Mutual exclusion does not prove a centre

Two structures, two spectra uses the rule of mutual exclusion to tell a linear XY₂ from a bent one: carbon dioxide has a centre of inversion, so no vibration appears in both its infrared and its Raman spectrum — the same theorem selection rules are one theorem derives once and this field has been spending ever since — and finding a band in common would settle the shape without any further argument.

That is the rule used forwards, and it is sound. Used backwards it is not, and the backwards use is so common that it usually goes unremarked: no coincidences, therefore centrosymmetric.

The forward direction is a two-line theorem. With a centre of inversion, every representation is even or odd under it. A coordinate — $x$, $y$ or $z$ — is odd, because inversion reverses it. A product of two coordinates is even, because two sign changes cancel. So no representation can carry both, no vibration can be both infrared and Raman active, and the spectra share nothing.

Nothing in that runs backwards. It says a centre is sufficient, and says nothing whatever about necessary.

The census #

Fifteen groups, and the census is one line of arithmetic per representation: does it have a linear function attached, does it have a quadratic, do any have both.

Eight of them have a centre — Ci, C2h, D2h, D4h, D5d, D6h, Oh — and every one excludes. For each, the parity argument is checked rather than described: every representation carrying a coordinate has character $-d$ under the inversion class and every one carrying a quadratic has $+d$, where $d$ is its dimension. That is the proof, evaluated on the tables themselves.

Six of the seven without a centre show coincidences, and the coincidences are the ordinary case. In C2v the totally symmetric representation carries $z$ and also $x^2$, $y^2$ and $z^2$ — every A₁ vibration of water is active in both spectra, which is why water’s spectra look alike. In Td the T₂ representation carries $(x, y, z)$ and also $(xy, xz, yz)$, so methane’s triply degenerate stretches and bends appear in both — the same T₂ that fewer bands than electrons counts in a photoelectron spectrum. In D3h it is E′ that carries $(x, y)$ and $(x^2 - y^2, xy)$ together.

And one group without a centre excludes anyway.

D5h, and why five is the smallest axis that can do it #

Take an $n$-fold axis and ask where the coordinates go and where their products go.

The pair $(x, y)$ rotates by one full turn of the angle as the axis is applied: it belongs to the representation labelled E₁, the one whose character under $C_n$ is $2\cos(2\pi/n)$. The pair $(x^2 - y^2,\ xy)$ rotates twice as fast — a quadratic in the coordinates picks up twice as much phase — so it belongs to E₂, whose character is $2\cos(4\pi/n)$.

E₁ and E₂ are different representations as long as they have different characters. With a threefold axis they do not: $2\cos(4\pi/3) = 2\cos(2\pi/3)$, so E₁ and E₂ are the same thing and the coordinates sit with their own products. With a fourfold axis, $4\pi/4 = \pi$ and the “E₂” combination splits into two one-dimensional representations that land on the same rows as other things. Only from five onwards are the first-order and second-order pairs genuinely distinct representations.

That is the whole mechanism, and it says the counterexample could not have been smaller. Every group in this file with an axis of order five or more and no centre excludes; every group without one, and without a centre, does not.

A molecule that does it #

Ferrocene is the case, and it is a good one because it comes in two conformations that differ by a twist of thirty-six degrees.

Staggered — the two rings offset — is D5d, which has a centre of inversion. Mutual exclusion is expected.

Eclipsed — the rings aligned — is D5h, which has no centre. Mutual exclusion is expected by nothing, and holds.

Both structures here are built from two measured bond lengths: an Fe–C distance of 2.064 Å and a ring C–C of 1.430 Å, from which the ring radius and the ring height follow by trigonometry rather than by quotation. The point group is then recovered from the coordinates by the same search that finds water’s C2v, and it returns D5h and D5d respectively. A fivefold axis also sets a trap that no smaller axis can: a rotation by 144° is the square of a fivefold rotation and has order five, while dividing 2π by its angle gives two and a half, which rounds to three — so an operation’s order has to be found by applying it until it returns to the identity, not read off its angle.

The vibrational representation is then computed the standard way: the character of the 3N Cartesian displacements under each operation, less the translations and rotations, reduced in the group’s own table — the method of how many frequencies, not how many modes, applied to a molecule five times the size of the ones it was first shown on. Fifty-seven vibrations for twenty-one atoms, which is $3N - 6$ exactly.

For the eclipsed form: 16 infrared-active vibrations, in six E1′ pairs and four A2″ singletons, and 26 Raman-active ones in four A1′, six E2′ and five E1″. For the staggered form: 16 infrared-active in four A2u and six E1u, and 26 Raman-active in four A1g, five E1g and six E2g. Neither list overlaps its partner, and fifteen of the fifty-seven vibrations in each conformer appear in neither spectrum at all.

Two molecules, one with a centre and one without, both showing the pattern that is supposed to prove a centre.

The fifteen that appear in neither #

Adding the two lists gives forty-two of the fifty-seven vibrations. The other fifteen are in neither spectrum, and they are worth a paragraph because the rule is usually stated as though a mode had to be in one or the other.

For the eclipsed conformer they are the single A2′ mode, the two A1″ modes and the six doubly degenerate E2″ pairs. None of those representations carries a coordinate, so no infrared band; none carries a product of coordinates, so no Raman band. They are silent — not weak, not overlapped, but required to have zero intensity in both experiments by the same theorem that forbids anything else.

Mutual exclusion is therefore a statement about a partition into three parts and not two, and a molecule with a centre has the same three: gerade modes in the Raman, ungerade modes in the infrared, and whatever falls in a representation with neither kind of function attached. Benzene has four such modes out of thirty and is the standard example.

The practical consequence is that counting bands is not counting modes. A spectroscopist who sees sixteen infrared bands and twenty-six Raman bands for eclipsed ferrocene has seen every mode that either experiment can show and is still fifteen short of the molecule’s vibrations, and no amount of better apparatus will close the gap. That is the same distinction how many frequencies, not how many modes draws for benzene, arriving here from the other end: there the modes outnumbered the frequencies because of degeneracy, and here the modes outnumber the observable frequencies because of a selection rule.

What the rule is actually good for #

None of this makes mutual exclusion useless. It makes it an implication rather than a test, and the two are used differently.

A coincidence proves the absence of a centre, and that is the contrapositive of the theorem, so it is as solid as the theorem is. A band at the same frequency in both spectra means there is no inversion centre, full stop — and that direction is the one two structures, two spectra actually uses when it distinguishes a bent XY₂ from a linear one, because the bent molecule shows the coincidence.

No coincidence proves nothing on its own. It is consistent with a centre and consistent with a fivefold axis and — as the next section says — consistent with a molecule whose coincident bands are simply too weak to see.

The distinction is the same one what an absence proves makes for a single band: a symmetry-forbidden intensity is exactly zero and an unobserved one is merely below a detection limit, and the two look identical on a spectrum.

Three practical reasons a real spectrum shows no coincidence #

Beyond symmetry, the experimental situation supplies at least three ways to see no coincidence in a molecule that has one.

Intensities are unrelated. A band’s infrared intensity depends on how the dipole changes and its Raman intensity on how the polarisability changes, and those are different derivatives of different quantities. A mode allowed in both may be strong in one and invisible in the other, and why a d–d band is weak is the same effect in a different corner: an allowed transition can be feeble for reasons that have nothing to do with what is allowed.

Resolution. Coincidence means “at the same frequency” — and a frequency here is a harmonic one, which the frequency is not the bond strength is careful about — and two bands within a wavenumber of each other are not distinguishable in a solid-state spectrum with bands twenty wavenumbers wide.

The molecule may not be in the conformation it was assigned. Ferrocene is exactly this case: it is D5d in the crystal and D5h in the gas phase, with a barrier between them of about four kilojoules a mole, which is small enough that at room temperature the rings turn freely. Which spectrum belongs to which conformer is not settled by the spectrum.

The dipole selection rules in D₅ₕ are computed the way every selection rule is computed — the product of two representations reduced, and the answer read off from whether it contains the totally symmetric species — so what the exclusion census reports is a consequence of the same arithmetic rather than a separate rule to be remembered alongside it.

How the rule is actually used, and a better version of it #

Working spectroscopists do not use mutual exclusion as a test for a centre, and it is worth setting out what they do instead, because it is the same information used more carefully.

The rule is used forwards, as a consistency check on an assignment. Having proposed a structure, compute its vibrational representation, sort the modes into infrared-active, Raman-active and silent, and check that the counts match what the two spectra show. That uses the whole reduction rather than the one bit of it the exclusion rule extracts, and it is far more discriminating: a wrong structure usually gets the counts wrong somewhere even when it gets the exclusion right.

Coincidences are used to eliminate. A band at the same frequency in both spectra rules out every centrosymmetric structure at once, which is often several of the candidates.

And the absence of coincidences is used as weak evidence. It is consistent with a centre, and a spectroscopist will say so and go and look for something else.

The census in this essay sharpens the third of those. Among the groups a small molecule is likely to have, the absence of coincidences narrows the field to the centrosymmetric ones plus anything with a fivefold or higher axis — which for most molecules is a short list, and for a sandwich compound, a fullerene fragment or a metallocene is exactly the case that matters. The rule is not wrong; it is a rule with an exception, and the exception is a class of molecules that did not exist when the rule was formulated.

What does settle it #

The section above ends with a spectroscopist going to look for something else, and the something else is worth naming, because two of the obvious candidates are weaker than they appear and one is decisive.

Diffraction is the obvious answer and it has a famous catch. The intensities an X-ray experiment measures are proportional to the squared modulus of the structure factor, and that quantity is unchanged by reversing the sign of every index — so the measured pattern is centrosymmetric whether or not the crystal is. A structure without a centre and its inverse produce the same diffraction intensities to first order, which is why deciding between a centrosymmetric space group and its non-centrosymmetric subgroup is a standard difficulty rather than a formality. The distinction is recoverable, from anomalous scattering near an absorption edge, but it comes from a small correction rather than from the main signal.

A dipole moment settles it in one direction only. A measurable permanent dipole rules a centre out immediately, since inversion would reverse the vector and symmetry requires it to be unchanged. But a molecule can have no dipole for other reasons — carbon tetrachloride has none and no centre either — so a null result proves nothing, and this is the same converse error the essay is about, committed with a different observable.

Second-harmonic generation is the positive test. The second-order susceptibility that converts two photons into one of twice the frequency is a third-rank tensor, and a third-rank tensor is odd under inversion: in a centrosymmetric medium every component of it is required to vanish identically. So a sample that doubles the frequency of light passed through it has no centre, and the conclusion runs the direction the exclusion rule cannot — a signal that is present rather than one that is absent.

That is the general shape of the repair. The exclusion rule fails backwards because its evidence is a missing band, and a missing band has many causes. The test that works is one whose evidence is a present signal that a centre would forbid, and the parity argument that makes the exclusion rule true is exactly what makes second-harmonic generation such a test.

Where the model stops #

Symmetry supplies no intensity. Every statement above is about whether a quantity is required to vanish. Nothing here says how large a permitted band is, and no intensity is computed.

Harmonic, and one electronic state. Overtones and combination bands follow different rules and can appear where a fundamental cannot; a resonance Raman spectrum is a different experiment with different selection rules again.

The fifteen groups here are not all of them. The census is over fifteen tables, which cover the molecules drawn here. The claim it supports is that a counterexample exists and needs a fivefold axis, not that D5h is the only one — D5, C5h, D7h and every larger analogue behave the same way, and the icosahedral group without a centre does too. The two fivefold groups the argument needs are generated and verified against the same four internal relations as the other thirteen, and the same tables serve a spectrum counts environments, not atoms without any change.

One further consequence of the census is worth recording because it is the kind of thing a table makes obvious and prose does not. Every group here with no centre puts its coincidence in a degenerate representation, except the three smallest — C1, Cs and C2 — where the representations are all one-dimensional and the coincidence is in the totally symmetric one.

So there are two quite different ways for a group to fail to exclude: the low-symmetry way, where nothing is separated because there is not enough symmetry to separate anything, and the medium-symmetry way, where a degenerate representation is large enough to hold both a coordinate and a product of coordinates at once. D5h escapes the second because its degenerate representations come in two kinds and the two functions land in different ones — which is the mechanism this essay is about, seen from the census rather than from the axis.

What running the rule backwards adds #

Counting frequencies rather than modes, distinguishing two structures by their spectra, deciding what an absence proves, counting environments rather than atoms, and watching a spectrum change when only a mass does are all uses of symmetry on spectra.

This one takes a rule all of those lean on and asks whether it runs backwards. It does not. The forward direction is proved from the parity of the representations, in every centrosymmetric group here; the backward direction is refused by a molecule that has been in every inorganic textbook for seventy years; and the reason is a piece of arithmetic about how fast a quadratic rotates compared with a coordinate, which makes five the smallest axis that can separate them.

The open question is intensities — the quantity symmetry declines to supply, and the one every argument in this field has so far been careful to say it is not computing.