168
7 Spherical Symmetry and Spins
This potential is invariant under the symmetry properties of the metal complex. As a
result, the operator part reduces to the totally symmetric components of the spherical
harmonics. Moreover, interactions with d electrons imply that ℓ must be limited to
four, and to six for f electrons. In the case of an octahedral field, the subduction
relations for spherical harmonics (see Sect. C.1) indicate that a totally symmetric
A 1g component can be subduced only from ℓ = 4 and ℓ = 6. Filling in the angular
positions of the ligands in an octahedron then yields
V O h =
6e 2 Z L
√
4πǫ 0 R L
Y 00 +
e 2 Z L r 4
√
4πǫ 0 R 5
L
√
35
6
√
2
Y 44 + Y 4−4 +
√
14
√
5
Y 40
(7.14)
This is the famous crystal-field operator for an octahedron, which splits the d-shell
into e g and t 2g subshells. The crystal-field interaction is usually parameterized by
the crystal-field parameter, 10Dq, which corresponds to the splitting of the e g and
t 2g orbitals. The term in brackets here is the octahedral invariant of rank 4. In the
multipole expansion this corresponds to a hexadecapole operator. In normalized
form it reads
|4A 1g |=
1
2
√
3
√
5
√
2
(Y 44 + Y 4−4 ) +
√
7Y 40
(7.15)
Here the notation between vertical bars indicates that this is an operator. We shall
now derive this expression with the aid of the coupling coefficients. Since the d orbitals transform as e g + t 2g , the squares of the d-orbitals yield two totally symmetric
results, which may be abbreviated as follows:
A 1g (e × e)
=
1
√
2
θ
2 + ǫ
2
A 1g (t 2 × t 2 )
=
1
√
3
ξ
2 + η
2 + ζ
2
(7.16)
These two invariants are at the origin of two spherical operators: one corresponds
to the constant scalar |0A 1g |, and the other to |4A 1g |. The former invariant may be
obtained by taking the norm of the entire d-manifold, which can be expressed in the
A 1g functions of Eq. (7.16) as follows:
|0A 1g |=
1
√
5
θ
2 + ǫ
2 + ξ
2 + η
2 + ζ
2
=
1
√
5
√
2
A 1g (e × e)
+
√
3
A 1g (t 2 × t 2 )
(7.17)
The |4A 1g | invariant must be orthonormal to this result and thus will be given by
|4A 1g |=
1
√
5
√
3
A 1g (e × e)
−
√
2
A 1g (t 2 × t 2 )
(7.18)
7 Spherical Symmetry and Spins
This potential is invariant under the symmetry properties of the metal complex. As a
result, the operator part reduces to the totally symmetric components of the spherical
harmonics. Moreover, interactions with d electrons imply that ℓ must be limited to
four, and to six for f electrons. In the case of an octahedral field, the subduction
relations for spherical harmonics (see Sect. C.1) indicate that a totally symmetric
A 1g component can be subduced only from ℓ = 4 and ℓ = 6. Filling in the angular
positions of the ligands in an octahedron then yields
V O h =
6e 2 Z L
√
4πǫ 0 R L
Y 00 +
e 2 Z L r 4
√
4πǫ 0 R 5
L
√
35
6
√
2
Y 44 + Y 4−4 +
√
14
√
5
Y 40
(7.14)
This is the famous crystal-field operator for an octahedron, which splits the d-shell
into e g and t 2g subshells. The crystal-field interaction is usually parameterized by
the crystal-field parameter, 10Dq, which corresponds to the splitting of the e g and
t 2g orbitals. The term in brackets here is the octahedral invariant of rank 4. In the
multipole expansion this corresponds to a hexadecapole operator. In normalized
form it reads
|4A 1g |=
1
2
√
3
√
5
√
2
(Y 44 + Y 4−4 ) +
√
7Y 40
(7.15)
Here the notation between vertical bars indicates that this is an operator. We shall
now derive this expression with the aid of the coupling coefficients. Since the d orbitals transform as e g + t 2g , the squares of the d-orbitals yield two totally symmetric
results, which may be abbreviated as follows:
A 1g (e × e)
=
1
√
2
θ
2 + ǫ
2
A 1g (t 2 × t 2 )
=
1
√
3
ξ
2 + η
2 + ζ
2
(7.16)
These two invariants are at the origin of two spherical operators: one corresponds
to the constant scalar |0A 1g |, and the other to |4A 1g |. The former invariant may be
obtained by taking the norm of the entire d-manifold, which can be expressed in the
A 1g functions of Eq. (7.16) as follows:
|0A 1g |=
1
√
5
θ
2 + ǫ
2 + ξ
2 + η
2 + ζ
2
=
1
√
5
√
2
A 1g (e × e)
+
√
3
A 1g (t 2 × t 2 )
(7.17)
The |4A 1g | invariant must be orthonormal to this result and thus will be given by
|4A 1g |=
1
√
5
√
3
A 1g (e × e)
−
√
2
A 1g (t 2 × t 2 )
(7.18)