70
4 Representations
Table 4.3 Subduction of T 1u in O h ↓ D 3d
O h
ˆ
E
8 ˆ
C 3
6 ˆ
C 2
6 ˆ
C 4
3 ˆ
C 2
ˆ
ı
6 ˆ
S 4
8 ˆ
S 6
3 ˆ
σ h
6 ˆ
σ d
T 1u
30
−11
−1
−3
−1011
D 3d
ˆ
E
2 ˆ
C 3
3 ˆ
C 2
ˆ
ı
2 ˆ
S 6
3 ˆ
σ d
T 1u
30
−1
−301
A 2u
11
−1
−1
−11
E u
2
−10
−210
polar symmetry of the applied field, as discussed in Sect. 3.9. In chemistry a common approach to external symmetry breaking is to substitute one or more atoms or
atomic groups by homologues, or to interchange sites, which may give rise to different stereo-isomers. Internal symmetry breaking is more subtle and may arise as
a consequence of the Jahn–Teller effect. In this case the presence of a degenerate
electronic state in the high-symmetry conformation of the molecule may provoke a
spontaneous geometric distortion of the nuclear frame, leading to a lower symmetry
in which the degeneracy is removed. This effect involves the coupling of representations and will be discussed in Sect. 6.6. Our concern here is what will happen to
the irreps of G when the symmetry is reduced to H . This can easily be decided on
the basis of the character theorem. We simply have to determine the character of
the representation in the subgroup. The procedure consists of three steps. One first
identifies the correspondence between the elements of G and the elements of H .
Then the characters of the irrep in G are transferred to the characters for the corresponding operations in the subgroup. Third, the character string in the subgroup is
reduced according to the standard procedure of the character theorem. Hence, let Γ
denote an irrep of the parent group, and γ an irrep of the subgroup. The number of
times that this subgroup representation occurs in the subduction G ↓ H is given by
c γ (Γ G ↓ H)=
1
|H |
h∈H
¯
χ
γ (h)χ
Γ (h)
(4.64)
In Table 4.3 we follow as an example the fate of the octahedral T 1u irrep for the
subduction O h ↓ D 3d . When the subduction is performed, the anchoring of the
correspondences in the first step of the procedure is very important. For instance,
the octahedron has two conjugacy classes of ˆ
C 2 axes. The 6 ˆ
C 2 class collects the
twofold axes which bisect the Cartesian directions, while the 3 ˆ
C 2 class is made up
of the ˆ
C 2
4 axes along the Cartesian directions. In the case of subduction to D 3d ,the
three axes perpendicular to the trigonal direction belong to the 6 ˆ
C 2 class. As the
table indicates, in D 3d the threefold-degenerate representation becomes reducible
by splitting into two trigonal irreps:
O h ↓ D 3d : T 1u → A 2u + E u
(4.65)
In some cases a subgroup can be reached via two different symmetry breakings. An
example is the subduction O h ↓ D 2h . Here, two pathways for symmetry breaking
4 Representations
Table 4.3 Subduction of T 1u in O h ↓ D 3d
O h
ˆ
E
8 ˆ
C 3
6 ˆ
C 2
6 ˆ
C 4
3 ˆ
C 2
ˆ
ı
6 ˆ
S 4
8 ˆ
S 6
3 ˆ
σ h
6 ˆ
σ d
T 1u
30
−11
−1
−3
−1011
D 3d
ˆ
E
2 ˆ
C 3
3 ˆ
C 2
ˆ
ı
2 ˆ
S 6
3 ˆ
σ d
T 1u
30
−1
−301
A 2u
11
−1
−1
−11
E u
2
−10
−210
polar symmetry of the applied field, as discussed in Sect. 3.9. In chemistry a common approach to external symmetry breaking is to substitute one or more atoms or
atomic groups by homologues, or to interchange sites, which may give rise to different stereo-isomers. Internal symmetry breaking is more subtle and may arise as
a consequence of the Jahn–Teller effect. In this case the presence of a degenerate
electronic state in the high-symmetry conformation of the molecule may provoke a
spontaneous geometric distortion of the nuclear frame, leading to a lower symmetry
in which the degeneracy is removed. This effect involves the coupling of representations and will be discussed in Sect. 6.6. Our concern here is what will happen to
the irreps of G when the symmetry is reduced to H . This can easily be decided on
the basis of the character theorem. We simply have to determine the character of
the representation in the subgroup. The procedure consists of three steps. One first
identifies the correspondence between the elements of G and the elements of H .
Then the characters of the irrep in G are transferred to the characters for the corresponding operations in the subgroup. Third, the character string in the subgroup is
reduced according to the standard procedure of the character theorem. Hence, let Γ
denote an irrep of the parent group, and γ an irrep of the subgroup. The number of
times that this subgroup representation occurs in the subduction G ↓ H is given by
c γ (Γ G ↓ H)=
1
|H |
h∈H
¯
χ
γ (h)χ
Γ (h)
(4.64)
In Table 4.3 we follow as an example the fate of the octahedral T 1u irrep for the
subduction O h ↓ D 3d . When the subduction is performed, the anchoring of the
correspondences in the first step of the procedure is very important. For instance,
the octahedron has two conjugacy classes of ˆ
C 2 axes. The 6 ˆ
C 2 class collects the
twofold axes which bisect the Cartesian directions, while the 3 ˆ
C 2 class is made up
of the ˆ
C 2
4 axes along the Cartesian directions. In the case of subduction to D 3d ,the
three axes perpendicular to the trigonal direction belong to the 6 ˆ
C 2 class. As the
table indicates, in D 3d the threefold-degenerate representation becomes reducible
by splitting into two trigonal irreps:
O h ↓ D 3d : T 1u → A 2u + E u
(4.65)
In some cases a subgroup can be reached via two different symmetry breakings. An
example is the subduction O h ↓ D 2h . Here, two pathways for symmetry breaking