanishes.

Where it came from

The connection between molecular symmetry and dipole moments was worked out in the 1920s and 1930s, and Debye’s work on measuring moments by dielectric constant made it experimentally useful — the unit is named after him.

At the time, dipole measurements were among the very few ways to get structural information about a molecule, long before routine diffraction or spectroscopy. That a symmetry argument could turn a single number into a structural conclusion made it disproportionately important, and the reasoning is unchanged.

Symmetry as a source of information

Stepping back, the general point about this kind of argument, because it is what makes symmetry worth learning at all.

A symmetry argument takes almost no input — only the shape — and produces an exact conclusion. It does not need to know what the atoms are, how electronegative they are, what the bonds are made of, or what the electron distribution looks like. It needs the arrangement.

That is an unusual amount of output for the input, and it is why symmetry arguments are used first in almost every branch of physical science. They eliminate possibilities cheaply, and what survives can then be examined by expensive means.

hydrogen cyanide — C∞vThe 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.HCNC∞vprincipal axis C82 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 5 Hydrogen cyanide, C∞ᵥ. Linear, with no inversion centre — so a direction is fixed by every operation, and the molecule may be polar. It is, strongly, and the symmetry argument establishes the possibility before anything about the bonding is considered.
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. 6 Ethene, D₂ₕ. An inversion centre, so the moment is exactly zero — regardless of the C–H bonds being polar and regardless of anything about the pi system.

The limitation is the other side of the same coin. Symmetry never predicts a magnitude, because a magnitude requires knowing the things symmetry ignores. Where a number is wanted, it has to be measured or computed, and the symmetry argument’s role is to say when the answer must be zero.

Where the ladder goes next

The companion property is chirality, decided by the same kind of argument with a different condition.

The critique of the alternative account is the dipole is not a sum of bonds.

And the machinery that produces the group is point groups from coordinates.

What the pictures here cannot show. A dipole moment is a property of the electron distribution, and these figures draw nuclei. That a moment must vanish is a statement about an integral over a density that no figure here computes or displays.

Where this goes next

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. BrClCHFC1no rotation axis0 mirror planesno inversion centremay be polarmay be chiralgroup recovered from the coordinates5 atoms Chirality is a symmetry statement A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several. 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.