Physical Chemistry Third Edition

1294 I Matrix Representations of Groups

The 3 by 3 matrix in Eq. (I-3) is a representative of̂C 2 , denoted byR(̂C 2 ). The

representative matrices of the other three operations in the group are

R(̂E)

⎡

⎣

100

010

001

⎤

⎦ (I-4)

R(̂σyz)

⎡

⎣

− 100

010

001

⎤

⎦ (I-5)

R(̂σxz)

⎡

⎣

100

0 − 10

001

⎤

⎦ (I-6)

Each of these matrices, when applied to the Cartesian coordinates of an arbitrary point

as in Eq. (I-3), gives the same result as operating on the point with the corresponding

symmetry operator. They also have the same multiplication table as the symmetry

operators and are a representation of the groupC 2 v.

Reducible and Irreducible Representations

The representation of theC 2 vgroup that we just obtained is not the only representation

of theC 2 vgroup. This representation is called areducible representation. This means

that it can be divided somehow into representations consisting of matrices with fewer

rows and columns (smaller dimension). This particular representation contains only

diagonal matrices. Such matrices have elements that act on only one coordinate at a time.

Because of this, we can make a set of 1 by 1 matrices by taking the upper left element

of each of the 3 by 3 matrices, and these matrices will have the same multiplication

table as the 3 by 3 matrices. The same thing is true of the set of center elements

and the set of lower right elements. We say that the representation can be divided into

threeirreducible representations, which cannot be further subdivided. A representation

consisting of 1 by 1 matrices is called a one-dimensional representation, a representation

consisting of 2 by 2 matrices is called a two-dimensional representation, and so on.

A one-dimensional representation is necessarily irreducible. A representation of higher

dimension might or might not be irreducible.

The representation obtained by taking the upper left elements is

R(̂E)[1] (I-7)

R(̂C 2 )[−1] (I-8)

R(̂σyz)[−1] (I-9)

R(̂σxz)[1] (I-10)

This set of 1 by 1 matrices consists of the only nonzero elements of the matrices that

act on thexcoordinate. For example, thêC 2 operator turnsxinto –x, andR(̂C 2 )isthe

constant that multipliesxto turn it into –x. The other 1 by 1 matrices are similar.