68
4 Representations
in ammonia. One has:
ˆ
P 11 |1s A =
1
3
2|1s A −|1s B −|1s C
=
√
2
√
3
|ψ x
ˆ
P 21 |1s A =
√
2
√
3
|ψ y
ˆ
P 12 |1s A =0
ˆ
P 22 |1s A =0
(4.57)
Note that the bracket ψ y |1s A vanishes because |1s A does not occur in the |ψ y
target, and this gives rise to the zeros in Eq. (4.57). If one wants to avoid the cumbersome construction of the irreducible representation matrices, one can construct
trace projectors by putting k = l and summing over all k:
k
ˆ
P
Γ
kk =
dim(Γ )
|G|
k
R
¯
D
Γ
kk (R) ˆ
R
=
dim(Γ )
|G|
R
¯
χ
Γ (R) ˆ
R
(4.58)
In this case, only the character tables are needed in order to construct such projectors. They will certainly destroy all parts of the function space that do not belong
to the irrep Γ , but, on the other hand, one loses the additional information in the
subrepresentation. As we will see in the subsequent chapters, these little auxiliary
indices are nonetheless valuable. A further remarkable property of a projector is that
if it is applied twice with inverted kl indices, one again obtains a projector:
ˆ
P
Γ
lk
ˆ
P
Γ
kl =
dim(Γ ) 2
|G| 2
RS
¯
D
Γ
lk (R) ¯
D
Γ
kl (S) ˆ
R ˆ
S
=
dim(Γ ) 2
|G| 2
RT
¯
D
Γ
lk (R) ¯
D
Γ
kl
R
−1 T
ˆ
T
=
dim(Γ ) 2
|G| 2
T
m
R
¯
D
Γ
lk (R) ¯
D
Γ
km
R
−1 ¯
D
Γ
ml (T ) ˆ
T
=
dim(Γ ) 2
|G| 2
T
m
R
¯
D
Γ
lk (R)D
Γ
mk (R)
¯
D
Γ
ml (T ) ˆ
T
=
dim(Γ )
|G|
T
m
δ m,l ¯
D
Γ
ml (T )
ˆ
T
= ˆ
P
Γ
ll
(4.59)