In other words, V
8 is decomposed into the above two R-invariant subspaces (see
Part III); one is the hydrogen-related subspace and the other is the carbon-related
subspace.
Correspondingly, a representation D comprising the above representation matrices should be reduced to a direct sum of irreducible representations D
(α)
. That is, we
should have
D ¼
X
α
q α D
α
ð Þ ,
where q α is a positive integer or zero. Here we are thinking of decomposition of (8, 8)
matrices such as (19.178) into submatrices. We estimate q α using (18.83). That is,
q α ¼
1
n
X
g
χ
α
ð Þ g
ð Þ
à χ g
ð Þ:
ð18:83Þ
With an irreducible representation A 1 of T d for instance, we have
q A 1 ¼
1
24
X
g
χ
A 1
ð Þ g
ð Þ
à χ g
ð Þ ¼
1
24
1 Á 8 þ 1 Á 8 Á 2 þ 1 Á 6 Á 4
ð
Þ ¼ 2:
As for A 2 , we have
q A 2 ¼
1
24
1 Á 8 þ 1 Á 8 Á 2 þ À1
ð ÞÁ6 Á 4
½
Š ¼ 0:
Regarding T 2 , we get
q T 2 ¼
1
24
3 Á 8 þ 1 Á 6 Á 4
ð
Þ ¼ 2:
For other irreducible representations of T d , we get q α ¼ 0. Consequently, we have
D ¼ 2D
A 1
ð Þ
þ 2D
T 2
ð Þ
:
ð19:183Þ
Evidently, both for the hydrogen-related representation D
(H ) and carbon-related
representation D
(C) , we have individually
Table 19.11 Characters for
individual symmetry
transformations of hydrogen
1s orbitals and carbon 2s and
2p orbitals in methane
T d
E
8C 3
3C 2
6S 4
6σ d
Γ
8
2
0
0
4
19.5 MO Calculations of Methane
781