Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.11 PHYSICAL APPLICATIONS OF GROUP THEORY

4 mm IQR,R′ mx,my md,md′

A 1 11 1 1 1 z;z^2 ;x^2 +y^2

A 2 11 1 − 1 − 1 Rz

B 1 11 − 11 − 1 x^2 −y^2

B 2 11 − 1 − 11 xy

E 2 −20 0 0 (x, y); (xz, yz); (Rx,Ry)

Table 29.5 The character table for the irreps of group 4mm(orC 4 v). The

right-hand column lists some common functions, or, for the two-dimensional

irrep E, pairs of functions, that transform according to the irrep against which

they are shown.

Function Irrep Classes

I 2 C 3 3 σv

xy E2− 10

x E2− 10

x^2 −y^2 E2− 10

product 8 − 10

Table 29.6 The character sets, for the groupC 3 v(or 3mm), of three functions

and of their productx^2 y(x^2 −y^2 ).

Function Irrep Classes

IC 2 2 C 6 2 σv 2 σd

xy B 2 11 − 1 − 11

x E2−20 0 0

x^2 −y^2 B 1 11 − 11 − 1

product 2 −20 0 0

Table 29.7 The character sets, for the groupC 4 v(or 4mm), of three functions,

and of their productx^2 y(x^2 −y^2 ).

multiplying together the corresponding characters for each of the three elements. Now, by
inspection, or by applying (29.18), i.e.

mA 1 =^16 [1(1)(8) + 2(1)(−1) + 3(1)(0)] = 1,

we see that irrep A 1 does appear in the reduced representation of the product, and soJ
is not necessarily zero.
Case(ii). From table 29.5 we find that, under the groupC 4 v,xyandx^2 −y^2 transform
as irreps B 2 and B 1 respectively and thatxis part of a basis set transforming as E. Thus
the calculation table takes the form of table 29.7 (again, chemical notation for the classes
has been used).
Here inspection is sufficient, as the product is exactly that of irrep E and irrep A 1 is
certainly not present. ThusJis necessarily zero and the dipole matrix element vanishes.