122
6 Interactions
ΓγΓ b γ b |Γ a γ a =
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)
Γγ
′ Γ b γ
′
b |Γ a γ
′
a
(6.28)
These equations form a system of homogeneous linear equations from which the
coupling coefficients can be obtained. At present, we shall use this result only in
the simplified case where all components have been chosen to be real, so that Eqs.
(6.27) and (6.28) form the same system of equations. 1 From this it follows that the
corresponding coupling coefficients will be proportional to each other, independent
of the components; hence:
Γ a γ a Γ b γ b |Γγ=xΓγΓ b γ b |Γ a γ a
(6.29)
The proportionality constant can be determined by summing the square of the coefficients over all components and using the normalization result from Eq. (6.15).
γ a γ b γ
Γ a γ a Γ b γ b |Γγ
2 = x
2
γ a γ b γ
ΓγΓ b γ b |Γ a γ a
2
dim(Γ ) = x
2 dim(Γ a )
(6.30)
The permutation of irreps between bra and ket in the CG-coefficients thus requires
a uniform dimensional renormalization:
dim(Γ )
−1/2 Γ a γ a Γ b γ b |Γγ=±
dim(Γ a )
−1/2 ΓγΓ b γ b |Γ a γ a
(6.31)
The renormalization leaves a phase factor undetermined. This phase factor is the
same for the entire coupling table, and thus can be chosen in arbitrarity. As an
example, in the group O (see Appendix F), the coefficients T 2 ξT 1 x|Eθ and
EθT 1 x|T 2 ξ are related as follows:
1
√
2
T 2 ξT 1 x|Eθ=
1
√
3
EθT 1 x|T 2 ξ =−
1
2
(6.32)
Here the phase was chosen to be +1.
6.4 Product Symmetrization and the Pauli Exchange-Symmetry
In principle, the T 1g and T 2g coupled two-electron states, which we obtained in
Table 6.2 of the previous section, could apply to the case of the (t 2g ) 1 (e g ) 1 excited
states of a d 2 transition-metal ion in an octahedral ligand field, which splits the d
1 The general case with complex irreps is exemplified for the coupling of spin representations in
Sect. 7.4.
6 Interactions
ΓγΓ b γ b |Γ a γ a =
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)
Γγ
′ Γ b γ
′
b |Γ a γ
′
a
(6.28)
These equations form a system of homogeneous linear equations from which the
coupling coefficients can be obtained. At present, we shall use this result only in
the simplified case where all components have been chosen to be real, so that Eqs.
(6.27) and (6.28) form the same system of equations. 1 From this it follows that the
corresponding coupling coefficients will be proportional to each other, independent
of the components; hence:
Γ a γ a Γ b γ b |Γγ=xΓγΓ b γ b |Γ a γ a
(6.29)
The proportionality constant can be determined by summing the square of the coefficients over all components and using the normalization result from Eq. (6.15).
γ a γ b γ
Γ a γ a Γ b γ b |Γγ
2 = x
2
γ a γ b γ
ΓγΓ b γ b |Γ a γ a
2
dim(Γ ) = x
2 dim(Γ a )
(6.30)
The permutation of irreps between bra and ket in the CG-coefficients thus requires
a uniform dimensional renormalization:
dim(Γ )
−1/2 Γ a γ a Γ b γ b |Γγ=±
dim(Γ a )
−1/2 ΓγΓ b γ b |Γ a γ a
(6.31)
The renormalization leaves a phase factor undetermined. This phase factor is the
same for the entire coupling table, and thus can be chosen in arbitrarity. As an
example, in the group O (see Appendix F), the coefficients T 2 ξT 1 x|Eθ and
EθT 1 x|T 2 ξ are related as follows:
1
√
2
T 2 ξT 1 x|Eθ=
1
√
3
EθT 1 x|T 2 ξ =−
1
2
(6.32)
Here the phase was chosen to be +1.
6.4 Product Symmetrization and the Pauli Exchange-Symmetry
In principle, the T 1g and T 2g coupled two-electron states, which we obtained in
Table 6.2 of the previous section, could apply to the case of the (t 2g ) 1 (e g ) 1 excited
states of a d 2 transition-metal ion in an octahedral ligand field, which splits the d
1 The general case with complex irreps is exemplified for the coupling of spin representations in
Sect. 7.4.