Molecular Geometry & Bonding
what the shape decides
Concept

Conjugacy class — where it appears

A set of group operations related to one another by conjugation, which is what a character table's columns are. Computing them from the operations rather than looking them up is what makes a table checkable.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The Td character table. The irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.

Character tables and reduction

A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be.

symmetry · Representation Oh descending to D4h. Every irreducible representation of Oh restricted to D4h and reduced there — A1g, A2g, Eg, T1g, T2g, A1u, A2u, Eu, T1u, T2u. A representation that arrives in more than one piece is a degeneracy the lower symmetry cannot hold, so a level carrying it must split when the molecule distorts.

Descent in symmetry

Lower a molecule's symmetry and its labels stop being available. Which of them survive, which split, and into what, is decided by restricting characters to the operations that are left — arithmetic, not a table to be looked up.

symmetry · Point group How many groups a molecule can fall to. For each molecule, every subgroup of its point group, found by closing subsets of the operations recovered from its atom positions — beside the number the corresponding abstract group is known to have. The two agree in all 4 cases. The last column is how many of those subgroups this site holds a character table for, which is a minority in every row but the first.

Every group a molecule can fall to

A distortion takes a molecule's symmetry away, and what is left is not a free choice — it has to be a group. Closing subsets of methane's twenty-four operations returns thirty of them, which is exactly the number the symmetric group on four letters has.

symmetry · Point group Why a character table has the rows it has. All 20 groups this site defines, with the two counts that fix the shape of every character table: the number of representations equals the number of classes, and the sum of the squares of their dimensions equals the order of the group. Neither column pair differs anywhere in the table, and there is no room for another row in any of them.

Why a character table stops where it stops

A character table is square, and both of its dimensions are forced. The number of representations equals the number of classes, the squares of their dimensions sum to the order of the group, and between them there is no room for another row.

symmetry · Representation Three counts for each molecule, and the barrier between them. Four molecules, with the number of operations available to each: the point group, which is what a rigid rotation or reflection can do; the feasible group, which adds whatever a crossable barrier makes available; and the whole permutation-inversion group, which is every rearrangement of identical nuclei whether a molecule can reach it or not. The middle column is the one that describes a real spectrum, and it equals the first only when nothing moves.

The group of a molecule that will not hold still

A point group is a group of rotations of space, and it presupposes a structure for them to act on. Ethane has no one structure — its methyl groups turn billions of times a second — and the group that describes its spectrum has thirty-six elements where the point group has twelve, out of two thousand eight hundred and eighty conceivable.

symmetry · Point group Every set of point groups here that shares a table. Found by comparing character matrices rather than named: five sets among the tabulated groups, at orders 2, 4, 6, 8, 20. The smallest is the most startling — a mirror plane, a centre of inversion and a twofold axis all have the two-row table with entries 1, 1 and 1, −1, and one of the three describes a chiral molecule. The largest is the eclipsed and staggered conformers of ferrocene, which are one molecule at two temperatures.

One table, three groups

A mirror plane, a centre of inversion and a twofold axis have the same character table — two rows, entries 1, 1 and 1, −1 — and one of the three describes a chiral molecule. A character table is a property of an abstract group; a point group is that plus an action on space, and the action is what a molecule has.

symmetry · Representation

Named alongside it

The objects these essays reach for when they reach for this one.

Group orderPoint groupSymmetry operationCharacter tableIrreducible representationsDegeneracyReduction formulaSubgroupChiralityJahn–Teller distortionPolarityBasis

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