Goodenough-Kanamori rules

A pictorial explanation for the ferromagnetic nature of the semicovalent exchange between filled and half-filled orbitals is given in Fig. 6.15. In the upper part, we can see how the subsequent electron transfer from py to b and from a to py leads to a 

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 Ms = 0  

To simplify the derivation of the singlet and triplet energy, we first construct spin-symmetry adapted CSFs by forming linear combination of the above-listed determinants with unpaired electrons  

The plus (minus) combinations are triplet (singlet) functions, except for the combination of closed-sheU determinants 0i3 and 0i4, which are both singlets. Now we can construct the 6 x 6 configuration interaction matrix for the triplet functions and an 8 X 8 matrix for the singlet and then determine the energy either by diagonalizing the matrices or (simpler) with perturbation theory. 

Remember that the intersite exchange interactions have been neglected, otherwise the zeroth-order triplet-singlet gap would be 2Kab- The second-order correction to the energy of the triplet and singlet are 

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TABLE 8.4. The Goodenough-Kanamori Rules for Magnetic Exchange... 

The Goodenough-Kanamori rules are given in Table 8.4, which lists the four types of interatomic interactions between a pair of cations, denoted as A and B ... 

Based on the Goodenough-Kanamori rules, what would be the expected sign of ... 

Fig. 6.11 Representation of the two virtual electron exchange mechanisms that are the basis of the Goodenough-Kanamori rules. Above the virtual exchange between two atoms (A and B) with non-zero spin moment, known as superexchange and below the virtual exchange between two atoms and a shared anion, the so-caUed semi covalent exchange...

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