Physical Chemistry , 1st ed.

arbitrarily numbered Greek capital letters gamma,n, is an older system that

is occasionally still seen in the literature.

13.6 Wavefunctions and Symmetry

How does symmetry apply to wavefunctions? First of all, consider the mole-

cule itself. A molecule has a shape that can be described by one of a limited

number of groups of symmetry operations. Such groups contain anywhere

from one symmetry operation (C 1 , which has E) to many (Oh, which has 48)

to infinite (Chand Dv, for example, have an infinite number ofvplanes of

symmetry).

Wavefunctions of molecules span the entire molecule. This is true even

though we tend to picture localized electrons in molecules, as in a covalent

bond between two atoms. However, strictly speaking, wavefunctions cover—

and so the electron exists over—the entire molecule. (In approximations, many

molecular orbitals may have a very large, almost-unity value for a single atomic

orbital in a linear combination of atomic orbitals, but the basic truth is that

all orbitals are molecular orbitals.) If molecules have a shape, then the molec-

ular orbitals must have the same shape. This demands that the molecular wave-

functions must have the same symmetry properties as the molecule.We h a v e

noted that there are a limited number of combinations of characters for the

symmetry operations within a point group. Wavefunctions of a molecule must

also have some behavior with respect to the symmetry operations of the point

group. The symmetry behavior of a wavefunction must correspond to one of

the irreducible representations of the point group. It is typical to label a wave-

function with the symbol for that irreducible representation, just as it is to la-

bel the wavefunction with its quantum numbers. Further, the character of the

Esymmetry operation for a wavefunction is the same as the degeneracy of

the wavefunction. A review of the character tables in Appendix 3 shows that a

molecule must have at least a C 3 axis in order to have doubly degenerate wave-

functions, and more than one highest-order axis in order to have triply de-

generate wavefunctions (although this condition does not guarantee a Tirre-

ducible representation).

Example 13.8

Figure 13.19 shows a diagram of the bondingand antibondingorbitals of H 2 .

Assuming that the Eis 1 for both, determine the irreducible representations

for each molecular orbital. Assume the highest-order axis is the z-axis.

Solution

The symmetry of H 2 is Dh. Upon operation of the diagram of the bond-

ing orbital with all symmetry operations, one finds that the character for each

operator is 1 (that is, the same orientation of the orbital is reproduced). This

means that the characters for all operations are 1 and so the wavefunction’s

label is A1g(or g). For the antibonding orbital, inversion through the cen-

ter yields a wavefunction that has the negative value of the original . A sign

change also occurs upon operation of the C 2 and S operations. (You should

satisfy yourself that this is the case, using Figure 13.19.) Therefore, the anti-

bonding wavefunction can be labeled with the A1u(or u) irreducible

representation.

13.6 Wavefunctions and Symmetry 437

antibonding

H H

(+)

(+) bonding

(–)

Figure 13.19 A nonrigorous but illustrative
representation of the bonding and antibonding 
orbitals of H 2 . See Example 13.8.