Physical Chemistry , 1st ed.

Example 13.2

Identify the symmetry elements in ethylene, which has the structure

From the tables in Appendix 3, determine its point group.

Solution

Ethylene has E, three independent C 2 axes, three independent planes of sym-

metry, and a center of inversion. This total of eight symmetry elements com-

poses the D2hpoint group.

The number of individual symmetry operations in a point group is called

the orderof the group and is symbolized by the letter h(not to be confused

with Planck’s constant). In the above example, the D2hpoint group has an

order of 8. Symmetry operations in a point group are occasionally grouped

together for reasons we will see later. For example, the C3vpoint group treats

C 3 and C^23 together, as well as the three planes of symmetry. As such, it is com-

mon to see that the symmetry operations of the C3vpoint group are listed as

E,2C 3 , and 3v. Each collection of one or more symmetry operations is called

a class. Eis always in its own class. Only similar symmetry operations are

grouped together in classes, but not all of the same symmetry operations can

be grouped together in the same class. For example, the C2vpoint group lists

vand 

vseparately, indicating that they are not grouped together in the same

class. One can say that the C3vpoint group has three classes and an order of 6,

and the C2vpoint group has four classes and h4.

C

H

H

H

H

C

426 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

H

N

H

H* H

N

H*

H H*

N

H

H

Same

H*

N

H

H

C 3

H

N

H

H*

v''

v

Figure 13.12 The closure requirement for a point group means that any combination of mul-

tiple symmetry operations must be equivalent to a single symmetry operation of the group. Here,

the two symmetry operations C 3 and vare shown to be equivalent to the voperation of the

C3vpoint group.