Wavefunctions can be labeled with appropriate irreducible representation
labels. Therefore, the wavefunction of the bonding molecular orbital for H 2
would be labeled A1g(that is,A1g). Another phrase that means “irreducible
representations” is symmetry species.We say that the symmetry species of this
wavefunction is A1g.
As we saw briefly in the previous chapter, perhaps the simplest way to rep-
resent a molecular orbital is to base it on the atomic orbitals of the atoms in-
volved in the bonding, generally as a linear combination of atomic orbitals
(LCAO-MO). Because a wavefunction has symmetry restraints imposed on it,
the thought of taking a linear combination of atomic orbitals to represent a
molecular orbital becomes a little more tricky, now that linear combination
must have the correct symmetry. Such a requirement suggests that we should
not use just any linear combination of atomic orbitals, although we can. It
would be better to use some symmetry-adapted linear combinations(SALCs) of
atomic orbitals, in order to take advantage of group theory and symmetry con-
siderations. Before we construct SALCs, we must introduce one powerful tool
of group theory.13.7 The Great Orthogonality Theorem
We showed above that individual symmetry species are orthogonal to each
other. This is a consequence of a more general statement of group theory called
the great orthogonality theorem(GOT, sometimes called the grand orthogonal-
ity theorem). The GOT is a general relationship between allof the matrix ele-
ments of a representation of a symmetry operation (like the matrices in equa-
tion 13.2). Here, we will focus on the application of the GOT to the characters
of the irreducible representations, which will be much simpler than consider-
ing all of the matrix elements.
Since wavefunctions are being combined in linear combinations, one can
also take linear combinations of the irreducible representations and their char-
acters. However, it is more common to be able to determine the set of charac-
ters that represents the entire linear combination, instead of its constituent
parts. This set of characters is almost always a reducible representation. The
question is, what is the linear combination in terms of the irreducible repre-
sentations? How many A 1 ’s in the combination, how many B 1 ’s, how many E’s?
We can apply the great orthogonality theorem to determine the specific con-
tent (that is, the number of each irreducible representation) of a set of char-
acters that represent a linear combination. We will use the capital Greek letter
gamma,, to represent any one of the irreducible representations of a point
group. The number of contributions any particular has to a reducible repre-
sentation is given by the following formula:a
h1
(^)
all classes
Nlinear combo (13.6)
ofa
where ais the number of times the irreducible representation appears in
the linear combination,his the order of the group,Nis the number of oper-
ations in each class,is the character of the class of that irreducible repre-
sentation (from the character table), and linear combois the character of the
class for the linear combination. Note that this is similar to the expression in
equation 13.5. Example 13.9 shows how to apply this equation.
438 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics