Physical Chemistry , 1st ed.

Solution

Because wavefunctions for molecules must have the same symmetry as the

molecule, one must identify the molecule’s symmetries. Figure 13.14 can be

used to determine the following point groups of the molecules and, therefore,

their wavefunctions.

a.H 2 O:C2v

b.C 6 H 6 :D6h

c.Allene:D2d

d.CHBrClF:C 1

We will return to the application of symmetry to wavefunctions in a later

section. Symmetry in molecules has several immediate consequences. For ex-

ample, the dipole moment of a molecule depends in part on how the atoms in

the molecule are arranged. It can be shown that any molecule whose structure

has a point group symmetry ofCs,Cn,or Cnv, with n 1, is polar, and mol-

ecules that do not have such symmetry are nonpolar. Further, it can also be

shown that any molecule maybe chiral if it does notcontain an Sn(n1)

symmetry element. However, chirality is not guaranteed if the molecule does

not possess an improper axis of symmetry (for example, H 2 O does not have

an improper rotation axis but is not chiral). Chirality is an important issue in

organic chemistry and is the basis of stereochemistry.

13.5 Character Tables

Now that the importance of symmetry in wavefunctions has been established,

some of the utility of symmetry can be introduced. A water molecule, H 2 O, is

in the position indicated by Figure 13.17. H 2 O has all of the symmetry ele-

ments described by the C2vpoint group, so it has E,C 2 , and two v’s, which

we will designate (xz) and (yz). Remembering from above that each sym-

metry operation can be defined as a matrix, we can construct matrices to de-

fine the symmetry operations for H 2 O. However,each atomin the molecule has

430 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

yO

xO

zO

O

yH1

xH1

zH1

H

(yz)

(xz)

C 2

yH2

xH2

zH2

H

Figure 13.17 The H 2 O molecule, the Cartesian degrees of freedom of the atoms, and the sym-

metry operations of the C2vpoint group.