Magnetic exchange between orbitally degenerate ions:: A new development for the effective Hamiltonian

A new approach to the problem of the kinetic exchange for orbitally degenerate ions is developed. The constituent multielectron metal ions are assumed to be octahedrally coordinated, and strong crystal field scheme is employed, making it possible to take full advantage from the symmetry properties of the fermionic operators and collective electronic states. In the framework of the microscopic approach, the highly anisotropic effective Hamiltonian of the kinetic exchange is constructed in terms of spin operators and standard orbital operators (matrices of the unit cubic irreducible tensors). As distinguished from previous considerations, the effective Hamiltonian is derived for a most general case of the multielectron transition metal ions possessing orbitally degenerate ground states and for arbitrary topology of the system. The overall symmetry of the system is introduced through the restricted set of the one-electron transfer integrals implied by the symmetry conditions. All parameters of the effective Hamiltonian are expressed in terms of the relevant transfer integrals and fundamental parameters of the two moieties, namely crystal field and Racah parameters for the metal ions in their normal, reduced, and oxidized states. The developed approach is applied to two kinds of systems: edge-shared (D-2h) and corner-shared (D-4h) bioctahedral clusters. In the particular case of d(1) ions (T-2(2)-T-2(2) problem) the energy pattern in both cases consists of several multiplets splitted by the isotropic part of exchange. In both cases we have found a weak ferromagnetic splitting for several multiplets of the system. This splitting is due to the competition of ferro-and antiferromagetic contributions arising from the high-and low-spin reduced states in line with Anderson's considerations, Goodenough-Kanamori rules, and McConnell mechanism of ferromagnetic interaction. On the contrary, these weak ferromagnetic interaction are found to coexist with strong ferro-and antiferromagnetic contributions in which only high-spin and low-spin excited states are respectively involved. In addition to these unexpected results in both topologies the ferro-and antiferromagnenic contributions vanish separately for one of the level, the last being thus paramagnetic. These results are in a strike contradiction with the generally accepted point of view on the ferromagnetic role of orbital degeneracy in the magnetic exchange. They also show that the simple qualitative models have a restricted area of applications and that the peculiarities of the exchange problem in the case of orbital degeneracy are much more complicated. The energy pattern of the exchange levels is closely related to the topology of the system and to the network of the one-electron transfer intercenter connections forming effective parameters of the kinetic exchange in the case of orbital degeneracy.