What a dipole cannot tell apart

A dipole moment is one of the most-quoted numbers in structural chemistry and one of the least informative, in a precise sense that can be stated and measured. The charge distribution of a molecule is a function of three variables. The dipole moment is three numbers computed from it. The map from the first to the second discards essentially everything, and the interesting question is what survives.

The answer has a clean shape. The moments form a tower — charge, dipole, quadrupole, octupole — and the lowest one that does not vanish is a property of the molecule; every one above it is a property of the molecule together with a choice of origin. So a neutral polar molecule has a dipole and does not have a quadrupole until somebody says where the origin is, and a neutral non-polar molecule has a quadrupole that is as much its own as a dipole ever was.

That is not a subtlety about conventions. It is the reason tables of dipole moments are printed without an origin column and tables of quadrupole moments are printed with a footnote.

Why the lowest moment is the fixed one #

The proof is short enough to give and is more useful than the statement.

Shift the origin by a vector $\mathbf{d}$. Each position becomes $\mathbf{r}_i - \mathbf{d}$, and the dipole becomes

$$\boldsymbol{\mu}' = \sum_i q_i(\mathbf{r}_i - \mathbf{d}) = \boldsymbol{\mu} - q\,\mathbf{d}$$

where $q$ is the total charge. So the dipole is unchanged if and only if the molecule is neutral. A charged species has a dipole that can be made anything at all by choosing where to stand, which is why the dipole of an ion is not a quantity.

The same calculation for the quadrupole picks up terms proportional to $\mu_a d_b$ and $\mu_b d_a$. Those vanish exactly when the dipole vanishes. Carry it up the tower and the pattern repeats: each moment’s shift depends on the moments below it, so the first one whose predecessors are all zero is the first one that does not move.

There is a detail here that the table above had to be corrected for, and it is worth keeping because it is easy to be fooled by. Shifting perpendicular to the dipole changes only the off-diagonal parts of the quadrupole. Water’s dipole lies along its twofold axis; shifting along a perpendicular direction leaves every diagonal component of its quadrupole exactly where it was, so a figure reading off the diagonal alone would show a polar molecule’s quadrupole apparently fixed. The figure now reads the largest component of the whole tensor, and the change appears.

What symmetry forces, and where it stops #

Symmetry forbids a dipole settles when the first moment vanishes: the dipole is a vector, and it must be left alone by every operation of the group, so a group with no invariant direction forbids it outright. Carbon dioxide, benzene, boron trifluoride and methane all have such groups and all have exactly zero dipole, not approximately.

The quadrupole is a rank-two tensor and the same argument runs with a different answer. It must be left alone by every operation too, and a group can leave a tensor alone while leaving no vector alone — which is exactly what happens for the first three.

Methane is the one that goes further. Td forbids the quadrupole as well, and the table shows it as an exact zero rather than a small number. The first non-vanishing moment of methane is the octupole. That is why methane’s intermolecular interactions are so much weaker than carbon dioxide’s at the same molecular size, and why its boiling point is a hundred and thirty degrees lower: the leading electrostatic term is three ranks up and falls off far faster.

Boron trifluoride and benzene sit in between, with dipoles of zero and quadrupoles that are properties of the molecule. Benzene’s is the largest in the table by a factor of nearly two, and it is the reason benzene molecules stack edge-to-face rather than face-to-face — a fact about a solid that follows from a moment a dipole measurement cannot see.

The bond-dipole model, and the two molecules that break it #

The everyday use of a dipole moment is the reverse of everything above: given the measured moment, say something about the charge distribution. The standard route is to write the moment as a sum of bond contributions, and the dipole is not a sum of bonds sets out why that decomposition is not unique. The sharpest single counterexample is a pair of molecules with the same shape and the same central atom.

Ammonia has three N–H bonds at 107.8°. Nitrogen trifluoride has three N–F bonds at 102.2°. Both are pyramidal, both are C3v, both have their moment along the threefold axis.

Two things favour NF₃ having the larger moment. Its bonds are more polar: on the Pauling scale the difference across N–F is 0.94 and across N–H is 0.84. And its geometry sums the bond vectors more efficiently: three unit vectors at a bond angle of 102.2° make an angle of 64.0° with the axis and sum to 1.316, while at 107.8° they make 68.9° and sum to 1.080.

Multiply the two together and a bond-dipole model predicts NF₃’s moment to exceed ammonia’s by a factor of about 1.36.

The measured moments are 0.24 D for NF₃ and 1.47 D for ammonia. The model is wrong by a factor of eight and in the wrong direction.

What the missing term is, and roughly how large #

The term the model omits is the lone pair. Nitrogen’s non-bonding pair sits on the axis, opposite the three bonds, and carries a moment of its own.

In ammonia the bond moments and the lone-pair moment point the same way and add. In nitrogen trifluoride the bonds are polarised the other way — fluorine is the electronegative end rather than the central atom — so the bond contribution reverses while the lone pair does not, and the two nearly cancel.

The size of it can be extracted, with a stated assumption. Take the bond moment to be proportional to the electronegativity difference, with one constant of proportionality shared between the two molecules, and let the lone pair contribute the same amount in both. That is two equations in two unknowns:

$$1.080\,b_{\mathrm{H}} + L = 1.47, \qquad L - 1.316\,b_{\mathrm{F}} = 0.24$$

with $b_{\mathrm{H}} = 0.84c$ and $b_{\mathrm{F}} = 0.94c$. Solving gives $c = 0.574$ debye per unit of electronegativity difference and $L = 0.95$ debye for the lone pair.

That is a fit and not a computation, and it is worth being exact about what it does and does not show. It does not establish that a lone pair “has” a moment of 0.95 D — the decomposition into bond and lone-pair pieces is one of the arbitrary decompositions worth distrusting. What it shows is that the omitted term is of the same size as the whole bond contribution, so a model that leaves it out is not making a small error.

The other half of the problem with the bond-dipole route is that for six ordinary bonds the four electronegativity scales disagree about which end is positive. A decomposition whose signs are disputed cannot be checked against a measurement, because the measurement is a single number and the decomposition has as many versions as there are tables.

How many distributions give one dipole #

The many-to-one statement can be made concrete rather than left as an impression, and the arithmetic is elementary.

A molecule of $N$ atoms with a partial charge on each has $N$ numbers describing its charge distribution, subject to one condition — they sum to the total charge. That leaves $N - 1$ free. The dipole is three numbers. So for anything past a diatomic there are $N - 4$ directions in charge space along which the charges can be varied with the dipole staying exactly where it is.

For water that is a one-parameter family: raise the oxygen’s charge and lower both hydrogens’ in the right ratio and the moment does not move. For methane it is a one-parameter family too, and for benzene, with twelve atoms, it is eight-dimensional. Every member of each family reproduces the measured moment exactly, and they differ from one another at the quadrupole and above.

This is before any of the real difficulties. Atoms do not carry point charges; the density is continuous and most of it is not at a nucleus; and the assignment of a continuous density to atomic centres has no unique answer, which is why Mulliken, Löwdin, Hirshfeld and Bader analyses of the same wavefunction give different charges for the same molecule. The point-charge count above is therefore a lower bound on the underdetermination rather than a description of it.

What the count does establish is that fitting charges to a dipole is not a determination. It is a choice of one member of a family, made by whatever else the fitting procedure happens to prefer, and two procedures that prefer different things will disagree while both reproducing the measurement.

The moment that a rotation measures, and the one it does not #

There is a second reason a dipole is quoted more often than a quadrupole, and it is instrumental rather than conceptual.

A dipole moment is measured by watching rotational transitions shift in an electric field. The shift is first order in the field for a symmetric top and second order for an asymmetric one, and either way it is proportional to the moment and large enough to read off a spectrum. The rotational spectrum is a moment of inertia sets out the theory the measurement sits on.

A molecule with no dipole has no such shift, so the same apparatus returns nothing. Quadrupole moments are measured instead by watching something else entirely — the birefringence a gas acquires in a field gradient, or the second virial coefficient, or the collision-induced spectrum — and each of those is a harder measurement with a larger error bar. Benzene’s quadrupole has been quoted at values differing by twenty per cent between methods.

So the tower of moments is quoted with a precision that falls off sharply with rank, for reasons that have nothing to do with which moments matter. For methane, whose first two moments both vanish, the leading electrostatic quantity is the one measured worst.

That asymmetry is worth carrying whenever a table of dipole moments is being used to compare molecules. The comparison is between the best-measured property of one molecule and the best-measured property of another, and for a non-polar molecule that property is zero — which says something about the group and nothing about the distribution.

What a measured moment does determine #

Setting out the negative results without the positive one would be unfair to a quantity that is measured to five figures and used constantly. A dipole moment determines several things exactly.

It determines that the group has an invariant direction. A non-zero moment rules out every centrosymmetric group and every group with more than one axis, which for a small molecule is often enough to fix the structure. This is the argument two structures, two spectra makes with vibrations and it works with a dipole in the same way and more cheaply.

It determines the direction of the moment relative to the inertial axes, when it is measured by the Stark effect on a rotational spectrum, which is a far more informative measurement than the magnitude alone.

It constrains any model of the charge distribution. A calculation that gets the moment wrong is wrong. The one-way nature of that is the recurring caution — a calculation that gets it right may be wrong too, for exactly the reason this essay opened with.

Two molecules with equal moments and unequal everything else #

The clearest illustration of the many-to-one map is a pair whose moments happen to coincide, and chemistry supplies several.

Chlorobenzene and chloroethane both have moments near 1.7 debye. One is a planar aromatic ring with a substituent and a large quadrupole; the other is a small saturated molecule with a small one. Their charge distributions have almost nothing in common, their intermolecular behaviour is different in kind, and the one number they share is the one most likely to be quoted about either of them.

Sulfur dioxide and water are a sharper pair, because both are standard examples. Water’s moment is 1.85 debye and sulfur dioxide’s is 1.63 — close enough to be neighbours in a list. Both are C2v and both have their moment along the twofold axis. But water’s bond angle is 104.5° and sulfur dioxide’s is 119.3°, the bond lengths differ by more than half an ångström, and the two have three atoms arranged so differently that two structures, two spectra can tell them apart from a band count alone.

The moral is not that the dipole is useless. It is that a quantity computed by throwing away almost all of a function is a poor thing to reason backwards from, and reasoning backwards from it is what the bond-dipole model does.

The molecule where even the sign is not what it looks like #

One measured case makes the essay’s point without any algebra, and it is worth having because it fails in both of the ways a dipole can.

Carbon monoxide’s moment is 0.112 debye — very small for a bond between two elements whose electronegativities differ by a full unit — and it points with the carbon end negative, which is the opposite of what any electronegativity argument gives.

Both features have the same cause and neither is visible in the number. The σ and π transfer toward oxygen is real and large; the carbon carries a lone pair pointing away from the bond, which contributes a moment in the opposite direction; and the two nearly cancel. So a moment of a tenth of a debye is the residue of two contributions each worth several, and its sign is decided by which of them happens to be marginally larger.

A reader given only 0.112 debye would conclude that the molecule is barely polar and that the charge sits where the number says. The distribution is strongly polarised and the number is a near-cancellation — which is the whole of what a dipole cannot tell apart, in one molecule of two atoms.

Where this stops, and whose ground is next door #

Two boundaries are worth naming.

The charges here are quoted, not computed. The moments in the table are computed exactly, from point charges obtained by a stated electronegativity rule. Every claim made from them is about how moments transform under a shift of origin, which holds for any charges whatever — but the numbers themselves are only as good as the charges, and the charges are a correlation from the 1930s. A moment computed from a real density is a different quantity and none is computed here.

What the moments then do belongs elsewhere. How the field of a multipole falls off with distance, what torque one exerts on another, and what a polarisability is are field theory rather than molecular structure, and they are illustrated-physics.com’s. Nothing in this essay involves a second molecule, a field, or a force.

The half that is this site’s is what a molecule’s own charge distribution is, what a moment of it means, and which moments are properties of the molecule at all. That last is the one this essay exists for, and it is settled by three lines of algebra and a table that could have failed.

Still open: the octupole #

The obvious open question is the moment nobody quotes: the octupole, which is methane’s first non-vanishing one and therefore the leading term in every electrostatic argument about it. The calculation here extends to it without difficulty — the same sum with one more factor of position — and the interesting part would be the same as the interesting part here: which groups force it to zero, and what is left for a molecule whose first three moments all vanish.

The other direction is the harder one. Every number in this essay came from point charges on nuclei, and a real molecular charge distribution is continuous, has most of its weight between the nuclei rather than on them, and has no unique decomposition into atomic pieces at all. That is the same underdetermination hybridisation does not explain argues about in a different currency, and it is why a partial charge is a convention and a moment is not.