A.V. Palii

Orbitally dependent magnetic coupling between cobalt(II) ions: The problem of the magnetic anisotropy

A comprehensive theoretical study of the magnetic exchange between Co2+ ions is reported. Using the microscopic background we deduce the general Hamiltonian for a corner shared bioctahedral system involving kinetic exchange, spin–orbit coupling, and low-symmetry local crystal field. This Hamiltonian acting within the orbitally degenerate ground manifold (4T1g)A⊗(4T1g)B of the cobalt pair is expressed in terms of orbital and spin operators. The treatment of the Hamiltonian is performed with the use of the irreducible tensor operator technique. We elucidate the major electronic factors controlling the magnetic anisotropy in the Co(II) pairs. The degree of the exchange anisotropy is shown to depend on the strength of the cubic crystal field and on the relative efficiency of two kinds of electron transfer pathways (e–e and t2–t2) contributing to the kinetic exchange. An unusual role of spin–orbit interaction is revealed. This interaction tends to reduce the anisotropy caused by the orbitally dependent exchang...

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

Abstract A new effective Hamiltonian is reported for the kinetic exchange between two arbitrary terms 2S A +1 Λ A and 2S B +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 ΛA and ΛB 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.

Localisation vs. delocalisation in the dimeric mixed-valence clusters in the generalised vibronic model. Magnetic manifestations

Abstract The problem of localisation–delocalisation in the dimeric mixed-valence clusters is considered in the framework of the generalised vibronic model. The model takes into account both the local vibrations on the metal sites (Piepho–Krausz–Schatz model) and the multicenter (molecular) vibrations changing the intermetallic distances (as suggested by Piepho). In the framework of the semiclassical adiabatic approach the potential surfaces are analysed and different kinds of localised and delocalised states are found. On the basis of the calculated degrees of the localisation the conventional Robin and Day classification of mixed-valence compounds is reconsidered in view of the generalised vibronic model. The magnetic properties of the many-electron mixed-valence dimers are considered as well. The multicenter vibrations are shown to produce a ferromagnetic effect.

High‐nuclearity mixed‐valence magnetic clusters: A general solution of the double exchange problem

We report here a general solution of the double‐exchange problem in the high‐nuclearity mixed valence systems containing arbitrary number P of the electrons delocalized over the network of N (P<N) localized spins. The developed approach is based on the successive (chainlike) spin‐coupling scheme and takes full advantage from the quantum angular momentum theory. In the framework of this approach the closed‐form analytical expressions are deduced for the matrix elements of the double exchange interaction, two‐electron transfer, and three‐center interaction that can be referred to as the potential exchange transfer. For the arbitrary nuclearity mixed‐valence systems the matrix elements of all named interactions are expressed in terms of all relevant spin quantum numbers and 6j symbols and do not contain higher order recoupling coefficients. We describe also the combined approach taking into account both angular momentum consideration and advantages of point symmetry adapted basis set.

Magnetic exchange interaction in a pair of orbitally degenerate ions: Magnetic anisotropy of [Ti2Cl9]−3

The theory of the kinetic exchange in a pair of orbitally degenerate ions developed by the authors [J. Phys. Chem. A 102, 200 (1998)] is applied to the case of face-shared bioctahedral dimer (overall D3h-symmetry). The effective kinetic exchange Hamiltonian is found for a 2T2–2T2 system taking into account all relevant transfer pathways and charge-transfer crystal field states. The influence of different transfer integrals involved in the kinetic exchange on the energy pattern and magnetic properties of the system is examined. The role of other related interactions (trigonal crystal field, spin–orbit coupling) is also discussed in detail. Using the pseudoangular momentum representation and the technique of the irreducible tensor operators of R3-group we give a general outlook on the nontrivial symmetry properties of the effective Hamiltonian for the D3h-pair, and on the magnetic anisotropy arising from the orbital interactions specific for the case of orbital degeneracy. The magnetic properties of the bin...

Magnetic exchange interaction in clusters of orbitally degenerate ions. II. Application of the irreducible tensor operator technique

Abstract The irreducible tensor operator technique in R3 group is applied to the problem of kinetic exchange between transition metal ions possessing orbitally degenerate ground states in the local octahedral surrounding. Along with the effective exchange Hamiltonian, the related interactions (low-symmetry crystal field terms, Coulomb interaction between unfilled electronic shells, spin–orbit coupling and Zeeman interaction) are also taken into account within a unified computational scheme. Extension of this approach to high-nuclearity systems consisting of transition metal ions in the orbital triplet ground states is also demonstrated. As illustrative examples, the corner-shared D4h dimers 2S+1 T 2 – 2S+1 T 2 and 2S+1 T 1 – 2S+1 T 1 are considered. Finally, the developed approach is applied to the calculation of the energy levels and magnetic properties of the 2 T 2 – 2 T 2 corner-shared dimer. The problem of the magnetic anisotropy arising from the orbital interactions is discussed and the influence of different relevant interactions is revealed.

Anisotropic double exchange in orbitally degenerate mixed valence systems

Abstract The problem of the double exchange is considered for the mixed valence dimers in which one or both transition metal ions possess orbitally degenerate ground states. In the pseudo-angular momentum representation, the general formula is deduced for the matrix elements of double exchange involving the transfer integrals and all spin and orbital quantum numbers. The pairs 3 T 1 t 2 2 – 2 T 2 t 2 1 and 3 T 1 t 2 2 – 4 A 2 t 2 3 are considered in three high-symmetric topologies: edge-shared D2h, corner-shared D4h, and face-shared D3h bioctahedra. The double exchange in orbitally degenerate systems is shown to produce strong magnetic anisotropy of an orbital nature. The character of the anisotropy proved to be dependent on the overall symmetry of the system and on the ground terms of the metal ions.

Orbitally dependent kinetic exchange in cobalt(II) pairs: origin of the magnetic anisotropy

Abstract A comprehensive theoretical study of the magnetic exchange between Co 2+ ions is reported. Using the microscopic background we deduce the general Hamiltonian for a corner-shared bioctahedral system involving kinetic exchange, spin–orbit coupling and low-symmetry local crystal field. This Hamiltonian acting within orbitally degenerate ground manifold 4 ( T 1g ) A ⊗ 4 ( T 1g ) B of the cobalt pair is expressed in terms of orbital and spin operators. We elucidate the major electronic factors controlling the exchange anisotropy in the Co(II) pairs. The degree of the magnetic anisotropy is shown to depend on the strength of the cubic crystal field and on the relative efficiency of two kinds of electron transfer pathways ( e – e and t 2 – t 2 ) contributing to the kinetic exchange. An unusual role of spin–orbit interaction is revealed. This interaction tends to reduce the anisotropy caused by the orbitally dependent exchange. Finally, we discuss conditions of the applicability of the isotropic Lines’ model conventionally accepted in magnetochemistry of cobalt clusters.

EXCHANGE-TRANSFER IN MIXED-VALENCE CLUSTERS WITH ONE MIGRATING HOLE

Abstract A new mechanism for exchange-transfer specific to hole-type mixed-valence clusters is proposed. The intermediate state in the second-order exchange-transfer process is achieved by the jump of the electron from the spin-core of the d n +1 ion into the empty orbital of the d n ion. As distinguished from the mixed-valence clusters with one delocalized electron this intermediate state is the high-spin one giving rise to a ferromagnetic contribution to the ground manifold. On the basis of angular momentum theory a general solution of the exchange-transfer problem is given for arbitrary nuclearity mixed-valence hole-type clusters with many-electron paramagnetic spin-cores. The interplay between the double exchange, exchange-transfer and Heisenberg exchange is considered in detail for the simple case of a d 2 -d 2 -d 1 trimer.

Exchange transfer in high-nuclearity mixed valence magnetic clusters: Theoretical approach and expected manifestations

Abstract We report here a general solution of the exchange transfer problem in the high-nuclearity mixed valence clusters containing arbitrary number of itinerant electrons. The concept of two kinds of exchange transfer, namely kinetic and potential, is introduced by analogy with basic Andersons mechanisms of the magnetic exchange. The kinetic exchange transfer is treated as a second order transfer process between two centres through the excited state of a third centre. The potential exchange transfer is also considered as a three-centre interaction but in this case only the ground states of the constituent ions are involved. The actual parameters of the exchange transfer are expected to be of the same order of magnitude as those for the magnetic exchange. An efficient computational approach is proposed making it possible to build up the matrices of all kinds of exchange transfer interactions in a closed analytical form, expressing the matrix elements in terms of the relevant spin quantum numbers. As distinguished from the magnetic exchange the kinetic and potential contributions to the exchange transfer act independently and their parameters cannot be, in general, combined to give a joint parameter of the exchange transfer. The magnetic manifestations of the exchange transfer are considered for selected trimeric systems with different topologies (linear and trigonal) and different electronic shells involving either migrating electrons or holes. The magnetic nature of the ground state is found to depend on the topology of the system and on the sign of the double exchange parameter.