Zero dipole is not no interaction

A molecule with a centre of inversion has a dipole moment of exactly zero. This is not a measurement and not an approximation: it follows from the group, because the dipole is a vector and inversion sends every vector to minus itself, so a vector that is unchanged by a symmetry of the molecule must be zero. Benzene, carbon dioxide, nitrogen, acetylene, ethene, sulfur hexafluoride — all zero, exactly, at every origin.

The sentence usually appended to that is and therefore they have no electrostatic interaction. It is wrong, and for benzene it is wrong by an amount that decides how the solid packs.

The moment that survives #

A dipole is the first moment of the charge distribution. When it vanishes the second moment does not have to, and the second moment — the quadrupole — is a tensor rather than a vector, so inversion leaves it alone. A centrosymmetric molecule can have as large a quadrupole as it likes.

For benzene it is large. The measured value is $-8.7$ D Å, and the sign is a statement about where the electrons are: negative means the density is concentrated away from the molecular axis, which for a flat aromatic ring means the π clouds above and below the plane are negative and the rim of hydrogens is positive. That displacement is the same one delocalisation describes, seen as a charge distribution rather than as an energy.

The point charges used below are fitted to those measured numbers and to nothing else. That division is worth being explicit about: the shape of the field is decided by where the charges are put, which follows from the molecular geometry, and only the size of it comes from the measurement. A point-charge model that claimed to compute a charge distribution would be claiming far more than it can, and every model of this kind in use has the same status.

Where the charges go, and why the geometry does the work #

The models are deliberately crude and each is built the same way: put charges where the molecule’s own symmetry says the density is displaced, then scale them until the computed moment matches the measured one.

Benzene gets six charges in the ring plane, at the carbon positions, and two on the sixfold axis at $\pm 1.05$ Å for the π clouds. Carbon dioxide gets three charges on a line — negative at the oxygens, twice as positive in the middle. Nitrogen gets four: two positive inside the bond and two negative outside it, which is the arrangement that gives a negative moment and is the model’s way of saying the lone pairs stick out of the ends. Acetylene gets the same four charges with the signs reversed, because its ends are hydrogens.

Two checks come free and both are exact rather than approximate. The dipole of every model is zero to the last bit a double can hold — it has to be, since the charge sets are centrosymmetric by construction — and the quadrupole tensor is traceless, which is a property of the definition rather than of the molecule and would catch an arithmetic error in the tensor.

What the models cannot claim is any statement about the charges themselves. A partial charge is not an observable and there is no experiment that returns one; what is observable is the moment, and the moment is what these were fitted to. The right reading of a point-charge model is that it is a compact way of writing down a field with the right symmetry and the right leading strength, and nothing more.

The exponent, measured #

The textbook statement is that a dipole–dipole interaction falls off as $R^{-3}$ and a quadrupole–quadrupole one as $R^{-5}$. It is quoted everywhere and checked nowhere, and checking it costs one line: compute the energy at a range of separations far enough out that the multipole expansion is the whole answer, take logarithms, and fit a slope.

Two things make this a test rather than a formality. The first is the dipole curve: a fit that could not tell 3 from 5 would not be measuring anything, and it does. The second is that the same exponent, 5.00, comes out for nitrogen, whose quadrupole moment is smaller than benzene’s by a factor of six. The exponent is a property of which moments the two partners have, not of how large those moments are.

The third case is the refusal. Take a charge set with no dipole and no quadrupole — eight charges of alternating sign at the corners of a cube, which is a pure octopole — and the measured exponent comes out above 6.5. So the fit is reading the lowest surviving moment of each partner rather than returning a number that was assumed.

What that exponent costs #

A steeper exponent is not a smaller interaction; it is a shorter-ranged one, and the difference matters for what the interaction can do.

At contact distances a quadrupolar interaction can be as large as a dipolar one — benzene’s is, comfortably. At three times the distance it is down by a factor of 243 where the dipolar one is down by 27. So a quadrupolar molecule is strongly ordered by its neighbours and almost unaffected by anything further away, which is exactly the condition for a structure decided by local packing rather than by a long-range field.

That is why carbon dioxide and nitrogen form molecular crystals with definite orientational order and no long-range electrostatic cohesion to speak of, and why the lattice sums that a dipolar or ionic solid needs converge trivially here. The whole of the quadrupolar interaction is with the nearest shell.

The orientation, which is the consequence #

Now the question the whole essay is for. Two benzenes, five ångström apart, with one turned against the other.

Face to face repels. Edge to face attracts. And the two are not merely different in strength — they have opposite signs, so the preference cannot be dismissed as a modest bias.

This is the arrangement benzene’s crystal has: a herringbone, in which every molecule presents its rim to its neighbour’s face. It is the arrangement of the benzene dimer in the gas phase. And it is the arrangement of a pair of aromatic side chains in a folded protein, where the geometry has been surveyed often enough to be a design rule.

The usual explanation offered for all three is π stacking, a phrase that suggests the faces attract. On the electrostatics alone they repel, and the attraction that does exist between stacked aromatic rings is dispersion — a different interaction, with a different origin and a different distance dependence, which is not in this calculation at all.

What the model leaves out, and how much it matters #

Three things are missing, and each pulls a different way.

Dispersion. Always attractive, roughly as $R^{-6}$, and larger for a stacked pair than for a T-shaped one because the contact area is greater. This is the term that makes stacking competitive, and it is the reason a substituted benzene with a reduced quadrupole — hexafluorobenzene, whose moment has the opposite sign — stacks readily with benzene while benzene does not stack with itself.

Repulsion. The overlap of the closed shells, which is what fixes the contact distance and is why 5 Å was chosen rather than 3.

Higher moments. The quadrupole is the first surviving moment, not the only one. At contact distances the expansion is not converged, and a point-charge model handles that better than a truncated multipole series does — which is exactly why point charges are used here rather than a tensor.

None of the three changes the sign of the result at the angles that matter, and the direction of the preference is the claim.

The angle where it changes sign #

The crossing between 50° and 60° is the most useful single number in the scan, because it is what turns a preference into a rule.

An interaction that merely weakened with tilt would give a soft preference, easily overturned by anything else in the problem — packing, dispersion, a substituent. An interaction that changes sign partitions the orientations into two sets, and no amount of scaling the strength moves the boundary between them. The angle at which it happens is set by the shape of the angular dependence, which for two axial quadrupoles is a known function of the two tilt angles with a zero in it.

The consequence is that the herringbone is robust in a way a stacking preference would not be. Perturbing the strength of the quadrupole — by substituting the ring, by pressure, by putting it in a solvent — moves the depth of the minimum and leaves its position alone until the moment changes sign altogether.

Hexafluorobenzene is the experiment that does change the sign: replacing every hydrogen by fluorine reverses the displacement of the density, and its quadrupole moment is close to $+9$ D Å. Two hexafluorobenzenes therefore behave like two benzenes, and a benzene with a hexafluorobenzene does the opposite — the faces attract, and the mixed crystal is a stack of alternating rings with a melting point well above either pure compound’s. That is a prediction of the argument above, and it is a well-known fact of solid-state chemistry which is usually presented without the reason.

The dipole is not the whole of polarity #

There is a wider point here about what a single number can be asked to carry, and it connects to two other results.

A dipole moment cannot tell some structures apart: several distinct arrangements give the same magnitude, so the measurement under-determines the geometry. That is a limit on what the first moment reports.

A dipole is not a sum of bond dipoles: adding vectors along the bonds leaves a residue, and the residue is the lone pairs and the atomic polarisation. That is a limit on how the first moment is built.

This essay is the third limit and it is the simplest: the first moment can be zero and the molecule can still be strongly electrically structured. Benzene is not electrically featureless; it is electrically featureless to first order, which is a statement about a Taylor series rather than about a molecule.

Origin dependence, and why the sign is safe to quote #

That last point is worth its own paragraph, because it is what makes the numbers above quotable at all.

Move the origin and the moments change. A dipole is origin-independent only if the total charge is zero; a quadrupole is origin-independent only if the total charge and the dipole are both zero. So for a neutral molecule with no dipole the quadrupole is a genuine property, and for a molecule with a dipole it is not — it is a property of a molecule and a chosen point.

Benzene, carbon dioxide, nitrogen and acetylene all qualify — the group settles it in each case. Water does not, and the quadrupole moments quoted for water in the literature are always accompanied by a stated origin, usually the centre of mass, which is a convention rather than a physical fact. It is the same theorem that makes a bond dipole a construction rather than an observable, applied one order up.

Two consequences of the sign, one of them a melting point #

The quadrupole’s magnitude decides how strong the interaction is and its sign decides what the arrangement looks like. Two measured consequences follow from the sign alone, and the second is as clean a demonstration as this subject offers.

A cation sits on the face. Benzene’s quadrupole is negative in the sense that puts electron density above and below the ring and leaves the rim relatively positive. So a positive ion approaching the face meets an attraction, and the attraction is not small: a potassium ion binds to a benzene in the gas phase by of order eighty kilojoules a mole, which is comparable to a hydrogen bond and larger than most people would guess for an interaction with a molecule that has no dipole at all. That interaction is the reason aromatic side chains line the potassium channels and the acetylcholine binding sites of proteins, and it is entirely a consequence of a moment that a table of dipole moments records as zero.

And reversing the sign reverses the packing. Replace benzene’s six hydrogens with fluorines and the ring’s quadrupole changes sign: hexafluorobenzene has its electron density pulled to the rim and its faces left relatively positive. Two hexafluorobenzenes therefore prefer the same T-shaped arrangement two benzenes do, for the mirror-image reason.

Put one of each together and the two faces attract. A benzene and a hexafluorobenzene stack face to face, at a separation and an orientation neither pure compound adopts, and the consequence is visible without any instrument: benzene melts at 5.5 °C, hexafluorobenzene at 5.1, and the one-to-one complex of the two at 23.7. A mixture of two liquids that freezes eighteen degrees above either of them.

That is a strong result to obtain from a moment nobody quotes. Nothing about the two molecules’ dipoles distinguishes them — both are exactly zero, both for the same reason, both at every origin. Nothing about their sizes or their polarisabilities predicts a stacked complex, since dispersion is indifferent to orientation. What predicts it is one sign, and the sign is a property of the charge distribution that the lowest moment of that distribution cannot record.

It also sharpens the essay’s title. Zero dipole is not no interaction is the weak form. The strong form is that two molecules with identical zero dipoles can attract each other in an arrangement that neither will adopt with its own kind, and the difference between them is invisible in every quantity a dipole table contains.

What this says about cohesion #

The four molecules here are all gases at room temperature and all condense at accessible ones, and the quadrupole is a real part of why. But it is a part, and the accounting is instructive.

Benzene’s enthalpy of vaporisation is about 34 kJ/mol. The quadrupolar interaction with a single well-oriented neighbour is a few kJ/mol at contact, and a molecule in the liquid has a dozen neighbours, most of them badly oriented for it. Dispersion supplies the bulk of the cohesion; the quadrupole supplies the orientation.

That division — one interaction setting the magnitude and another setting the geometry — is the same division solid-state cohesion keeps meeting from the other side, and it is why a cohesive energy and a crystal structure are so nearly independent pieces of evidence about the same material. The same split appears in how a bond weakens as neighbours multiply: a total and a geometry answer to different terms.

Still open: competing moments, and polarisation #

The obvious open question is the molecule where the two moments compete: something with a dipole and a large quadrupole, where the arrangement that optimises one frustrates the other. Water is the standard example and the standard difficulty, since its structure is decided by a directional interaction — the hydrogen bond — that no multipole expansion converges to.

The other open question is the one already within reach. The interaction computed above is between two rigid charge distributions, and a real molecule polarises: benzene’s quadrupole is modified by its neighbour’s field, and the induced part is always attractive. Adding it needs a polarisability, which is a response rather than a property, and it is the point at which a model of a molecule stops being a picture of one and starts being a parametrised object with terms in it.

The claim that stands without any of that is small. Whether a molecule has a dipole moment is a question about its point group, and the answer is often no. Whether it has an electric field around it is a different question, and the answer for benzene is emphatically yes — large enough to sort two rings into a T rather than a stack, and to do it with a sign rather than a preference.