Molecular Symmetry and Group Theory

Lecture 3 — Character tables

Lecture Summary

An irreducible representation is a set of numbers — one for each class of operation in a given point group — that ‘represents’ the effect of the group operations on various directional properties. These properties, listed in the right hand columns, may be identified with particular features such as atomic orbitals. For example, x, y and z indicate the representations appropriate to the x, y and z directions, and therefore px, py and pz orbitals. The full set of possible irreducible representations constitutes the character table of the point group. The character table for the C2v point group is given below:

Point group symbol C2v E C2 σ(xz) σ(yz)    
Mulliken symbols A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x,Ry xz
B2 1 -1 -1 1 y,Rx yz

From the table we can see that a py orbital centred on the O atom in H2O, for example, is modified by the operations in accordance with the B2 irreducible representation. The characters tell us that this orbital is inverted (character -1) by the C2 and σ(xz) operations, but unaltered by σ(yz) and E.

Before proceeding, convince yourself that what the table tells us about the effects of the operations on all the O 2p orbitals in the H2O molecule is correct.

In the course of applying group theory to chemical problems we commonly generate sets of numbers that are not irreducible representations, but linear combinations of these known as reducible representations. The set of numbers 3 3 1 1, for example, is clearly not one of the four irreducible representations of the C2v point group, but is equivalent to 2A1 + A2 (check that this is the case).

It is usually necessary to ‘reduce’ a reducible representation to its constituent irreducible representations. This may be achieved through the reduction formula (Workshop 3, Exercise 2).