Physical Chemistry Third Edition

21.9 Groups of Symmetry Operators 901

Table 21.4 Examples of Some Point Groups

Schoenflies Symbol Symmetry Operations Example Molecule

C 1 E CIBrClF

Cs E,σ H 2 CCClBr

Ci E,i HClBrC−CHClBr (staggered conformation)

C 2 E,C 2 H 2 O 2

C 2 v E,C 2 ,2σ H 2 O, SO 2

C 3 v E,2C 3 ,3σ NH 3

C 4 v E,2C 4 ,C 2 ,2σv,2σd XeOF 4

C∞v E,C∞,σv HF

D 3 h E,2C 3 ,3C 2 ,3σv,σh BF 3

D 6 h E,2C 6 ,2C 3 ,C 2 ,C′ 2 ,3C′′ 2 ,i,2S 3 ,2S 6 ,σh,3σd,3σv C 6 H 6

D∞h E,C∞,C 2 ,σh,σv,i H 2

Td E,8C 3 ,3C 2 ,6S 4 ,6σd CCl 4

Oh E,8C 3 ,6C 2 ′,6C 4 ,3C 2 ,i,6S 4 ,8S 6 ,3σd,3σh SF 6

mirror plane. For example, the groupC 4 vhas aC 4 axis, four vertical mirror planes, and

the identity operationE. The groupC 1 contains only the identity operationE. TheCs

group contains only a reflection plane andE. TheTdgroup is the group of tetrahedral

molecules such as CH 4 , and the Ohgroup is the group of octahedral molecules such as

SF 6. These groups contain many symmetry operations, which the interested reader is

invited to list.

Exercise 21.19

Think of a molecule belonging to each of the point groups:

a.C 4 v

b.C 3 v(other than NH 3 )

c.D 3 h

Figure 21.12 represents a scheme for assigning a molecule to a point group. This

type of diagram is called adecision treeor aflow chart. It is assumed that the user

can assign linear molecules, tetrahedral molecules, and octahedral molecules to their

groups without use of the decision tree. To use the decision tree, one starts at the top

of the diagram with knowledge of the equilibrium nuclear conformation. One looks at

the questions at each branching point and answers yes or no to whether a particular

symmetry element is present. After each answer one proceeds along the appropriate

branch to the next question. There are two places where there is a choice between three

alternatives, which must be considered from left to right. The use of this decision tree

is illustrated in the following example:

EXAMPLE21.17

Assign the benzene molecule to a point group.

Solution

The molecule in its equilibrium conformation is in the shape of a regular hexagon of 6

carbon atoms, with a hydrogen atom bonded to each carbon atom, forming a second regular