Why a character table has the rows it has

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.

Seven 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.

Descent in symmetry

Every representation of the octahedral group restricted to D₄ₕ and reduced there. The five degenerate species all arrive in more than one piece, because D₄ₕ has no representation larger than two-dimensional and Oh has two that are three-dimensional. Each row is a sum over classes rather than a table entry.

Methane to chloromethane. The two singly-degenerate species survive unchanged, E survives as E because C₃ᵥ has a two-dimensional representation to hold it, and both threefold species split into a single and a pair — because C₃ᵥ has nothing three-dimensional. Four hydrogen atoms become three plus one, and the t₂ orbital set becomes a₁ ⊕ e.

Benzene to a bond-alternated ring. Both of D₆ₕ’s degenerate pairs split, since D₂ₕ has only one-dimensional representations. In benzene itself the argument goes the other way: the ring’s π shell is closed, there is no partly filled degenerate level, and so nothing is gained by distorting — which is the symmetry statement of why benzene has equal bonds and cyclobutadiene does not.

Every group a molecule can fall to

Four molecules, and every subgroup of each point group, found by taking the operations recovered from the atom positions and closing subsets of them under multiplication. The right-hand column is how many of those subgroups this site holds a character table for, which is a minority in three rows out of four.

Three molecules and the number of groups each can fall to. Lagrange’s theorem is the whole of the bound: a subgroup’s order divides its parent’s, so ammonia’s group of order six leaves room for orders one, two, three and six only — and how many subgroups there are of each permitted order is a count rather than a consequence.

A species label is ambiguous in every one of the seventeen groups considered here. That is the practical consequence of the lattice being large: a molecule that falls into a subgroup acquires new labels, and the labels do not say which of the parent’s species they came from unless the correlation is worked out.

Why a character table stops where it stops

The three groups of the paragraph above, with the two counts laid out. C2v: order four, four classes, four representations, all one-dimensional, and 1 + 1 + 1 + 1 = 4. C3v: order six, three classes, three representations of dimensions 1, 1 and 2, and 1 + 1 + 4 = 6. Td: order twenty-four, five classes, five representations of dimensions 1, 1, 2, 3 and 3, and the squares sum to twenty-four. The classes column and the representations column are the same column, and the last column is the order.

Every group this site defines, with the two counts. The classes column and the representations column are the same column in every row. The last column is the sum of the squares of the dimensions and it is the order in every row. Neither is checked against a reference: both are read off the tables the site generates and uses.

The three largest groups this site holds, where the room is tightest. Oh: forty-eight operations, ten classes, ten representations of dimensions 1 1 2 3 3 and the same five again, squaring to forty-eight. D6h: twenty-four operations in twelve classes, twelve representations, and the largest dimension anywhere in it is two. D4h: sixteen operations in ten classes, and again nothing above two. The dimensions are not chosen; each row’s list is the only way to write its order as a sum of that many squares.

The tolerance is a decision

What a descent in symmetry does to the labels, for the case the stretch produces. Every representation of the parent goes somewhere in the child, and the map is fixed by the groups alone — so once a verdict has been reached, its consequences follow. The difficulty this essay is about is entirely in reaching the verdict.

The group of a molecule that will not hold still

The counting theorems for the common point groups: the sum of the squares of the dimensions is the order, and the number of representations is the number of classes. Both hold for a molecular symmetry group as well — a group is a group — and the representations of the larger group are what label a torsional spectrum.

How far, and along which coordinate

The tabulated descent for comparison: what each species of D6h becomes in D2h. That table is a statement about representations; the one above is a statement about coordinates, and each is a check on the other.

A label that prices nothing

The share of each molecule’s vibrations belonging to a species that appears more than once. Every bar is above zero and the smallest is two in five.

Each molecule’s vibrations against what its group could hold without repeating a species. The shorter bars are the molecules that repeat with room to spare.

How many groups a molecule can fall to, which is the other half of what a label does not price. A species label says which representation a mode belongs to; it does not say how many ways the molecule could lose symmetry, and the two counts are not related — benzene has ten species and far more subgroups to descend into.