Magnetic Exchange between Orbitally Degenerate Ions:  A New Development for the Effective Hamiltonian

Abstract

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 2 h ) and corner-shared (D 4 h ) bioctahedral clusters. In the particular case of d ions (2T2−T2 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 antiferromagnetic 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.

Discrete Fulleride Anions and Fullerenium Cations

“…Calculations which include configuration interaction typically find the singlet state lower or equal in energy to the triplet state. 60,[300][301][302][303][304] The magnetic susceptibility data for [PPN] 3 [C 60 ] (see Figure 15), 21 a necessarily magnetically dilute system because of the three large cations, and for [Fe-(C 5 H 5 )(C 6 Me 6 )] 3 [C 60 ] 143 are consistent with an isolated S ) 1/2 system, not the S ) 3/2 quartet expected from a Hund's rule (t 1u ) 3 configuration. This is perhaps the most unexpected result that has arisen from the study of the spin states of discrete fulleride ions.…”

MVPACK: A package to calculate energy levels and magnetic properties of high nuclearity mixed valence clusters

“…This result was initially attributed to a very strong antiferromagnetic coupling via a multiroute superexchange mechanism, 39 but more recently it was suggested [17][18][19] that the electron delocalization can stabilize the diamagnetic ground state and it was proved 18,19,40,41 (see also review 2 ) that a combination of Coulomb repulsion between the electrons and electron delocalization is an efficient mechanism of a spin pairing in the ground state. The cited works 18,19,40,41 dealt with the case of 2e-reduced polyoxoanion containing delocalized electronic pair. Here we present the calculations of the energy patterns for a series of polyoxoanions with Keggin structure containing different even numbers P 5 2, 4, 6, 8, 10, 12 of itinerant electrons.…”

“…As a result these compounds can be viewed as model systems to study the interplay between electron transfer and exchange interactions. 11,[13][14][15][16][17][18][19] The second class contains the MV chain compounds 20 in which we also face the problem of the interaction between localized and delocalized electrons.…”

Highly Anisotropic Orbitally Dependent Superexchange in Cyano‐Bridged Clusters Containing Mn(III) and Mn(II) Ions

“…Below we apply this general approach to the problem of the magnetic exchange between orbitally degenerate ions developed in our recent articles [14][15][16] in which the effective Hamiltonian of the exchange interaction is derived and its use is illustrated in detail. Application of this approach leads to the following expression for the kinetic exchange Hamiltonian operating within the ground…”

“…The parameters F k ð Þ G G 0 in Equation (3) can be found with the aid of an existing approach, [14][15][16] therefore we have omitted the details of the calculation and give only the final result in Equation (5): À Á term of the Mn(III) ion (ion B) as a state possessing fictitious orbital angular momentum.…”