6.4 Goodenough–Kanamori Rules
201
Fig. 6.15 Semi covalent exchange between filled and half-filled orbitals as operative in the magnetic
coupling between two copper ions with an oxygen bridge forming an angle of 90 ◦ . Upper part for
antiferromagnetic coupling, and below for ferromagnetic interaction
non-Hund determinant with two anti-parallel electrons on center A. This unfavorable
electronic configuration is avoided in the case of a ferromagnetically coupled initial
state shown in the lower part of the figure.
Estimation with perturbation theory: The ferromagnetic nature of the interaction
between two magnetic centers with S =
1
2 through a ligand under an angle of 90 ◦
can also be rationalized with perturbation theory in the same way as discussed in
Sect. 5.1.1 for the magnetic interactions in a linear geometry or in Sect. 5.4.3 to derive
the equations to estimate the magnitude of the four-center interactions with single
determinant methods. The derivation is similar to the one presented by Koch in Ref.
[19] with this difference that we here work within a spin restricted formalism.
There are four orbitals involved in the coupling as can be seen in Fig. 6.14.The
six electrons can be distributed in sixteen different ways over the orbitals under the
restriction of M S = 0:
Φ 1 =|a 1 p x p x p y p y b 2 |
Φ 2 =|a 1 p x p x p y p y b 2 |
Φ 3 =|a 1 p x p y p y b 2 b 2 |
Φ 4 =|a 1 p x p y p y b 2 b 2 |
Φ 5 =|a 1 a 1 p x p x p y b 2 |
Φ 6 =|a 1 a 1 p x p x p y b 2 |
Φ 7 =|a 1 a 1 p x p y p y b 2 |
Φ 8 =|a 1 a 1 p x p y p y b 1 |
Φ 9 =|a 1 p x p x p y b 2 b 2 |
Φ 10 =|a 1 p x p x p y b 2 b 2 | Φ 11 =|a 1 a 1 p x p y b 2 b 2 | Φ 12 =|a 1 a 1 p x p y b 2 b 2 |
Φ 13 =|a 1 a 1 p x p x p y p y | Φ 14 =|p x p x p y p y b 2 b 2 | Φ 15 =|a 1 a 1 p x p x b 2 b 2 |
Φ 16 =|a 1 a 1 p y p y b 2 b 2 |
Taking the energy of Φ 1 and Φ 2 as reference, the other determinants lie at ∆E CT
(Φ 3 ...Φ 6 ), ∆E ′
CT (Φ 7 ...Φ 10 ), ∆E 2CT (Φ 11 and Φ 12 ), U d (Φ 13 and Φ 14 ) and
∆E 2CT + U p (Φ 15 and Φ 16 ). The last two determinants are high in energy and
will be neglected. The subscript p refers to the O-2p x or 2p y orbital and the subscript
d to the a or b Cu-3d orbital.
201
Fig. 6.15 Semi covalent exchange between filled and half-filled orbitals as operative in the magnetic
coupling between two copper ions with an oxygen bridge forming an angle of 90 ◦ . Upper part for
antiferromagnetic coupling, and below for ferromagnetic interaction
non-Hund determinant with two anti-parallel electrons on center A. This unfavorable
electronic configuration is avoided in the case of a ferromagnetically coupled initial
state shown in the lower part of the figure.
Estimation with perturbation theory: The ferromagnetic nature of the interaction
between two magnetic centers with S =
1
2 through a ligand under an angle of 90 ◦
can also be rationalized with perturbation theory in the same way as discussed in
Sect. 5.1.1 for the magnetic interactions in a linear geometry or in Sect. 5.4.3 to derive
the equations to estimate the magnitude of the four-center interactions with single
determinant methods. The derivation is similar to the one presented by Koch in Ref.
[19] with this difference that we here work within a spin restricted formalism.
There are four orbitals involved in the coupling as can be seen in Fig. 6.14.The
six electrons can be distributed in sixteen different ways over the orbitals under the
restriction of M S = 0:
Φ 1 =|a 1 p x p x p y p y b 2 |
Φ 2 =|a 1 p x p x p y p y b 2 |
Φ 3 =|a 1 p x p y p y b 2 b 2 |
Φ 4 =|a 1 p x p y p y b 2 b 2 |
Φ 5 =|a 1 a 1 p x p x p y b 2 |
Φ 6 =|a 1 a 1 p x p x p y b 2 |
Φ 7 =|a 1 a 1 p x p y p y b 2 |
Φ 8 =|a 1 a 1 p x p y p y b 1 |
Φ 9 =|a 1 p x p x p y b 2 b 2 |
Φ 10 =|a 1 p x p x p y b 2 b 2 | Φ 11 =|a 1 a 1 p x p y b 2 b 2 | Φ 12 =|a 1 a 1 p x p y b 2 b 2 |
Φ 13 =|a 1 a 1 p x p x p y p y | Φ 14 =|p x p x p y p y b 2 b 2 | Φ 15 =|a 1 a 1 p x p x b 2 b 2 |
Φ 16 =|a 1 a 1 p y p y b 2 b 2 |
Taking the energy of Φ 1 and Φ 2 as reference, the other determinants lie at ∆E CT
(Φ 3 ...Φ 6 ), ∆E ′
CT (Φ 7 ...Φ 10 ), ∆E 2CT (Φ 11 and Φ 12 ), U d (Φ 13 and Φ 14 ) and
∆E 2CT + U p (Φ 15 and Φ 16 ). The last two determinants are high in energy and
will be neglected. The subscript p refers to the O-2p x or 2p y orbital and the subscript
d to the a or b Cu-3d orbital.