118
6 Interactions
Table 6.2 The coupling
coefficients for the direct
product E g × T 2g in O h
symmetry
E g × T 2g
T 1g
T 2g
xy
zξ
η
ζ
|θ|ξ −
√
3
2
00−
1
2
00
|θ|η
0
√
3
2
00−
1
2
0
|θ|ζ
00
00
0
1
|ǫ|ξ −
1
2
00
√
3
2
00
|ǫ|η
0
−
1
2
00
−
√
3
2
0
|ǫ|ζ
00
10
0
0
factorization of the coupling coefficients, but, express the important symmetry properties of the couplings at the level of the brackets. Two guidelines will thereby be
used: when dealing with coupling coefficients it is important to bear in mind that
the coupling is based on the formation of a product, as we have illustrated in the
preceding section, and, secondly, that we should treat the coupling coefficients as
far as possible as ordinary brackets.
A direct consequence of the latter viewpoint is that the rules for complex conjugation of brackets apply:
Γ a γ a Γ b γ b |Γγ==Γγ|Γ a γ a Γ b γ b
(6.14)
Being expansion coefficients of SALCs, the coupling coefficients also obey
two orthogonality rules. Column-wise orthonormality results from the orthonormal
properties of the coupled states.
γ a γ b
Γ
′ γ
′ |Γ a γ a Γ b γ b
Γ a γ a Γ b γ b |Γγ=δ ΓΓ ′ δ γγ ′
(6.15)
In addition, the scalar products along rows are orthonormal, because of the orthonormal properties of the basic kets. Note that the summation runs over the irreps of the
entire product space: Γ ∈ Γ a × Γ b .
Γγ
Γ a γ
′
a Γ b γ
′
b |Γγ
Γγ|Γ a γ a Γ b γ b =δ γ ′
a γ a δ γ ′
b γ b
(6.16)
The permutational properties of the CG-coefficients refer to interchange of the
bra and ket irreps. If Γ a and Γ b are not equivalent, their ordering will not affect the
symmetry of the coupled state, since the factors in the direct product commute:
Γ a × Γ b = Γ b × Γ a
(6.17)
We can therefore define the coupling coefficients in such a way that interchange of
the coupled irreps leaves the coefficient invariant:
Γ a = Γ b ::Γ a γ a Γ b γ b |Γγ≡≡Γ b γ b Γ a γ a |Γγ
(6.18)
6 Interactions
Table 6.2 The coupling
coefficients for the direct
product E g × T 2g in O h
symmetry
E g × T 2g
T 1g
T 2g
xy
zξ
η
ζ
|θ|ξ −
√
3
2
00−
1
2
00
|θ|η
0
√
3
2
00−
1
2
0
|θ|ζ
00
00
0
1
|ǫ|ξ −
1
2
00
√
3
2
00
|ǫ|η
0
−
1
2
00
−
√
3
2
0
|ǫ|ζ
00
10
0
0
factorization of the coupling coefficients, but, express the important symmetry properties of the couplings at the level of the brackets. Two guidelines will thereby be
used: when dealing with coupling coefficients it is important to bear in mind that
the coupling is based on the formation of a product, as we have illustrated in the
preceding section, and, secondly, that we should treat the coupling coefficients as
far as possible as ordinary brackets.
A direct consequence of the latter viewpoint is that the rules for complex conjugation of brackets apply:
Γ a γ a Γ b γ b |Γγ==Γγ|Γ a γ a Γ b γ b
(6.14)
Being expansion coefficients of SALCs, the coupling coefficients also obey
two orthogonality rules. Column-wise orthonormality results from the orthonormal
properties of the coupled states.
γ a γ b
Γ
′ γ
′ |Γ a γ a Γ b γ b
Γ a γ a Γ b γ b |Γγ=δ ΓΓ ′ δ γγ ′
(6.15)
In addition, the scalar products along rows are orthonormal, because of the orthonormal properties of the basic kets. Note that the summation runs over the irreps of the
entire product space: Γ ∈ Γ a × Γ b .
Γγ
Γ a γ
′
a Γ b γ
′
b |Γγ
Γγ|Γ a γ a Γ b γ b =δ γ ′
a γ a δ γ ′
b γ b
(6.16)
The permutational properties of the CG-coefficients refer to interchange of the
bra and ket irreps. If Γ a and Γ b are not equivalent, their ordering will not affect the
symmetry of the coupled state, since the factors in the direct product commute:
Γ a × Γ b = Γ b × Γ a
(6.17)
We can therefore define the coupling coefficients in such a way that interchange of
the coupled irreps leaves the coefficient invariant:
Γ a = Γ b ::Γ a γ a Γ b γ b |Γγ≡≡Γ b γ b Γ a γ a |Γγ
(6.18)