72
4 Representations
Table 4.5 Subduction of T 2g
under O h ↓ D ′
2h
O h
ˆ
E
6 ˆ
C 2
3 ˆ
C 2
ˆ
ı
3 ˆ
σ h
6 ˆ
σ d
↓↓ ց
↓↓
↓↓ 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
311
−13
−111
A g
1111
1111
B 2g
1
−11
−11
−11
−1
B 3g
11
−1
−11
−1
−11
irreps of the parent group. The outcome of the induction is determined by the reciprocity theorem due to Frobenius.
Theorem 7 The number of times that a given irrep Γ of a parent group G occurs
in the induction H ↑ G of a subgroup irrep γ is equal to the number of times that γ
is present in the subduction G ↓ H of that irrep Γ .
We shall present the proof here since it introduces the important concept of the
ground representation [2, 3]. This concept is especially useful when considering
a polyhedral molecular cluster or complex consisting of several equivalent sites.
Typically, these sites could be the atoms in a network covering a hollow cage, or
ligands in a metal complex. In the case of ammonia, the sites are simply the three
hydrogen atoms. Usually, the site group is of type C nv . We choose site a as the
starting site, which is stabilized by the subgroup H A . The group G is expanded in
cosets of this subgroup, with coset representatives ˆ
g κ :
G =
κ
ˆ
g κ H A
(4.67)
As we have seen in the previous chapter, the coset representatives each address a
copy of site a, which we shall label as κ. The site group that stabilizes this site
is isomorphic to H A and is denoted by H κ . We thus have the following mappings:
κ ˆ
g κ a
H κ =ˆ g κ H A ˆ
g
−1
κ
(4.68)
The mapping of the stabilizer, H A → H κ , is recognized as a similarity transformation of the whole subgroup. Two different sites can share the same site group. As
an example, in a square pyramidal complex, with parent group C 4v and site groups
C s , two ligands, trans to each other, have the same site group. With reference to the
square in Fig. 4.3a, the cosets may be generated as follows:
C 4v =
3
k=0
ˆ
C
k
4 { ˆ
E, ˆ
σ 1 }={ ˆ
E, ˆ
σ 1 }+{ ˆ
C 4 , ˆ
σ 4 }+
ˆ
C
2
4 , ˆ
σ 2
+
ˆ
C
3
4 , ˆ
σ 3
(4.69)
4 Representations
Table 4.5 Subduction of T 2g
under O h ↓ D ′
2h
O h
ˆ
E
6 ˆ
C 2
3 ˆ
C 2
ˆ
ı
3 ˆ
σ h
6 ˆ
σ d
↓↓ ց
↓↓
↓↓ 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
311
−13
−111
A g
1111
1111
B 2g
1
−11
−11
−11
−1
B 3g
11
−1
−11
−1
−11
irreps of the parent group. The outcome of the induction is determined by the reciprocity theorem due to Frobenius.
Theorem 7 The number of times that a given irrep Γ of a parent group G occurs
in the induction H ↑ G of a subgroup irrep γ is equal to the number of times that γ
is present in the subduction G ↓ H of that irrep Γ .
We shall present the proof here since it introduces the important concept of the
ground representation [2, 3]. This concept is especially useful when considering
a polyhedral molecular cluster or complex consisting of several equivalent sites.
Typically, these sites could be the atoms in a network covering a hollow cage, or
ligands in a metal complex. In the case of ammonia, the sites are simply the three
hydrogen atoms. Usually, the site group is of type C nv . We choose site a as the
starting site, which is stabilized by the subgroup H A . The group G is expanded in
cosets of this subgroup, with coset representatives ˆ
g κ :
G =
κ
ˆ
g κ H A
(4.67)
As we have seen in the previous chapter, the coset representatives each address a
copy of site a, which we shall label as κ. The site group that stabilizes this site
is isomorphic to H A and is denoted by H κ . We thus have the following mappings:
κ ˆ
g κ a
H κ =ˆ g κ H A ˆ
g
−1
κ
(4.68)
The mapping of the stabilizer, H A → H κ , is recognized as a similarity transformation of the whole subgroup. Two different sites can share the same site group. As
an example, in a square pyramidal complex, with parent group C 4v and site groups
C s , two ligands, trans to each other, have the same site group. With reference to the
square in Fig. 4.3a, the cosets may be generated as follows:
C 4v =
3
k=0
ˆ
C
k
4 { ˆ
E, ˆ
σ 1 }={ ˆ
E, ˆ
σ 1 }+{ ˆ
C 4 , ˆ
σ 4 }+
ˆ
C
2
4 , ˆ
σ 2
+
ˆ
C
3
4 , ˆ
σ 3
(4.69)