Magnetic exchange between metal ions with unquenched orbital angular momenta: Basic concepts and relevance to molecular magnetism

This review article is a first attempt to give a systematic and comprehensive description (in the framework of the unified theoretical approach) of the exchange interactions in polynuclear systems based on orbitally degenerate metal ions in the context of their relevance to the modern molecular magnetism. Interest in these systems is related to the fundamental problems of magnetism and at the same time steered by a number of impressive potential applications of molecular magnets, like high-density memory storage units, nanoscale qubits, spintronics and photoswitchable devices. In the presence of orbital degeneracy, the conventional spin Hamiltonian (Heisenberg-Dirac-van Vleck model) becomes inapplicable even as an approximation. The central component of this review article constitutes the concept of orbitally-dependent exchange interaction between metal ions possessing unquenched orbital angular momenta. We present a rigorous procedure of derivation of the kinetic exchange Hamiltonian for a pair of orbitally degenerate transition metal ions that is expressed in terms of the orbital matrices and spin operators. The microscopic background reveals the interrelations between the parameters of the Hamiltonian and the internal parameters of the system including all relevant transfer integrals and fundamental intracenter interactions. The developed formalism integrated with the irreducible tensor operator (ITO) technique makes it possible to describe the exchange coupling and all relevant interactions (crystal fields, spin-orbit (SO) and Zeeman couplings) in terms of the ITOs of the full spherical group, and in this way to develop anunified and efficient computational tool. The orbitally-dependent exchange was shown to lead to an anomalously strong magnetic anisotropy that can be considered as a main physical manifestation of the unquenched orbital angular momentum in metal clusters of orbitally-degenerate ions. The theoretical background is illustrated by the following applications. The magnetic properties of the binuclear face-shared unit [Ti₂Cl₉]³- in Cs₃Ti₂Cl₉ are discussed with the emphasis on the observed magnetic anisotropy and on the non-trivial symmetry properties of the exchange Hamiltonian. The major electronic factors controlling the magnetic anisotropy in Co(II) pairs are discussed. The degree of the exchange anisotropy was shown to depend on the strength of the cubic crystal field, on the relative efficiency of the electron transfer pathways between unfilled d-shells and SO coupling. Provided strong SO coupling, the effective Hamiltonian was projected onto the subspace of low-lying Kramers doublets and similarly a pseudo-spin-1/2 Hamiltonian was derived. The described procedure allows to establish the interrelation between idem parameters of the system and the parameters of the pseudo-spin-1/2 Hamiltonian. Pseudo-spin-1/2 approach is illustrated by the study of the inelastic neutron scattering spectra and magnetic susceptibility of polyoxometalates encapsulating Co(II) clusters: Keggin derivative K₈[Co₂(D₂O)(W₁₁O₃₉)] · nD₂O, [Co₄(H₂O)₂(PW₉O₃₄)₂]^10- and [Co₃W(D₂O)₂(CoW₉O₃₄)₂]^12- clusters. In the consideration of the cyanide-bridged Mn(III)-CN-Mn(II) pair, it was demonstrated that under certain conditions the orbitally-dependent exchange is able to produce a barrier for the reversal of magnetisation. This seems to be instructive for the controlled design of cyano-based single molecule magnets with high-blocking temperatures.