An infinite group, worked in a finite one

Carbon dioxide’s point group is D∞h. It has a rotation by every angle about its axis, a mirror plane containing the axis at every orientation, and a two-fold axis perpendicular to it in every direction — a continuum of operations rather than a list.

Everything computed about a group starts by generating it. Closing a set of operations under multiplication terminates when the set is finite, and there is nothing here for it to terminate at. Worse, the reduction formula

$$n_i = \frac{1}{h}\sum_{\text{classes}} g_c\, \chi(c)\, \chi_i(c)$$

divides by h, the number of operations. So the arithmetic character tables and reduction are built on refuses a linear molecule outright.

The device, and what makes it legitimate #

The standard treatment, in every textbook, is to work in the largest finite subgroup that preserves the molecular axis: D2h for D∞h, C2v for C∞v. The molecule genuinely has all of those operations, so nothing false is being claimed about it; what is being given up is the operations of the full group that the subgroup leaves out.

The construction here is explicit rather than searched. The molecular axis is found from the coordinates, a right-handed frame is built on it, and the eight matrices of D2h — or the four of C2v — are written down in that frame and rotated back. Each one is then checked against the molecule: every constructed operation must carry every atom onto an atom of the same element, and one that does not stops the figure.

That check is the whole of what makes this different from simply declaring a group. This is the one place here where a group is constructed rather than found from the coordinates, and it has to be, because the search that finds groups elsewhere would never terminate on an axis of infinite order. So the construction is verified against the structure instead.

The count is then checked too. A linear molecule has 3N−5 vibrations rather than 3N−6, because it has only two rotations: turning it about its own axis moves no atom, so that motion is not a rotation of the molecule and never was. Subtracting a rotation basis computed in the finite subgroup takes away one representation too many, and the multiplicity that goes negative is exactly the one to restore — and at most one ever does.

What comes out for carbon dioxide #

The reduction gives ag ⊕ b₁u ⊕ b₂u ⊕ b₃u in D2h, which is four vibrations for three atoms as required. Translated into the linear labels those are Σg⁺ ⊕ Σu⁺ ⊕ Πu, with b₂u and b₃u, which came out separately, being the two components of the doubly degenerate bend.

The activity conclusions carry over exactly. The ag mode carries quadratic functions and no linear one — Raman active, infrared forbidden. The three u species carry x, y or z and no quadratic function — infrared active, Raman forbidden. No mode is active in both, which is the rule of mutual exclusion, and it appears here as a computed count in a group the molecule has rather than as a remembered slogan about a group it also has.

The spectrum the assignment belongs to has four fundamentals and three distinct frequencies, because two of them are a degenerate pair. Nothing in the working group changes any of those numbers; what it supplies is which of the bands can appear in which technique.

Where the device loses something, computed #

A subgroup has fewer operations, so an integral has fewer chances to be forced to vanish: a subgroup forbids less than the group does. The error is therefore always in the permissive direction, and the question is whether any case arises where the difference matters.

One does, and it is computed rather than warned about. The linear representation Δu is infrared forbidden in D∞h. Correlating it into D2h gives Au ⊕ B₁u, and B₁u carries z. So the working group calls infrared-active a mode the real group forbids.

That is the honest end of the device, and it is checked rather than mentioned: the correlation is looked up, the working group’s table is consulted, and the claim that the subgroup is more permissive here is required to hold.

The reason it does not bite for any molecule in this collection is a count. A linear triatomic or a linear four-atom molecule has only Σ and Π vibrations — delta modes need more atoms and more ways to bend — so no vibration of carbon dioxide, carbonyl sulfide or hydrogen cyanide is of a delta species, and every activity conclusion above is safe. A larger linear molecule would need the caution, and would get it.

What still works in the infinite group #

It is worth being precise about how much is actually lost, because the answer is: only the things that need to count operations.

Which operations exist is decidable directly. A linear molecule’s axis is found by checking whether every atom lies on one line, which is a test on the coordinates, and point groups from coordinates does exactly that before anything else — the linear case is caught first and reported as C∞v or D∞h according to whether inversion is present.

Whether the molecule can be polar or chiral needs only the symbol. C∞v permits a dipole and D∞h does not, by the same rule as every other group, and symmetry forbids a dipole applies unchanged.

Whether a particular integral vanishes can be settled in the infinite group by inspection, because the argument is a parity argument rather than a sum: a function odd under inversion integrates to zero over a centrosymmetric molecule whatever the group’s order. Nothing needs to be divided by anything.

What a basis’s character is under any given operation is a count of what stayed put, which is well defined for every operation of an infinite group individually.

The single thing that fails is the reduction — turning a character into a list of multiplicities — because that is where the order appears. Everything above is a way of getting the reduction done in a group where the formula applies.

The count that has to be repaired, and why #

The subtraction of translations and rotations needs one correction for a linear molecule, and following it is worth the paragraph because the correction is not arbitrary.

The rotational basis used in the reduction has three components, because it is built from the three Cartesian rotations. A linear molecule has only two rotations: the third — about the molecular axis — moves no atom whatever, so it is not a motion of the molecule and never was. Subtracting all three therefore removes one representation the molecule did not have.

The arithmetic says which one, and says it by going negative. After subtracting, exactly one multiplicity comes out at −1, and that is the species the spurious rotation was assigned to. The calculation requires that at most one multiplicity goes negative — more than one would mean something else was wrong — sets it to zero, and then checks the total dimension against 3N−5.

That check is the reason the repair is safe rather than a fudge. If the wrong representation had been restored, or two had been, the dimension would come out at 3N−6 or 3N−4 and the check would fail.

What is exact, what is constructed and what is tabulated #

The essays around this one make a point of separating those three, and the linear case has all three in it at once. Setting them out explicitly is the honest summary of the device.

Exact, from the coordinates. That the molecule is linear; which finite operations it has; that each constructed matrix carries every atom onto an atom of the same element; the count 3N−5; and the activity of each species in the working group.

Constructed rather than found. The eight matrices of D2h, or the four of C₂ᵥ, written down in a frame built on the molecular axis. This is the exception to the site’s practice of generating groups by closure, and it exists because closure would never terminate here.

Tabulated. The characters of D2h and C₂ᵥ, as everywhere else on the site; and — uniquely — the correlation from Σ, Π and Δ onto the working group’s species, which cannot be computed because there is no finite table for the infinite group to restrict from.

That last line is the only place in this collection where an answer is looked up rather than derived, and it is marked as such in the figures: the column headed called is data, and the column headed by the working group’s name is arithmetic.

Read back into the infinite group, the same four fundamentals carry the labels a spectroscopist uses — Σg+, Σu+ and a doubly degenerate Πu — and the activities are the ones the finite working group returned. The translation runs both ways and loses nothing.

The relation to lowering symmetry on purpose #

The correlation used here — which species of the big group maps onto which of the small — is the same object descent in symmetry computes, and the difference is instructive.

There, the descent is a physical change: a molecule is distorted or placed in a lower-symmetry environment, and the question is which labels survive and which split. The characters are restricted to the operations that remain, and the restriction is computed rather than looked up, because both groups are finite and both tables are available.

Here, the descent is a computational device: the molecule is unchanged and only the analysis moves. And the correlation cannot be computed the same way, because there is no table for the infinite group to restrict from — it is data rather than arithmetic, and this site marks it as such. That is the one place in this collection where a table is consulted rather than derived, and it is worth flagging, because character tables and reduction makes a point of the characters being the only tabulated ingredient anywhere.

A descent in symmetry is the same operation used for a different purpose: there the molecule changes and the analysis follows it; here the molecule is unchanged and only the analysis moves. The correlation table is the same kind of object in both cases, which is why the same arithmetic serves.

The same device, elsewhere in physics #

Working in a subgroup because the full group is intractable is not peculiar to linear molecules, and naming the pattern makes the trade easier to remember.

A molecule in a crystal is analysed in its site group rather than its own, because the environment has destroyed some of its operations. Modes forbidden in the free molecule become allowed, and the correlation between the two sets of labels is computed exactly as descent in symmetry computes it.

A distorted molecule is analysed in the group it has rather than the one it nearly has, and the near-symmetry survives as an approximate selection rule: bands that would be forbidden in the ideal structure appear weakly.

A degenerate state that distorts moves into a subgroup on its own, which is the Jahn–Teller effect — the vibration that lowers the symmetry computes the coordinate it moves along.

All three share the direction of the error, and it is the direction this essay measured: a subgroup forbids less than a group. Every one of them makes forbidden things allowed and never the reverse, so a conclusion of the form this is forbidden is safe under the substitution and a conclusion of the form this is allowed is not.

The summary worth carrying is one sentence. Everything that does not divide by the group order works in the infinite group directly; only the reduction has to move, and the subgroup it moves into forbids less than the group it left.

How large the finite subgroup has to be #

The trade is worth pricing precisely rather than accepting, because there is a rule for how big the subgroup must be and it depends on the problem rather than on taste.

A linear molecule’s representations are labelled by how the function changes as it is rotated about the axis: unchanged, changing sign once per revolution, twice, three times — the labels Σ, Π, Δ, Φ and onwards, corresponding to $|m| = 0, 1, 2, 3$.

A finite subgroup with an $n$-fold axis cannot tell those apart indefinitely. Rotating by $2\pi/n$ multiplies a function of index $m$ by a phase factor, and two values of $m$ differing by $n$ acquire the same factor — so the subgroup aliases them, exactly as a sampled signal aliases frequencies above half the sampling rate.

That fixes the requirement. To keep the labels distinct up to some highest index $m_{\max}$, the subgroup needs

$$n > 2\,m_{\max},$$

and any smaller choice will report two different species as one.

The practical consequence is that the required size depends on what is being counted.

Vibrations of a linear molecule span only Σ and Π — a stretch along the axis and a bend perpendicular to it — so $m_{\max} = 1$ and a threefold axis already suffices. That is why the substitution is harmless for counting modes.

Electronic states reach further. A Δ state has $m = 2$ and a Φ state $m = 3$, so a treatment that includes them needs a sixfold or eightfold axis before the labels stop merging.

So the trade is not a uniform approximation with an error to be estimated. It is exact below the aliasing threshold and catastrophic above it — two species reported as one, which is a wrong integer rather than an inaccurate number.

Which makes the choice checkable rather than conventional. Ask what the highest index appearing in the problem is, and take a subgroup more than twice as large, and the finite calculation is not an approximation to the infinite one at all: it is the same answer, obtained in a group small enough to divide by.

The aliasing analogy is worth keeping rather than discarding once the rule is stated, because it predicts the failure’s shape as well as its threshold. An undersampled signal does not degrade gracefully — it reports a high frequency as a low one, confidently and with no indication that anything is wrong. A subgroup that is too small does the same: it returns a species from its own character table, the count comes out an integer, the arithmetic is consistent, and the answer is a different species from the true one.

So there is no diagnostic inside the calculation. The check has to be made before it, by asking what indices the problem contains — which is why the rule above is worth stating as a rule rather than left as a caution to be exercised when something looks odd.

What this does not solve #

Delta and higher species are handled by correlation only. For a molecule with such vibrations, the working group’s activity conclusions would need checking against the correlation case by case, and none of the molecules here has one.

Rotational structure is untouched. A linear molecule’s rotational levels do not need the finite subgroup at all — the rotational spectrum is a moment of inertia computes them from the moments directly — so the device is confined to the vibrational and electronic analysis.

Nothing here computes an infinite group’s characters. They exist and are simple, indexed by the angular momentum about the axis; they are not tabulated here; the finite subgroup is used instead, and said so wherever the substitution is made — the figures print the group and the working group together for exactly that reason.

For this collection the practical upshot is small and worth recording: three linear molecules — carbon dioxide, carbonyl sulfide and hydrogen cyanide — are analysed this way, none of them has a delta vibration, and every activity statement made about them is therefore exactly what the infinite group would have given.

Who does it this way #

Everybody. The device is in Cotton, in Harris and Bertolucci, in Wilson, Decius and Cross, and in every course. What is usually missing is the second half: what the substitution costs, and whether the case where it costs something has arisen.

That omission is the small failure this essay is against, and it is the same shape as several others. A method is taught with its justification and without its boundary; the boundary then has to be rediscovered by whoever first runs into it. Computing the Δu case takes one line of arithmetic and turns a footnote into a fact.

Still open: a symmetry a molecule cannot keep #

That takes point groups from recovering a group out of coordinates to working around one that cannot be enumerated. One spectroscopic case is left, where a molecule finds itself in a symmetry it cannot keep: the vibration that lowers the symmetry computes what happens to a degenerate electronic state, and finds the distortion that removes it in a direct product.