Physical Chemistry Third Edition

21.9 Groups of Symmetry Operators 899

the molecule arêE,̂C 2 ,̂σvyz, reflection through theyxplane and̂σvxz, reflection through the

xzplane.

Condition 1.The product of any two members of the group must be a member of the group.

The multiplication table of the group is shown in Table 21.3, in terms of the symmetry

operations. The operations listed in the first row of the table are to be used as the right factor

in a product, and the operations in the first column of the table are to be used as the left factor

in the product. The productσyzC 2 is obtained as follows: TheC 2 operation interchanges the

positions of the hydrogen nuclei, as shown in Figure 21.11b. Theσyzreflects through the

plane of the nuclei and does not move the nuclei. The nuclei are now in the positions that

either theC 2 or theσxzwould have put them, and we have to decide which of these operations

is equivalent to the productσyzC 2. In order to decide, we consider a coordinate system that

moves with the nuclei. A reflection turns a right-handed coordinate system into a left-handed

coordinate system. Since the two operations reverse the handedness of the coordinate system

as wouldσxz, we conclude that

σyzC 2 σxz (21.9-9)

A simpler analysis is to notice thatσyzis not the same asE, so thatσyzC 2 is not the same as

C 2. The action of theσxzoperation is shown in Figure 21.11c. The products of other pairs of

operations are obtained similarly.

Condition 2.The group does contain the identity operation,E.

Condition 3.The inverse of every member is in the group, because each member of this

group is its own inverse.

Condition 4.The group operation is associative, because operator multiplication is asso-

ciative.

Exercise 21.18

Verify several of the entries in Table 21.3, the multiplication table for this group.

Point Groups and Schoenflies Symbols

A group that consists of point symmetry operations is called apoint group. A unique

symbol called aSchoenflies symbolis assigned to each point group. The Schoenflies

symbol of the point group of the H 2 O molecule isC 2 v. This symbol indicates that the

molecule has aC 2 rotation axis and two vertical mirror planes. You can specify the

symmetry properties of a molecule like H 2 OorSO 2 quickly to a knowledgeable person

Table 21.3 Multiplication Table for

the H 2 O Molecule

EC 2 σyz σxz

E EC 2 σyz σxz

C 2 C 2 Eσxz σyz

σyz σyz σxz EC 2

σxz σxz σyz C 2 E