6.3 Symmetry Properties of the Coupling Coefficients
121
Eq. (6.22) then leads to the conclusion that the unique totally-symmetric irrep belongs to the symmetrized part of the direct square:
1
|G|
χ
Γ 0 |χ
[Γ a ] 2
=
1
2|G|
R
χ
Γ 0 (R)
χ
Γ a
2
(R) + χ
Γ a
R
2
= 1
(6.26)
For irreps with real characters, but transformation matrices that cannot be all real,
the unique totally-symmetric product appears in the antisymmetrized part. This is
the case for spin representations, which will be dealt with in Chap. 7. In summary,
as far as complex-conjugation properties are concerned, we have three kinds of irreps:
1. Irreps with real characters, and for which all D(R) transformation matrices can
be put in real form. In this case: Γ 0 ∈[Γ a ] 2 .
2. Irreps with real characters, but which cannot be represented by transformation
matrices that are all real. In this case Γ 0 ∈{Γ a } 2 .
3. Irreps with complex characters. In this case there is always a complex-conjugate
irrep, and Γ 0 ∈ Γ × ¯
Γ .
Equation (6.23) further exemplifies a case of product multiplicity. This is when
an irrep occurs more than once in the decomposition of a direct product. Both the H g
and G g irreps appear twice in the direct product H g × H g . In the point groups product multiplicity is quite rare. It occurs only in the icosahedral group for the products
G × H and H × H , as well for spin representations in cubic and icosahedral symmetries. Product multiplicity means that there are different coupling schemes for arriving at the product states. Each of these “channels” corresponds to a separate set of
CG coupling coefficients. There are several ways of obtaining linearly-independent
sets of coupling coefficients. For the separation of the two G g irreps in Eq. (6.23)
symmetrization of the product space is sufficient, since the symmetrized and antisymmetrized parts each contain one G g . This strategy does not work for the two H g
irreps, which are both the result of symmetrized coupling. In this case, more elaborate splitting schemes have been constructed, based inter alia on higher symmetries
[4, 5].
Last but not least, we should consider the relationship between coupling coefficients where irreps from bra and ket parts are interchanged. We shall limit the
discussion here to the simplified case in which all ingredients of the coupling are
taken to be real. A case with complex irreps will be treated in Chap. 7. Consider two
related couplings: Γ a × Γ b = Γ and Γ × Γ b = Γ a . The corresponding expansion
coefficients are scalar matrix elements and are thus invariant under the group action. By importing the group action inside the brackets, as we have frequently done
before, we obtain a set of equations in the CG-coefficients:
Γ a γ a Γ b γ b |Γγ=
γ ′
a γ ′
b γ
1
|G|
R
¯
D
Γ a
γ ′
a γ a
(R) ¯
D
Γ b
γ ′
b γ b
(R)D
Γ
γ ′ γ (R)
Γ a γ
′
a Γ b γ
′
b |Γγ
′
(6.27)
121
Eq. (6.22) then leads to the conclusion that the unique totally-symmetric irrep belongs to the symmetrized part of the direct square:
1
|G|
χ
Γ 0 |χ
[Γ a ] 2
=
1
2|G|
R
χ
Γ 0 (R)
χ
Γ a
2
(R) + χ
Γ a
R
2
= 1
(6.26)
For irreps with real characters, but transformation matrices that cannot be all real,
the unique totally-symmetric product appears in the antisymmetrized part. This is
the case for spin representations, which will be dealt with in Chap. 7. In summary,
as far as complex-conjugation properties are concerned, we have three kinds of irreps:
1. Irreps with real characters, and for which all D(R) transformation matrices can
be put in real form. In this case: Γ 0 ∈[Γ a ] 2 .
2. Irreps with real characters, but which cannot be represented by transformation
matrices that are all real. In this case Γ 0 ∈{Γ a } 2 .
3. Irreps with complex characters. In this case there is always a complex-conjugate
irrep, and Γ 0 ∈ Γ × ¯
Γ .
Equation (6.23) further exemplifies a case of product multiplicity. This is when
an irrep occurs more than once in the decomposition of a direct product. Both the H g
and G g irreps appear twice in the direct product H g × H g . In the point groups product multiplicity is quite rare. It occurs only in the icosahedral group for the products
G × H and H × H , as well for spin representations in cubic and icosahedral symmetries. Product multiplicity means that there are different coupling schemes for arriving at the product states. Each of these “channels” corresponds to a separate set of
CG coupling coefficients. There are several ways of obtaining linearly-independent
sets of coupling coefficients. For the separation of the two G g irreps in Eq. (6.23)
symmetrization of the product space is sufficient, since the symmetrized and antisymmetrized parts each contain one G g . This strategy does not work for the two H g
irreps, which are both the result of symmetrized coupling. In this case, more elaborate splitting schemes have been constructed, based inter alia on higher symmetries
[4, 5].
Last but not least, we should consider the relationship between coupling coefficients where irreps from bra and ket parts are interchanged. We shall limit the
discussion here to the simplified case in which all ingredients of the coupling are
taken to be real. A case with complex irreps will be treated in Chap. 7. Consider two
related couplings: Γ a × Γ b = Γ and Γ × Γ b = Γ a . The corresponding expansion
coefficients are scalar matrix elements and are thus invariant under the group action. By importing the group action inside the brackets, as we have frequently done
before, we obtain a set of equations in the CG-coefficients:
Γ a γ a Γ b γ b |Γγ=
γ ′
a γ ′
b γ
1
|G|
R
¯
D
Γ a
γ ′
a γ a
(R) ¯
D
Γ b
γ ′
b γ b
(R)D
Γ
γ ′ γ (R)
Γ a γ
′
a Γ b γ
′
b |Γγ
′
(6.27)