The Td character table

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

Nine essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Point groups from coordinates

Methane’s character table, with the class headings carrying the number of operations the generated group actually put in each. Those counts came out of the coordinates. The characters below them are the one tabulated ingredient on this page, and they are checked against the counts above before anything uses them.

Character tables and reduction

Methane’s character table, with each class heading carrying the number of operations found by generating the group from the coordinates. Those counts were produced before this table was opened, and the table is checked against them.

Degeneracy is a group theorem

The character table of the tetrahedral group, generated here by closing the operations found in methane’s coordinates under multiplication and sorting them into conjugacy classes by conjugation. The first column is each representation’s dimension: 1, 1, 2, 3, 3. Those five numbers are the complete list of degeneracies a tetrahedral molecule can have.

C₂ᵥ, water’s group, where every representation is one-dimensional. A molecule in this group has no degenerate levels available to it at all, which is a stronger statement than any calculation could make and is available before one is attempted.

D₆ₕ, the largest table this collection generates, where the dimensions run to two. Every degeneracy any benzene-shaped object can have is in the second column of this table and nowhere else — the levels a calculation returns must fall into these species, and no calculation can produce a threefold degeneracy in a group whose largest dimension is two.

The vibration that lowers the symmetry

Eg × Eg in D4h, reduced. It comes out a₁g ⊕ a₂g ⊕ b₁g ⊕ b₂g. The a₁g part is the breathing mode, which changes no symmetry; a₂g is a rotation; and b₁g is the rectangular distortion — one pair of opposite bonds lengthening while the other pair shortens. That is the coordinate a square molecule with a half-filled degenerate pair moves along.

Descent in symmetry

D₄ₕ, the group an octahedron becomes when it is stretched along one axis — here recovered from xenon tetrafluoride’s coordinates, which reach the same group by being square planar rather than by being a stretched octahedron. The class sizes above each column were counted on those coordinates before the table was opened. Ten representations, all of dimension one except two, which is why nothing three-dimensional can survive the descent. The table is verified before use against four internal relations — orthogonal rows, orthogonal columns, squared dimensions summing to the order, and one representation per class.

An infinite group, worked in a finite one

The working group doing its ordinary job. The product of carbon dioxide’s ground state with its antisymmetric stretch, reduced in D2h, contains a species carrying z — so the mode is infrared active, which is the same conclusion the infinite group reaches. Every activity statement in this essay is one of these products, and the trade only bites where a delta species is involved.

Why a character table stops where it stops

C3v as this site generates it: three classes recovered by conjugating the operations, so three representations and no more. The dimensions are 1, 1 and 2, and 1 + 1 + 4 is the order.

Td: twenty-four operations, five classes, five representations, dimensions 1, 1, 2, 3 and 3. The sum of squares is twenty-four, and the table is full.

One table, three groups

Water’s character table, generated from the twenty-four operations of its own structure by closure and sorted into classes by conjugation. Everything in the matrix here is shared with C₂ₕ and D₂; the assignments in the last two columns are not.

The ring's levels are its group's characters

Benzene’s full character table, generated from the twenty-four operations of its own structure. The cyclic subgroup used above is six of these twenty-four; the rest are what turn a pair of degenerate levels into a named species with a stated infrared and Raman activity.