would be found if a larger C2vmolecule were used as an example. That set
would be (1, 1,1,1). Together, these four sets of numbers represent the
simplest way of defining the symmetry properties of any object that has C2v
symmetry. They are the irreducible representationsof the C2vpoint group.
The numbers themselves are called characters.These characters are not al-
ways 1 or 1. They can be zero, a larger integer (2 and 3 are common for
higher-symmetry point groups), fractions, or exponential functions. Every
point group has a limited number of irreducible representations, each of
which is given some label that refers to that irreducible representation. These
representations are tabulated in character tables.Appendix 3 contains char-
acter tables that list not only the symmetry operations of the point groups,
but the characters that represent the simplest representation of the effects of
those symmetry operations.
Each irreducible representation is labeled with a letter (such as A,B,E,T,
depending on the character of the identity operation), which sometimes has a
subscript, superscript, or primes ( or ) with it. All of the irreducible repre-
sentations within a point group have different labels. Each label represents the
set of characters in that row of the table. Sometimes irreducible representations
from different point groups have the same label, but it is important to under-
stand that a certain label represents a specific set of characters for a particular
point group. Hence, the A 2 irreducible representation in C2vis a different set
of characters from the A 2 irreducible representation in D4d. It is necessary that
you be aware of what point group you are working in when you use the labels
of the irreducible representations.
In many point groups, some of the symmetry operations have the same
characters for all of the irreducible representations. They are grouped together
into the same class.For example, in C3v(the point group that describes the
symmetry of NH 3 ), the two C 3 symmetry operations are grouped as a class,
and the three v’s are also grouped in a class.A point group has only as many
irreducible representations as it has classes.Therefore,C2vhas four and only four
irreducible representations.C3vhas only three. Understand that there are a total
of six symmetry operations in C3v, so one canwrite a 3 6 C3vcharacter table.
(It won’t be 6 6 because C3vonly has three irreducible representations.)
However, many of the columns will be duplicates, so it is easier to list symme-
try operations by class.Example 13.6
Write the 3 6 character table for C3v.Solution
Using the 3 3 character table from Appendix 3 and separating the opera-
tions of each class:EC 3 C^23 v
v v
A 1 1 1 1 1 1 1
A 2 1 1 1 1 1 1
E 2 1 1 0 0 0As you can see, it is simply more efficient to group the classes together in the
character table.434 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics