Physical Chemistry Third Edition

898 21 The Electronic Structure of Polyatomic Molecules

Exercise 21.17

a.List the symmetry operations that belong to the ammonia molecule, NH 3.

b.List the symmetry operations that belong to the ethane molecule, C 2 H 6 , in its eclipsed

conformation. Place the molecule with the C–C bond on thezaxis with two hydrogens in

thexzplane.

Group Theory

A mathematicalgroupis a set of objects called members with a single rule for com-

bining two members of the group to produce another member of the group. We call

the member that is produced theproductof the first two members. In the mathemat-

ics of group theory, any well-defined rule for combining two members of the group

can be used. In our application of group theory the members of the groups are sym-

metry operators and the rule for combining two members of the group is operator

multiplication. The product of two operators is a third operator that is equivalent to

successive application of the two operators with the operator on the right operating

first.

The following requirements must be met for the set of members to be a group:

1. IfAandBare members of the group, and ifCis the productAB, thenCmust be a
   member of the group.


2. The group must contain the identity,E, which is defined such that ifAis a member
   of the group
   AEEAA (21.9-5)


3. IfAis a member of the group its inverseA−^1 must be a member of the group. The
   inverse ofAis defined such that

AA−^1 A−^1 AE (21.9-6)

4. The associative law must hold:

A(BC)(AB)C (21.9-7)

It is not necessary that the members of the group commute with each other. It is possible

that

ABBA (possible, but not necessary) (21.9-8)

Abeliangroups arenamedafter Niels If all the members of the group commute with each other, the group is calledabelian.
HenrikAbel,1802–1829,a great
Norwegianmathematicianwho was the
first to show that a general fifth-degree
algebraic equationdoesnotnecessarily
havearadical expressionas a solution.

It is a fact that the symmetry operators belonging to any symmetrical object form a

mathematical group.

EXAMPLE21.16

Show that the symmetry operations belonging to the H 2 O molecule form a group.

Solution

Figure 21.11a shows the H 2 O nuclear framework in theyzplane oriented as in Example

21.15. The hydrogen atoms are labeled HAand HB. The symmetry operators that belong to