6.3 Symmetry Properties of the Coupling Coefficients
117
basis orbitals.
c
Γ
γ a γ b
=
Γ a γ a (1)Γ b γ b (2)|Γγ(1, 2)
≡≡Γ a γ a Γ b γ b |Γγ
(6.11)
This coefficient is known as a Clebsch–Gordan (CG) coupling coefficient and denoted by the 3Γ bracket Γ a γ a Γ b γ b |Γγ. It indicates how the orbital irreps Γ a and
Γ b have to be combined to yield a product ket that transforms as |Γγ.T h eC G -
coefficients can be determined by using projection operators. The results are listed
in Appendix F. It is often possible to obtain these results by a simpler procedure.
We illustrate this for the components of the T 1g two-electron state, obtained in Eq.
(6.9). The z-component of this state is the only component that is totally symmetric
under the ˆ
C 4 splitting field. It is clear that this symmetry can be obtained only by
multiplying the |e g ǫ and |t 2g ζ components, since these are both antisymmetric
and thus will form a symmetric product. From here on we will adopt for the product
functions the usual notation of small letters for the orbitals and capital letters for the
coupled states. Hence:
|T 1g z=|e g ǫ|t 2g ζ
(6.12)
The coupling coefficient E g ǫT 2g ζ |T 1g z is thus equal to 1. The x and y components may then immediately be obtained by applying the cyclic ˆ
C 3 generator. As an
example for the x-component:
|T 1g x= ˆ
C 3 |T 1g z
=
ˆ
C 3 |e g ǫ
ˆ
C 3 |t 2g ζ
=
−
√
3
2
|e g θ −
1
2
|e g ǫ
|t 2g ξ
=−
√
3
2
|e g θ |t 2g ξ −
1
2
|e g ǫ|t 2g ξ
(6.13)
Thus: θξ|x=−
√
3/2;;ǫξ|x=− 1/2. The resulting coupling coefficients are
shown in Table 6.2.
6.3 Symmetry Properties of the Coupling Coefficients
The CG-coefficients in the finite point groups stem from Wigner’s celebrated coupling coefficients for the spherical symmetry group [1]. Wigner proposed reformulating these coefficients in terms of more primitive 3j symbols, which contain, in
a uniform way, the permutational properties of the spherical coupling coefficients.
Several attempts have been made to define similar 3Γ symbols for the point group,
but this requires the introduction of quite detailed phase conventions, which limits the efficiency of this formalism [2, 3]. We shall therefore not engage in a further
117
basis orbitals.
c
Γ
γ a γ b
=
Γ a γ a (1)Γ b γ b (2)|Γγ(1, 2)
≡≡Γ a γ a Γ b γ b |Γγ
(6.11)
This coefficient is known as a Clebsch–Gordan (CG) coupling coefficient and denoted by the 3Γ bracket Γ a γ a Γ b γ b |Γγ. It indicates how the orbital irreps Γ a and
Γ b have to be combined to yield a product ket that transforms as |Γγ.T h eC G -
coefficients can be determined by using projection operators. The results are listed
in Appendix F. It is often possible to obtain these results by a simpler procedure.
We illustrate this for the components of the T 1g two-electron state, obtained in Eq.
(6.9). The z-component of this state is the only component that is totally symmetric
under the ˆ
C 4 splitting field. It is clear that this symmetry can be obtained only by
multiplying the |e g ǫ and |t 2g ζ components, since these are both antisymmetric
and thus will form a symmetric product. From here on we will adopt for the product
functions the usual notation of small letters for the orbitals and capital letters for the
coupled states. Hence:
|T 1g z=|e g ǫ|t 2g ζ
(6.12)
The coupling coefficient E g ǫT 2g ζ |T 1g z is thus equal to 1. The x and y components may then immediately be obtained by applying the cyclic ˆ
C 3 generator. As an
example for the x-component:
|T 1g x= ˆ
C 3 |T 1g z
=
ˆ
C 3 |e g ǫ
ˆ
C 3 |t 2g ζ
=
−
√
3
2
|e g θ −
1
2
|e g ǫ
|t 2g ξ
=−
√
3
2
|e g θ |t 2g ξ −
1
2
|e g ǫ|t 2g ξ
(6.13)
Thus: θξ|x=−
√
3/2;;ǫξ|x=− 1/2. The resulting coupling coefficients are
shown in Table 6.2.
6.3 Symmetry Properties of the Coupling Coefficients
The CG-coefficients in the finite point groups stem from Wigner’s celebrated coupling coefficients for the spherical symmetry group [1]. Wigner proposed reformulating these coefficients in terms of more primitive 3j symbols, which contain, in
a uniform way, the permutational properties of the spherical coupling coefficients.
Several attempts have been made to define similar 3Γ symbols for the point group,
but this requires the introduction of quite detailed phase conventions, which limits the efficiency of this formalism [2, 3]. We shall therefore not engage in a further