4.2 Character Theorems
61
In this case, the only contribution to the trace comes from the distortion of A, which
is mapped onto minus itself; hence, χ(σ 1 ) =−1. The resulting character string also
has norm 12, but in this case it decomposes into A 2 + E. As shown in Fig. 4.2,the
A 2 component is the sum of the three distortions, which corresponds to a bodily
rotation of the molecule around its vertical axis. The E components are given by
Q E x =
1
√
2
((φ C − φ B )
Q E y =
1
√
6
(2φ A − φ B − φ C )
(4.39)
Perhaps the most surprising result is that the symmetry eigenfunctions of the C 3v
group can be only of three different types: A 1 ,A 2 , and E. In fact, this is another
result derived by Schur.
Theorem 5 The number of irreps in a group is equal to the number of conjugacy
classes.
The fact that there are only a few canonical ways of representing a symmetry
group can be rationalized in a general way as follows. Consider a molecule with
symmetry group G and a function that is localized on an arbitrary point in the
molecule, i.e., a point which is lying neither on an axis of symmetry, nor in a reflection plane, nor in the inversion center or the center of a rotation–reflection axis.
We shall denote this function as |f E since it is stabilized by, and only by, the unit
element. Any other element would move this function to some other point that is a
copy of the original one. We write this as
ˆ
R|f E =|f R
(4.40)
In this way the entire group generates a set of |G| functions, which all are different.
Indeed, suppose that, for ˆ
R = ˆ
S, the two corresponding functions are the same; then
the product operation ˆ
S −1 ˆ
R maps |f E onto itself. This contradicts the assumption that |f E is stabilized only by the unit element. Furthermore, the closure of the
group also guarantees that this set of functions forms an invariant function space.
This function space transforms according to a reducible representation, which is
called the regular representation, Γ reg . This representation describes the most general function basis that one can consider since it is based on functions that have no
symmetry whatsoever. Now let us determine the symmetry ingredients of this space
using the standard character procedure. The character of the regular representation
is equal to |G| for the unit element and zero for all other elements since none of the
functions is stabilized:
χ
Γ reg (R) =|G|δ ER
(4.41)
Inserting this result into the expression for the multiplicity coefficients yields
c k =
1
|G|
χ
Γ k |χ
Γ reg
=¯ χ
Γ k (E) = dim(Γ k )
(4.42)
61
In this case, the only contribution to the trace comes from the distortion of A, which
is mapped onto minus itself; hence, χ(σ 1 ) =−1. The resulting character string also
has norm 12, but in this case it decomposes into A 2 + E. As shown in Fig. 4.2,the
A 2 component is the sum of the three distortions, which corresponds to a bodily
rotation of the molecule around its vertical axis. The E components are given by
Q E x =
1
√
2
((φ C − φ B )
Q E y =
1
√
6
(2φ A − φ B − φ C )
(4.39)
Perhaps the most surprising result is that the symmetry eigenfunctions of the C 3v
group can be only of three different types: A 1 ,A 2 , and E. In fact, this is another
result derived by Schur.
Theorem 5 The number of irreps in a group is equal to the number of conjugacy
classes.
The fact that there are only a few canonical ways of representing a symmetry
group can be rationalized in a general way as follows. Consider a molecule with
symmetry group G and a function that is localized on an arbitrary point in the
molecule, i.e., a point which is lying neither on an axis of symmetry, nor in a reflection plane, nor in the inversion center or the center of a rotation–reflection axis.
We shall denote this function as |f E since it is stabilized by, and only by, the unit
element. Any other element would move this function to some other point that is a
copy of the original one. We write this as
ˆ
R|f E =|f R
(4.40)
In this way the entire group generates a set of |G| functions, which all are different.
Indeed, suppose that, for ˆ
R = ˆ
S, the two corresponding functions are the same; then
the product operation ˆ
S −1 ˆ
R maps |f E onto itself. This contradicts the assumption that |f E is stabilized only by the unit element. Furthermore, the closure of the
group also guarantees that this set of functions forms an invariant function space.
This function space transforms according to a reducible representation, which is
called the regular representation, Γ reg . This representation describes the most general function basis that one can consider since it is based on functions that have no
symmetry whatsoever. Now let us determine the symmetry ingredients of this space
using the standard character procedure. The character of the regular representation
is equal to |G| for the unit element and zero for all other elements since none of the
functions is stabilized:
χ
Γ reg (R) =|G|δ ER
(4.41)
Inserting this result into the expression for the multiplicity coefficients yields
c k =
1
|G|
χ
Γ k |χ
Γ reg
=¯ χ
Γ k (E) = dim(Γ k )
(4.42)