One table, three groups

A character table is a matrix of numbers with a list of class sizes beside it. It can be generated from a molecule’s own operations rather than looked up, and almost every use of symmetry in chemistry passes through one: how many bands a spectrum can have, which transitions vanish, what a set of orbitals reduces to.

The table is the object. It is not, however, the object it is usually treated as, and the difference is the subject here. A character table is a property of an abstract group — the multiplication table, considered without any reference to what the elements do. A point group is an abstract group together with an action on three-dimensional space, and the action is the part a molecule has.

Several point groups can therefore share a table entry for entry and disagree about every physical question asked of them.

Found rather than named #

The search is mechanical. Two groups share a table if they have the same order, the same list of class sizes, and the same set of character rows — sorted, because the order in which the representations are printed is a convention. Comparing signatures over the tables used here turns up four sets.

The smallest is the most startling and the least remarked on. The groups of order two are Cs — a molecule with a mirror plane and nothing else — Ci, a molecule with a centre of inversion and nothing else, and C₂, a molecule with a twofold axis and nothing else. All three have the same two-row table, with entries 1, 1 and 1, −1.

They could hardly describe more different molecules. One of them is chiral.

The order-four set #

The set worth working through in full is the next one, because all three of its members are common and the differences between them are all in different directions.

C₂ᵥ has a twofold axis and two mirror planes containing it. C₂ₕ has a twofold axis, a horizontal mirror plane, and — as their product — a centre of inversion. D₂ has three mutually perpendicular twofold axes and no improper operation at all.

Every one is the Klein four-group: four elements, each its own inverse, all commuting. As abstract groups they are the same group. As symmetries they are three different things.

Polarity. A molecule may have a permanent dipole moment when one of the coordinates $x$, $y$, $z$ transforms as the totally symmetric representation. In C₂ᵥ, $z$ does, so water is polar. In C₂ₕ, $z$ transforms as Aᵤ and no coordinate is totally symmetric, so trans-dichloroethene is not. In D₂ the coordinates go to three different non-symmetric representations, and it is not either.

Chirality. A molecule may be chiral when its group contains no improper operation. D₂ contains none; C₂ᵥ and C₂ₕ both do. So one of the three describes molecules that come in non-superimposable mirror images and the other two do not — and the criterion is the group rather than a list of substituents.

Mutual exclusion. C₂ₕ has a centre of inversion, so no representation carries both a coordinate and a product of coordinates, and no vibration is both infrared and Raman active. C₂ᵥ and D₂ have no centre and show coincidences.

Where z lives. A₁ in C₂ᵥ, Aᵤ in C₂ₕ, B₁ in D₂. Three different labels for the representation a z-polarised transition connects to, on identical tables.

Every one of those is decided by which coordinate is attached to which row, and the attachment is what the action on space supplies. The numbers know nothing about it.

What the numbers are and are not #

It is worth being precise about what has and has not been shown, because the natural reading — character tables are unreliable — is wrong.

The table is a complete description of one thing: how the representations of the abstract group compose and decompose. Every calculation that uses only that is safe. Reducing a reducible representation, counting how many times a species appears, applying the orthogonality relations, computing a product — all of these use only the matrix, and all give identical answers in any two groups sharing it.

What is not in the matrix is the correspondence between representations and physical quantities. The right-hand columns of a printed character table — the ones listing $x$, $y$, $z$, $R_z$, $xy$, $x^2 - y^2$ — are not part of the matrix. They are the extra data, and they are exactly what differs between the groups sharing it.

This is why a printed table has those columns at all. It is easy to read them as a convenience, a reminder of which functions belong where. They are the content: without them the table cannot be used for a selection rule, and with them it is no longer shared.

The selection rule, computed three ways #

The abstract statement above is worth turning into the concrete calculation, because a selection rule is where the difference actually bites.

The vanishing-integral theorem says a transition between two states is allowed when the product of their representations with the representation of the operator contains the totally symmetric one. For an electric dipole transition out of a totally symmetric ground state, that reduces to: the excited state must transform as one of $x$, $y$, $z$.

Run it in all three order-four groups and the reduction agrees in each case — the z-allowed transition out of the ground state is allowed, which it has to be, since $z$ is what makes it so. What differs is where it goes. In C₂ᵥ it reaches A₁, which is the same representation the ground state is in. In C₂ₕ it reaches Aᵤ. In D₂ it reaches B₁.

So a spectroscopist handed a set of energy levels labelled A, B₁, B₂, B₃ and told the molecule’s table cannot say whether a transition to B₁ is z-polarised, y-polarised, or forbidden. The labels are shared; the polarisations are not. That is not an edge case — polarised spectroscopy on oriented samples is how vibrational assignments are made, and the whole method depends on the half of the table that the matrix does not contain.

The same argument applies to every reduction this collection performs. Reducing a set of hydrogen 1s functions gives a₁ ⊕ t₂ in methane, and that reduction is pure matrix arithmetic — the same numbers would come out of any group with the same table. What makes a₁ ⊕ t₂ a statement about a photoelectron spectrum is knowing that t₂ is the species the p orbitals span, and that is a basis assignment.

The order-eight set #

The same phenomenon one order up, with a degenerate representation in it.

C₄ᵥ, D₄ and D₂d all have order eight, five classes of sizes 1, 2, 1, 2, 2, and four one-dimensional representations plus one two-dimensional one. All three are the dihedral group of order eight.

The physical differences are the same in kind. C₄ᵥ may be polar and the other two may not. D₄ is chiral and the other two are not. And $z$ goes to A₁, A₂ and B₂ respectively — three different representations, so a z-polarised transition out of the ground state is allowed to a different species in each.

The chemistry here is common enough to be worth naming. A square pyramidal complex is C₄ᵥ and polar — the geometry a d-orbital splitting is computed for in half the coordination chemistry there is. A square planar complex with a twist — a chelate ring puckering, say — can be D₄ and chiral. Allene and its substituted relatives are D₂d, and their chirality when substituted is a textbook case that has nothing to do with a stereocentre.

The pair at order twenty #

The last set is two rather than three, and it is one molecule.

D₅h and D₅d are the groups of ferrocene in its eclipsed and staggered conformations. The two rings turn against each other over a barrier of a few kilojoules per mole, so a real ferrocene molecule is in both groups and neither — which is precisely the situation a rigid point group cannot describe.

Both are non-polar, so polarity does not separate them. What does is the centre: D₅d has one and D₅h does not. So the eclipsed conformer shows coincidences between its infrared and Raman spectra and the staggered one does not — and that is the whole of how mutual exclusion is used, applied to two conformers with the same character table.

The first version of the check behind this section required every shared set to disagree about polarity, and was refused here: both of these are non-polar. What has to hold is weaker and is the right statement — that something physical differs, or the table would determine the group after all.

A cautionary case: two conformers, one table #

The ferrocene pair deserves more than the census, because it is the only entry among the four where the two groups belong to the same substance.

Ferrocene’s rings turn against each other over a barrier of about 4 kJ per mole. In the gas phase the eclipsed conformer is the more stable one and the molecule is effectively free-rotating at room temperature; in the crystal the packing decides, and different salts and different temperatures give different answers. So a single compound has been assigned to D₅h and to D₅d in different papers, and both assignments can be right for the sample described.

If the two groups had different character tables, the disagreement would show up immediately in any calculation using one: a different number of representations, a different reduction, a different count of allowed bands. They do not, so almost every group-theoretical conclusion about ferrocene is the same either way, and the disagreement is invisible to the arithmetic.

Where it is not invisible is exactly the one place the basis assignments differ: the centre of symmetry. D₅d has one, D₅h does not, and mutual exclusion follows. So the whole of the experimental discrimination between two conformers of one molecule rests on the half of the character table that is not part of the character table.

Where this comes from #

The underlying fact is a piece of group theory that predates its chemical use. The character table of a group is determined by the group’s conjugacy classes and the dimensions of its representations, and none of that knows how the group is realised. Two realisations of the same abstract group — two different faithful actions on three-dimensional space — are as different as the actions are, and share everything the table contains.

The more famous statement in this area is stronger and does not arise for point groups: there exist non-isomorphic groups with identical character tables, the standard pair being the dihedral group of order eight and the quaternion group. That is a genuine failure of the table to determine even the abstract group. Nothing so dramatic is needed here. Every set found above consists of groups that are isomorphic; they differ only in how they act, and that is enough.

What the search would not identify #

A search comparing signatures could go wrong in an obvious way, so it is worth saying what it refuses. Two groups of different order never share a signature — the order is part of it. Two groups whose class sizes differ never share one either, and that is the less obvious half: D₃h and D₄h both have real character tables of a familiar shape, and their class structures are different, so no permutation of rows makes them agree.

The rows are sorted before comparison and the classes are not, and that asymmetry is deliberate. The printing order of the representations is arbitrary and two tables differing only in it are the same table. The classes are ordered by their sizes, which are part of the signature, so permuting the columns independently would identify tables that are genuinely different.

The columns that carry the chemistry #

Three groups sharing a table has a consequence for how a character table should be read, and it is the reverse of how one is usually presented: the numbers are the part that does not distinguish them, and the columns beside the numbers are the part that does.

Every character table carries, to the right of the characters, a list of basis functions for each row — which of $x$, $y$, $z$ belongs to which representation, and which of the quadratic products $x^2$, $xy$, $z^2$ and the rest. Those columns are where the group’s action on space is recorded, and they are exactly what the three groups with one numeric table do not share.

In the group with a mirror plane, one row carries two of the three coordinates and the other carries the third. In the group with a twofold axis, the split is different. In the group with a centre of inversion, all three coordinates are in the same row and every quadratic product is in the other.

Those differences decide everything a chemist wants from the table.

Whether a dipole is permitted depends on which row the coordinates fall in. Whether a vibration is infrared active depends on the same. Whether it is Raman active depends on where the quadratic products fall — and in the centrosymmetric case the coordinates and the products are in different rows, which is mutual exclusion, and in the other two they are not.

So two molecules with identical character tables have different selection rules, different activities, and in one case a chirality the other lacks — all of it recorded in columns that a reader skimming a table for its numbers passes over.

The practical form is short. A character table’s numbers are a property of an abstract group and its basis-function columns are a property of a point group, and every chemical conclusion drawn from a table comes from the second.

What to do about it #

Nothing, in the sense of a repair — the tables are correct and complete for what they contain. What changes is how a table is read.

A character table has two halves that are usually printed as one object. The matrix is a property of an abstract group and is shared. The basis assignments are a property of a point group, and every physical conclusion drawn here — a selection rule, a polarity, a chirality, an infrared activity — comes from them.

That distinction is easy to state and hard to keep in mind while working, because the two halves sit on the same page and look like one table. The clearest reminder is the smallest set: Cs, Ci and C₂, one two-row table between them, describing a molecule with a mirror, a molecule with a centre, and a molecule that has an enantiomer.

There is a second reminder in what the tables here already refuse. A character table stops where it stops because a group has only as many representations as it has classes, and that limit is a property of the abstract group too. So the tables above are the same size for the same reason, and being the same size is the first thing that makes them confusable.

Still open: matrices, and checking the basis columns #

Two directions follow directly.

The first is the sharpening: for a group of order two the table cannot distinguish three cases, but a representation can — the three groups have different faithful three-dimensional representations, and writing those out is what would separate them without appealing to the basis columns. That is a step from characters to matrices, and it is the same step the projector needed for a different reason.

The second is the practical one. A character table can be verified against four internal relations, and none of those relations can detect a table entered under the wrong name — because a table entered under a name it shares with two other groups is right, as a matrix, and wrong as a description. What would catch it is a check on the basis assignments against the operations, and that is a check on the half of the table nobody verifies.