J.J. Borrás-Almenar

MAGPACK1 A package to calculate the energy levels, bulk magnetic properties, and inelastic neutron scattering spectra of high nuclearity spin clusters

M agnetic molecular clusters, i.e., molecular assemblies formed by a finite number of exchange-coupled magnetic moments, are currently receiving much attention in several active areas of research as molecular chemistry, magnetism, and biochemistry. A reason for this interest lies in the possibility to use simple molecular clusters as magnets of nanometer size exhibiting unusual magnetic properties as superparamagnetic like behavior or quantum tunneling of magnetization.2 – 4 Organic molecules of increasing sizes and large number of unpaired electrons are being explored as a means of obtaining building blocks for molecule-based magnets.5 Magnetic clusters of metal ions are also relevant in biochemistry.6 This area between molecule and bulk will require new theoretical concepts and techniques for investigation of their peculiar properties. Still, the theoretical treatment required to understand the magnetic and spectroscopic properties of this wide variety of compounds is a challenging problem in molecular magnetism.7 For a long time, this problem has been mostly restricted to treat comparatively simple clusters comprising a reduced number of exchange-coupled centers and special spin topologies, for which solutions can be obtained either analytically or numerically. However, on increasing the spin nuclearity of the cluster, the problem rapidly becomes unapproachable because the lack of translational symmetry in the clusters. An additional complication is the spin anisotropy of the cluster. Until now only the isotropic-exchange case has been treated, so as to take full advantage of the spin symmetry of the cluster.8 In this article we present a very powerful and efficient computational approach to solve the exchange problem in high nuclearity spin clusters with all kind of exchange interactions (isotropic and anisotropic), including the single-ion anisotropic effects. The clusters are formed by an arbitrary number of exchangecoupled centers that combine different spin values and arbitrary topology. This approach is based on the use of the irreducible tensor operators (ITO) technique.7, 9 – 12 It allows evaluation of both eigenvalues and eigenvectors of the system, and then, calculation of the magnetic susceptibility, magnetization, or heat capacity, and also the inelastic neutron scattering spectra. In the following sections we will present both the theory and the four different implemented FORTRAN programs that integrate a package called MAGPACK . In the last section some examples are presented in order to show the possibilities of the programs.

Mixed-valence polyoxometalate clusters. I. Delocalization of electronic pairs in dodecanuclear heteropoly blues with keggin structure

Abstract The problem of delocalization of a pair of electrons over dodecanuclear polyoxometalate clusters with the Keggin structure is considered with the aim of explaining the spin pairing in these multi-nuclear mixed-valence systems. A general approach that considers the Coulomb interactions between the two delocalized electrons, as well as the single and double electron transfer processes which can be operative in delocalization of the electronic pairs is developed. The new approach is based on the site-symmetry concept which makes possible a group theoretical classification for the delocalized states of electronic pairs. This procedure proves to be very efficient in the calculation of the transfer matrices which are expressed in terms of the Coulomb energy, and the single- and double-transfer parameters. The influence of these electronic parameters on the spectrum of the low-lying energy levels of the cluster is discussed, and the conditions giving rise to the stabilization of a singlet ground spin state for the electronic pair are elucidated.

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...

Mixed-valence polyoxometalate clusters. III. Vibronic problem for the 2-electron reduced heteropoly blue with the Keggin structure

Abstract A general approach to the vibronic problem of delocalized electronic pairs in mixed-valence compounds is developed and applied to understand the ways of electron delocalization in dodecanuclear polyoxometalate clusters containing two moving electrons. The interplay between electronic and vibronic interactions is examined. The electronic spectrum is shown to consist of two spin triplets 3 T 1 and 3 T 2 and three spin singlets 1 A 1 , 1 E and 1 T 2 levels determined by the double-transfer processes (parameter P ). Jahn-Teller and pseudo-Jahn-Teller problems ( 3 T 1 + 3 T 2 ) ⊗ ( e + t 2 ) and ( 1 A 1 + 1 E + 1 T 2 ) ⊗ ( e + t 2 ) have been considered in the framework of the Piepho-Krausz-Schatz model dealing with the only vibronic parameter. Several kinds of spatial electronic distribution have been found corresponding to the stable points of the energy surfaces. For spin-triplet states, potential surfaces contain six minima in e space corresponding to partially delocalized electronic pairs over four sides of the T d structure (limiting case of weak coupling), or delocalized over two opposite sides (limiting case of strong coupling). The former situation restricts electron delocalization to two of the three metal octahedra of each M 3 O 12 triad in such a way that each electron moves over a tetrameric unit in which the metal sites are alternatively sharing edges and corners. In the t 2 space the electronic pair can be either delocalized over three sides, giving rise to a trigonal-type distortion of the cluster and a partial electron delocalization over two opposite M 3 O 12 triads (four trigonal minima in the case of strong transfer or relatively weak vibronic interaction), or be completely localized (case of strong vibronic coupling). For spin-singlet states the system possesses a stable point in the high-symmetrical nuclear configuration, corresponding to a full delocalization of the electronic pairs in the Keggin cluster. The influence of vibronic interaction on the nature of the spin of the ground states is considered.

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.

Mixed-valence polyoxometalate clusters. II. Delocalization of electronic pairs in 18-site heteropoly blues with Wells-Dawson structure

Abstract The problem of delocalization of two electrons in the 18-site Wells-Dawson polyoxometalate is examined from a general approach that takes into account both single- and double-transfer processes, as well as the Coulomb interactions between the two delocalized electrons. The electronic energy levels of this mixed-valence cluster are calculated and the conditions giving rise to the stabilization of a singlet ground spin state for the electronic pair are elucidated. It is shown that the spin pairing results from the simultaneous effects of single- and double-electron transfer processes, which are operative even when the two delocalized electrons are fairly widely separated in the Wells-Dawson structure.

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.