120
6 Interactions
Table 6.3 Direct square H g × H g in I h symmetry
I h
ˆ
E
12 ˆ
C 5
12 ˆ
C 2
5
20 ˆ
C 3
15 ˆ
C 2
ˆ
i
12 ˆ
S 10
12 ˆ
S
3
10
20 ˆ
S 6
15 ˆ
σ
χ Hg
500 −115
0
0 −11
χ Hg 2
25
0
0
1
1
25
0
0
1
1
χ Hg (R 2 )
500 −155
0
0 −15
χ [Hg ] 2
15
0
0
0
3
15
0
0
0
3
χ {Hg } 2
10
0
0
1
−21 0
0
0
1
−2
The trace for the symmetrized product is then found by subtracting the trace in Eq.
(6.21) from the total trace for the direct product.
χ
[Γ a ] 2
(R) = χ
Γ a
2
(R) − χ
{Γ a } 2
(R)
=
1
2
χ
Γ a (R)
2 + χ
Γ a
R
2
(6.22)
In Table 6.3 these quantities are given for the direct product H g × H g in icosahedral
symmetry. The product resolution is as follows:
H g × H g =[A g + G g + 2H g ]+{T 1g + T 2g + G g }
(6.23)
Note that this product contains one totally-symmetric irrep, notably in the symmetrized part. In general, for irreps with real characters the totally-symmetric irrep, Γ 0 , appears in a direct square only once. This can easily be derived from Eq.
(6.8). When Γ a is an irrep with real characters, one has:
χ
Γ 0 |χ
Γ a ×Γ b
=
R
¯
χ
Γ 0 (R)χ
Γ a (R)χ
Γ b (R)
=
R
χ
Γ a (R)χ
Γ b (R)
=|G|δ Γ a Γ b
(6.24)
In the case of irreps that can be represented by real transformation matrices,i t
is possible to show that this totally-symmetric irrep will belong to the symmetrized
part. In order to apply the character theorem to Eq. (6.22), the following intermediate
result is needed:
R
χ
Γ a
R
2
=
R
i
D
Γ a (R) × D
Γ a (R)
ii
=
R
i
j
D
Γ a
ij (R)D
Γ a
ji (R)
=
|G|
dim(Γ a )
i
j
δ ij =|G|
(6.25)
In order to arrive at this result we have made use of the GOT, on the assumption
that the D matrices are real. Combining the results of Eqs. (6.24) and (6.25) with