In this critical review we review the problem of exchange interactions in polynuclear metal complexes involving orbitally degenerate metal ions. The key feature of these systems is that, in general, they carry an unquenched orbital angular momentum that manifests itself in all their magnetic properties. Thus, interest in degenerate systems involves fundamental problems related to basic models in magnetism. In particular, the conventional Heisenberg–Dirac–Van Vleck model becomes inapplicable even as an approximation. In the first part we attempt to answer two key questions, namely which theoretical tools are to be used in the case of degeneracy, and how these tools can be employed. We demonstrate that the exchange interaction between orbitally degenerate metal ions can be described by the so-called orbitally-dependent exchange Hamiltonian. This approach has shown to reveal an anomalously strong magnetic anisotropy that can be considered as the main physical manifestation of the unquenched orbital angular momentum in magnetic systems. Along with the exchange coupling, a set of other interactions (such as crystal field effects, spin–orbit and Zeeman coupling), which are specific for the degenerate systems, need to be considered. All these features will be discussed in detail using a pseudo-spin-1/2 Hamiltonian approach. In the second part, the described theoretical background will be used to account for the magnetic properties of several magnetic metal clusters and low-dimensional systems: (i) the dinuclear face-sharing unit [Ti₂Cl₉]³⁻, which exhibits a large magnetic anisotropy; (ii) the rare-earth compounds Cs₃Yb₂Cl₉ and Cs₃Yb₂Br₉, which, surprisingly, exhibit a full magnetic isotropy; (iii) a zig-zag Co^II chain exhibiting unusual combination of single-chain magnet behavior and antiferromagnetic exchange coupling; (iv) a trigonal bipyramidal Ni₃Os₂ complex; (v) various Co^II clusters encapsulated by polyoxometalate ligands. In the two last examples a pseudospin-1/2 Hamiltonian approach is applied to account for the presence of exchange anisotropy (150 references).

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.

We report a combined experimental characterization and theoretical modeling of the hexa-coordinated high-spin Co( ii ) complex cis -[Co(hfac) ₂ (H ₂ O) ₂ ] ( I ).

The low temperature EPR spectra of the unique nanometer-scale molecular magnet V15 are analyzed within the three-spin model preserving trigonal symmetry. The Hamiltonian includes isotropic and antisymmetric (AS) exchange interactions introduced by Dzyaloshinsky (Zh. Eksp. Teor. Fiz. 32, 1547 (1957) [Sov. Phys. JETP 5, 1259 (1957)]) and [Moria Phys. Rev. 120, 91 (1960)]. With the aid of pseudoangular momentum representation the exact selection rules for the EPR transitions are deduced. AS exchange in the frustrated triangular system gives rise to a first order zero-field splitting of two low lying spin doublets and to a second order splitting of the excited quadruplet. This leads to a peculiar series of strong intramultiplet transitions and weak intermultiplet transitions. We show that the intramultiplet transitions are allowed only when the vector of the AS exchange is normal to the plane of vanadium triangle, but the series of weak intermultiplet transitions are a consequence of the in-plane part of the AS exchange. We have revealed a special temperature dependence of the EPR pattern with a peculiar shape of the full spectrum. Experimental data on high-frequency EPR of V15 at ultralow temperature are discussed.

We show here that mixed-valence (MV) magnetic molecules with a significant electron delocalization are extremely sensitive to an external electric field. In particular, we focus on the symmetric many-electron MV binuclear complexes that are on the borderline between Robin and Day classes II and III. In these molecules, the double-exchange, which has been shown to lead to the ferromagnetic ground spin state, competes with the electric field, which tends to localize the spin, thus creating an electric dipole and stabilizing the spin states with lower multiplicities. This provides an efficient and easy way to control the ground spin state of the molecule through the double-exchange mechanism. Thus, we predict that the application of an external electric field will lead to a strong stepwise decrease of the magnetic susceptibility and to a simultaneous increase of the electric polarization. The reverse effect, consisting of a sharp decrease of the electric polarization under the action of an external magnetic field, is also predicted. The results demonstrate that MV dimers of this class can be regarded as single-molecule analogs of multiferroics with promising potential to create a functional magnetoelectric unit in one molecule.