Physical Chemistry Third Edition

902 21 The Electronic Structure of Polyatomic Molecules

C`hD`hTd,Oh,Ci,C 1 ,Cs,Dnd,Dnh,Dn,Cnh,Cnv,Cn,Sn

Linear molecules Special symmetries Proper axis, Cn

n 5 maximum

Dnh,Dnd,Dn,Cnh,Cnv,Cn,Sn

Dnh,Dnd,Dn,Cnh,Cnv,Cn

Dnh,Dnd,Dn Cnh,Cnv,Cn

Cs,Ci,C 1

Yes

Yes

Yes No

Yes No

No No

No

Yes

Yes No

D`h,C`v

D`h

`C 2 C`

nC 2 Cn

C`v

Ci C 1

Cs Ci,C 1

i

S 2 n(alone or with i)

Dnh Dnd Dn Cnh Cnv Cn

Sn(n 5 even)

sh nsv sh nsv

TdOh

Figure 21.12 Flow Chart (Decision Tree) for the Assignment of a Molecule to a Point Group.To use this chart, start at the top and
follow the path that corresponds to the answer to each question.

hexagon in the same plane. Starting at the top of the figure, we eliminate the categories of

linear, tetrahedral or octahedral molecules, and move to the right side. There is a proper

C 6 axis so we answer “yes” and move to the left branch. Although there is anS 6 axis, is

does not occur alone or with inversion so we answer “no” and move to the left branch.

There are sixC 2 axes perpendicular to theC 6 axis so we answer “yes” and move to the

left branch. Next we have three choices, which are to be examined from left to right. The

molecule does have aσhplane of symmetry, so we arrive at the point groupD 6 h. If there

had not been aσhplane of symmetry, we would have asked if there were sixσvplanes. If

so, the group would beD 6 d. The benzene molecule does have sixσvplanes, but you must

ask about theσhplane before asking about theσvplanes, and the presence of theσhplane

means that it belongs to theD 6 hgroup.

Exercise 21.20

Using the decision tree of Figure 21.12, assign the following molecules to point groups:

a.1,1-Dichloroethene

b.trans-1,2-Dichloroethene

c.cis-1,2-Dichloroethene

Some Elementary Applications of Group Theory

Once a molecule has been assigned to a point group we can draw some conclusions

about it. If a molecule has a dipole moment, any symmetry operation belonging to the