4.6 Subduction and Induction
71
Fig. 4.3 (a) generation of equivalent sites in a square starting from a,(b) substitutional symmetry lowering of O h ↓ D 2h in the MX 2 Y 2 Z 2 complex isomer, and (c) O h ↓ D ′
2h symmetry lowering
by bidentate ligands in trans-M(L − L) 2 X 2
Table 4.4 Subduction of T 2g
under O h ↓ D 2h
O h
ˆ
E
3 ˆ
C 2
ˆ
ı
3 ˆ
σ h
↓
ւ↓ց
↓
ւ↓ց
D 2h
ˆ
E
ˆ
C
x
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
ı
ˆ
σ xy
ˆ
σ xz
ˆ
σ yz
T 2g
3
−1
−1
−13
−1
−1
−1
B 1g
1
−1
−11
11
−1
−1
B 2g
1
−11
−11
−11
−1
B 3g
11
−1
−11
−1
−11
are present. The three ˆ
C 2 axes of the orthorhombic symmetry are either based on the
3 ˆ
C 2 class, { ˆ
C x
2 , ˆ
C
y
2 , ˆ
C
z
2 }, or on a mixture of the two classes, as in { ˆ
C
z
2 , ˆ
C
xy
2 , ˆ
C
¯
xy
2 }.
We shall denote the latter group as D ′
2h . Simple molecular examples of both are
shown in Fig. 4.3. Tables 4.4 and 4.5 present the splitting of the T 2g irrep over these
two subduction paths. The corresponding splitting schemes are as follows:
O h ↓ D 2h : T 2g → B 1g + B 2g + B 3g
O h ↓ D
′
2h : T 2g → A g + B 2g + B 3g
(4.66)
Subduction tables are available in Appendix D.
The opposite process to subduction is induction. Here, we start from an irrep in
a subgroup H . By coset expansion, this subgroup is put on an orbit inside a higher
symmetry group. This leads to an extension of the function space and generates