Magnetic exchange interaction in clusters of orbitally degenerate ions. I. Effective Hamiltonian

A new effective Hamiltonian is reported for the kinetic exchange between two arbitrary terms ^2SA+1Λ_A and ^2SB+1 Λ_B that can be ground or excited in octahedrally coordinated transition metal ions. This Hamiltonian is applicable to both homo- and heterometallic clusters. For the homonuclear cluster the resonance part of the effective Hamiltonian is also presented for the case when one of the ions is excited. The operator part of the exchange Hamiltonian contains symmetry adapted products of the cubic irreducible tensors acting in orbital spaces Λ_A and Λ_B and scalar product of site spin operators. The parameters of the Hamiltonian are defined by the relevant intercenter transfer integrals and the fundamental intracenter interactions, namely, crystal field and Racah parameters for the constituent metal ions in their ground, oxidized and reduced electronic configurations. These parameters contain also the reduced matrix elements of the creation (annihilation) operators linking the ground state of the many-electron ions with their reduced and oxidized states (fractional parentage coefficients), W-symbols and 6j-symbols. The approach is discussed in context of the existing exchange models.