Symmetry forbids a dipole

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.

The argument #

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

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.

“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.

The same rule, applied to what a spectrum sees #

The dipole rule as stated above is a statement about a molecule in its equilibrium geometry. Applied instead to a molecule’s motions, the identical theorem says which vibrations absorb infrared light, and following it that far is the fastest way to see that nothing about the dipole case was special.

The counting runs like this. The atoms have $3N$ ways to move, which span a representation; three of those ways are translations of the whole molecule and three are rotations of it, and both of those span representations too. Subtracting leaves the vibrations, and the count must come out at $3N - 6$ — which is checked rather than assumed, because a subtraction that produced anything else would mean the characters were wrong.

Then the activity. A vibration absorbs infrared light when the transition moment integral survives, and that integral survives when the mode’s representation carries one of $x$, $y$ or $z$ — the same three functions the dipole rule turns on. It appears in a Raman spectrum when its representation carries a quadratic function instead.

Water’s three modes come out of the same subtraction, and all three are active in both the infrared and the Raman spectrum — which is what a molecule with no centre of inversion permits and is the ordinary case. The interesting case is the one where the two lists cannot overlap at all.

The mutual-exclusion result is the one worth pausing on. It is normally taught as a fact to remember, and it is a two-line consequence of the same integral theorem: $x$, $y$ and $z$ are odd under inversion, quadratics are even, and no representation is both. A molecule whose infrared and Raman spectra share no bands has a centre of inversion, and that conclusion needs no calculation of anything.

Both figures come from the same calculation, and it checks the converse as well: a molecule without a centre of inversion must have at least one mode active in both, or the reasoning is not doing what it claims.

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.

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.

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.

Three ways a moment can be zero #

A measured moment of zero is not one finding, and the distinction is the practical payoff of everything above.

Zero because the group forbids it. Carbon dioxide, methane, benzene, sulfur hexafluoride. The integral vanishes identically, at every geometry consistent with the symmetry, for every conceivable electron distribution the molecule could have. No experiment will ever find a permanent moment in one of these, and no improvement in the bonding model will produce one.

Zero because contributions happened to cancel. Nothing forbids a moment in a molecule of low symmetry, and nothing requires one either. A C₁ or Cs molecule whose moment measures as very small has an accidental cancellation, and it is accidental in the strict sense: substitute a distant atom, change the solvent, or warm it up and the cancellation goes away. This is the case the bond-vector method cannot distinguish from the first, because a vector sum that comes to nothing looks the same whichever reason it came to nothing.

Zero because it averaged away. A flexible molecule visits conformations with moments pointing in different directions, and what a bulk measurement returns is the average over them. 1,2-dichloroethane is the standard case: the anti conformer is centrosymmetric and has none, the gauche conformers have a substantial one, and the measured value depends on temperature because the population does.

Only the first is a statement about the molecule. The second is a statement about a coincidence and the third about an ensemble, and a group-theoretical argument is what separates the first from the other two without measuring anything.

What it costs #

The dipole rule as usually applied costs nothing at all: read the symbol, check it against a list of five families, done. That is genuinely the cheapest useful statement in the subject, and the list is short enough to memorise.

What costs something is making it a computation rather than a recollection, and the price is worth stating because this site pays it everywhere.

Deciding polarity from a symbol needs the symbol, and getting the symbol needs the search over candidate axes described in point groups from coordinates. Going further — to the vibrational counting above, or to a selection rule — needs the group as a set of matrices rather than a label, which means closing the operations under multiplication and sorting them into conjugacy classes. Methane closes at twenty-four operations, benzene at twenty-four by a different route, sulfur hexafluoride at forty-eight. All of it takes a few milliseconds.

The return on that cost is that every claim becomes checkable. The polarity of every molecule here is checked against what the group finds, and a molecule declared non-polar must additionally be shown to have an operation that forbids the moment — a check that would catch a molecule declared non-polar for the wrong reason. The vibrational counts are checked against $3N-6$. The mutual-exclusion result is checked in both directions. And the checks are fed deliberate errors, because one that has never rejected anything proves nothing: a forbidden transition declared allowed is refused, an allowed one declared forbidden is refused, and a molecule assigned the wrong point group is refused.

The cheap version and the expensive version give the same answers. Only the expensive one can be wrong in a way that anybody notices.

Where the model stops #

Four 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.

The rule is one-directional and stays that way. No amount of extra calculation turns “may be polar” into “is polar”. The vibrational analysis above is a good illustration of how far the symmetry argument can be pushed and where it still stops: it says exactly how many modes there are and exactly which of them can absorb, and it says nothing whatever about at what frequency or how strongly. Symmetry forbids. It never predicts a magnitude, and a treatment that appeared to would be smuggling in a model.

The same theorem, applied to other quantities #

The argument used here is about a vector, and nothing in it is about dipoles specifically: a quantity that transforms as a vector must vanish unless the group permits a vector to survive. That immediately supplies a list of other properties settled by the same one line.

Anything that can orient a molecule in a field. A first-order Stark shift requires a permanent dipole, so the same groups that forbid the moment forbid the shift, and a molecule in a group that permits no vector cannot be oriented by a static electric field however strong.

Piezoelectricity and pyroelectricity in a crystal. The same argument applied to a crystal’s point group rather than a molecule’s: a crystal can develop a polarisation when squeezed only if its class permits a vector, and can have a spontaneous one only if the class permits a unique direction. Both are tabulated as lists of allowed crystal classes, and both lists come out of the reduction rather than out of any measurement.

And second-order optical effects. Frequency doubling requires a property that transforms as a rank-three tensor, and the condition for it to survive is weaker than the dipole’s: it needs only the absence of a centre of inversion, not the presence of a unique direction. So there are materials that cannot be polarised and can double light, and the difference between the two conditions is the difference between two reductions.

That last comparison is the useful one to carry, because it shows the theorem is not a single test. Each property has its own transformation behaviour and its own condition, and centrosymmetric is the answer to one of the questions rather than to all of them.

Which is why the reduction is worth doing rather than remembering. A rule of thumb that a centre of inversion kills a property is right for dipoles and for frequency doubling and wrong for a great many other quantities — including every one that transforms as a rank-two tensor, which survives inversion untouched.

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.

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 to read on #

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 search 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.