Chemistry Reference
In-Depth Information
Fig. 4.1
Hydrogen SALCs in
ammonia. The size of the
circles is proportional to the
eigenfunction coefficients.
|
ψ
0
transforms as the totally
symmetric irrep
A
1
;
|
ψ
x
and
|
ψ
y
are components of the
degenerate
E
representation
irreducible blocks are not be related. We will return to this in much more detail
in Sect.
5.2
. The algebraic treatment also provides an insight into the meaning of
degeneracy. The two components of the
E
irrep are locked in the same function
space because it is not possible to diagonalize the representation matrices for both
generators simultaneously. If two operators commute, it is always possible to find
solutions that are simultaneous eigenfunctions of both. However, the two generators
of
C
3
v
do not commute as this group is not abelian. The fact that the correspond-
ing representation matrices also do not commute explains why it is impossible to
block-diagonalize the
E
irrep.
4.2 Character Theorems
When examining a function space from a symmetry point of view, we note that there
are two basic questions to be asked:
1. What are the symmetry ingredients of the function space; in other words, which
irreps describe the symmetry of this space?
2. What do the corresponding SALCs look like?
The present section on characters deals with the first question and provides an ele-
gant description of the symmetries of function spaces. In the subsequent sections,
matrix theorems are used for the construction of projection operators that will carry
out the job of obtaining the suitable SALCs. The intuitive algebraic approach that
we have demonstrated in the previous section has been formalized by Schur, Frobe-
nius,
1
and others into a fully fledged character theory, which reveals which irreps
1
The papers by Schur and Frobenius have been edited as C. Frobenius, The Collected Works of
Frobenius (1849-1917), J.-P. Serre (ed.), Springer, Berlin (1968), 3 vols.; I. Schur, Gesammelte
Abhandlungen, A. Brauer and H. Rohrbach (eds.) Springer, Berlin (1973), 3 vols.
