Physical Chemistry , 1st ed.

a.H 2 S:C2v, just like H 2 O.

b.SF 6 :Oh. This molecule has the shape of an octahedron.

c.C 2 H 2 :Dh, since it is linear and symmetric (if it had no center of inver-

sion, it would be Cv).

d.C 6 H 6 :D6h.

e.NO 3 :D3h.

The applicability of symmetry to molecules is deeper than just the shape of

the molecule. Mathematical equations also have symmetry properties. We have

already discussed the concept of odd and even functions. This is a symmetry

property. An even function implies that a plane of symmetry exists, typically a

plane that intersects the y-axis. You can verify this by looking at plots of co-

sine, an even function, and sine, an odd function.

As mathematical functions, quantum-mechanical wavefunctions can also

have certain symmetry properties. But what symmetry properties does a wave-

function have? Since a wavefunction determines the distribution of electron

probability in a molecule, and that distribution of electrons ultimately gives a

molecule its shape, we conclude that the wavefunction of a molecule must pos-

sess the same symmetry elements as the molecule itself. Thus, if the symmetry

elements of a molecule are identified, then the wavefunctions of the molecule

should have the same symmetry elements, and belong to the same point group,

as the molecule. It is this idea that makes symmetry a valuable tool in quan-

tum mechanics.

Example 13.4

a.What is the point group of the wavefunction for the 1sorbital of H?

b.What is the point group of the wavefunction for the 2pzorbital of H? (See

Figure 13.16.)

Solution

a.The 1swavefunction is spherically symmetric, having E,i, and an infinite

number ofC and S and symmetry elements. (The subscript on the C

and Smeans that the rotation can be any angle and still be a symmetry op-

eration.) It would belong to the special point group Rh(3).

b.The 2pzwavefunction is not spherically symmetric by itself, but it does

have the symmetry elements E, infinite v’s, and C. Because of the differ-

ence in sign of the wavefunction on either side of the nodal plane, it does not

have a center of inversion (despite appearances). Therefore it belongs to the

Cvpoint group. (This does not imply that the hydrogen atom itself has Cv

symmetry; only this particular 2pzatomic orbital.)

Example 13.5

Indicate what symmetries the wavefunctions of the following molecules

must have.

a.Wa t e r, H 2 O

b.Benzene, C 6 H 6

c.Allene, CH 2 CCH 2

d.Bromochlorofluoromethane, CHBrClF

13.4 Molecules and Symmetry 429

(C 6 H 6 )

(Planar; all N–O

bonds equivalent)

S

F (all S–F bonds

equivalent)

F

FF

FF

S

HH

HHC C

N

OO

O

(a) planar (like H 2 O)

(b) octahedral

(c) linear

(d) planar

(e) planar

Figure 13.15 What are the point groups of
these five molecules? See Example 13.3.

(–)

(+)

z

y

x

Figure 13.16 A 2pzorbital of H. See Example
13.4b.