Physical Chemistry , 1st ed.

tially. For our purposes, the closure requirement is more important. Any

combination of symmetry operations in a point group is equivalent to a single

operation of that group. Again, consider the molecule NH 3 , which has the

point group C3v. It consists ofE,C 3 ,C^23 ,v,

v, and v. The corresponding

symmetry elements are illustrated in Figure 13.11. What is the consequence of

operating an NH 3 molecule with C 3 and then v? Figure 13.12 shows these two

operations with the hydrogens labeled so we can keep track of their relative

positions. Figure 13.12 also shows that performing a single operation,v,will

exchange the atoms in NH 3 in the same way that the combination ofC 3 and

vdid. This is an example of the closure requirement for groups. All combi-

nations of symmetry operations in a group behave similarly.

13.3 The Mathematical Basis of Groups 425

H

N

H

H* H

N

H*

H

H

N

H

H

H*

N

H

H

C 3

C 3 C 3

Figure 13.10 C 3 is not its own inverse. The ammonia molecule needs another symmetry op-

eration to rotate to its original position.

H

N

H

H

v

C 3 , C 32

v' v'' and E

Figure 13.11 The six symmetry operations of ammonia, NH 3. One N–H bond lies in each

plane of symmetry. Collectively, these six symmetry operations compose the C3vpoint group.