66
4 Representations
indeed generates the y component of Eq. (4.39):
ˆ
P
E
21 Q E x =
1
3
√
3
2
ˆ
C 3 − ˆ
C
2
3 −ˆ σ 2 +ˆ σ 3
1
√
2
((φ C − φ B )
=
1
2
√
6
((φ A − φ C ) − ((φ B − φ A )
− (−φ A + φ B ) + (−φ C + φ A )
=
1
√
6
(2φ A − φ B − φ C )
= Q E y
(4.50)
For a given irrep, the number of projectors that can be constructed is equal to the
number of all possible {k,l} combinations, which equals dim(Ω ′ ) 2 . By varying the
row index, k, one can obtain all the components of the invariance space of a given
irrep. This is demonstrated by acting on the operators with an element ˆ
S:
ˆ
S ˆ
P
Ω ′
kl =
dim(Ω ′ )
|G|
R∈G
j
¯
D
Ω ′
kl (R) ˆ
S ˆ
R
=
dim(Ω ′ )
|G|
T ∈G
¯
D
Ω ′
kl
S
−1 T
ˆ
T
=
dim(Ω ′ )
|G|
T ∈G
m
¯
D
Ω ′
km
S
−1 ¯
D
Ω ′
ml (T )
ˆ
T
=
dim(Ω ′ )
|G|
m
D
Ω ′
mk (S)
T ∈G
¯
D
Ω ′
ml (T ) ˆ
T
=
m
D
Ω ′
mk (S) ˆ
P
Ω ′
ml
(4.51)
In this derivation we have used the substitution ˆ
S ˆ
R = ˆ
T . The result shows that the
set of projectors with fixed index l forms a complete basis set for the Ω ′ irrep. On
the other hand, by changing the column index l we have access to different sets
of SALCs with the same symmetry. This applies only when the function space has
multiplicities, c Γ , greater than one. This can be illustrated for the {|f R } function
space transforming as the regular representation that has the maximal degree of
freedom. Two projectors with the same k index, but different l indices, will project
out two functions that are linearly independent, as the following overlap calculation
shows:
ˆ
P
Ω
kl f E | ˆ
P
Ω
kl ′ f E
=
dim(Ω) 2
|G| 2
R
¯
D
Ω
kl (R) ˆ
Rf E |
S
¯
D
Ω
kl (S) ˆ
Sf E
=
dim(Ω) 2
|G| 2
R,S
D
Ω
kl (R) ¯
D
Ω
kl (S)f R |f S