Irreducible Tensor Operator Technique Research Articles

Transfer of a weakly bound electron in collisions of Rydberg atoms with neutral particles. I. Long-range interaction effects in the ionic-covalent coupling

Ion-pair formation processes are studied in collisions of Rydberg atoms with neutral particles possessing small electron affinities. Nonadiabatic transitions from a Rydberg covalent term to an ionic term of a quasi-molecule are considered using the modified Landau-Zener theory supplemented with calculation of survival factors of an anion decaying in the Coulomb field of a positive ion core. Using the technique of irreducible tensor operators and the momentum representation of the wavefunction of a highly excited atom, exact expressions are obtained for transition matrix elements and the ionic-covalent coupling parameter. The approach developed in the paper provides the description beyond the scope of a conventional assumption about a small variation of the wavefunction of the Rydberg atom on the range of electron coordinates determined by the characteristic radius of the wavefunction of the anion. This allows one to correctly consider long-range effects of the interaction between a weakly bound electron and the neutral core of a negative ion in processes under study. It is shown by the example of thermal collisions of Xe(nf) atoms with CH3CN molecules that this is very important for a reliable quantitative description of anion formation with a low binding energy. The results are compared with experiments and calculations performed within the framework of a number of approximate methods.

Numerically exact and approximate determination of energy eigenvalues for antiferromagnetic molecules using irreducible tensor operators and general point-group symmetries

For small-enough quantum systems numerical exact and complete diagonalization of the Hamiltonian enables one to evaluate and discuss all static, dynamic, and thermodynamic properties. In this article we consider Heisenberg spin systems and extend the range of applicability of the exact diagonalization method by showing how the irreducible tensor operator technique can be combined with an unrestricted use of general point-group symmetries. We also present ideas on how to use spin-rotational and point-group symmetries in order to obtain approximate spectra.

Spin tomography and star-product kernel for qubits and qutrits

Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.

Orbital Angular Momentum Contribution to the Magneto-Optical Behavior of a Binuclear Cobalt(II) Complex

We report magnetic and magnetic circular dichroism investigations of a binuclear Co(II) compound. The Hamiltonian of the system involves an isotropic exchange interaction dealing with the real spins of cobalt(II) ions, spin-orbit coupling, and a low-symmetry crystal field acting within the (4)T(1g) ground manifold of each cobalt ion. It is shown that spin-orbit coupling between this ground term and the low-lying excited ones can be taken into consideration as an effective g factor in the Zeeman part of the Hamiltonian. The value of this g factor is estimated for the averaged experimental values of Racah and cubic ligand field parameters for high-spin cobalt(II). The treatment of the Hamiltonian is performed with the use of a irreducible tensor operator technique. The results of the calculation are in good agreement with experimental observations. Both a large effective g factor for the ground state and a large temperature-independent part of the magnetic susceptibility arise because of a strong orbital contribution to the magnetic behavior of the Co(II) dimer.

Spin dynamics and tunneling of the Néel vector in theFe10magnetic wheel

The spin dynamics of a ring-shaped molecule comprising 10 iron(III) ions $(s=5∕2)$ is studied by inelastic neutron scattering. Exchange integrals and single-ion anisotropy parameters are determined by studying several intermultiplet excitations. Difficulties associated with the dimension of the Hilbert space are overcome by exploiting both the irreducible tensor operator technique and the Lanczos algorithm. By calculating the time correlation function of the N\'eel vector we show that the low-temperature spin dynamics of ${\mathrm{Fe}}_{10}$ is only approximately described by the semiclassical picture of tunneling of the N\'eel vector.

Inelastic neutron scattering study of the molecular grid nanomagnet Mn-[3×3]

The spin Hamiltonian of a supramolecular planar grid formed by nine Mn(II) ions in an almost square geometry has been determined from the analysis of inelastic neutron scattering spectra. The experimental data have been interpreted in the framework of a theoretical model containing all the main interactions present in the system. Difficulties related with the dimension of the Hilbert space have been overcome by exploiting both the irreducible tensor operator technique and the Lanczos algorithm. The energy spectrum consists of level multiplets with an almost definite value of the total spin S. The ground multiplet has S=5/2, and the lowest-lying excited multiplets have (in order of increasing energy) S=7/2, S=3/2, S=3/2, and S=9/2. These multiplets are separated by the isotropic exchange and split by the anisotropic interactions. The two lowest manifolds of the exchange part of the spin Hamiltonian belong to the same irreducible representation of the molecular symmetry group and substantial mixing between them is demonstrated.

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 exchange. Special attention is paid on the topical case when spin–orbit coupling exceeds the exchange interaction. In this case the effective Hamiltonian in its general form is projected onto the subspace of low-lying Kramers doublets and in such a way the pseudo-spin-12 Hamiltonian is derived. Unlike the commonly accepted phenomenological approaches based merely on the symmetry arguments the proposed procedure is grounded on the microscopic consideration and hence allows us to establish the interrelation between the idem parameters of the system and the effective parameters of the pseudo-spin-12 Hamiltonian. Finally, we discuss the applicability of the conventional Lines’ model.

Magnetic properties of dinuclear cobalt complexes

We report the magnetic investigation of dinuclear Co(II) compounds on the base of hamiltonian which takes into account exchange interaction, zero-field splitting and Zeeman perturbation. The treatment of this hamiltonian is performed with the use of the irreducible tensor operator technique and symmetry of the system. A good agreement of the experimentally observed magnetic susceptibility and the results of calculation is obtained.

Non-Heisenberg magnetic behavior of a triangular bridged heterometallic Fe2(III)Co(II) complex: Evidence of strong orbital contributions

We report the synthesis, magnetization and ac susceptibility of the triangular μ3-oxo-bridged heterometallic complex [Fe2CoO(CH3COO)6(3-Cl-Py)3], in powder form. The model Hamiltonian is elaborated involving isotropic exchange interactions Fe(III)-Fe(III) and Co(II)-Fe(III) dealing with the real spins of high-spin Co(II) and Fe(III) ions, and also spin-orbit coupling and low-symmetry crystal field (C2v) acting within the T1g4 ground manifold of Co(II) ion. The outlined general model takes into account strong anisotropic orbital contribution to the magnetic characteristics of the whole system arising from Co(II) ion with unquenched orbital angular momentum. The treatment of the Hamiltonian is performed with the use of the irreducible tensor operator technique. The influence of the exchange and spin-orbital interactions on the magnetic properties of Fe2(III)Co(II) clusters is revealed. The result of calculations in the framework of the isotropic model (neglecting low-symmetry crystal field) are in a good agreement with the experimentally observed magnetization versus magnetic field, as well as with the unusual temperature behavior of the magnetic susceptibility of the powder samples (1.8–250 K). The states of the system are enumerated by the quantum number of the total angular momentum J, these hybrid states are combined from the sets of full and intermediate spins of Fe2Co triangle and orbital angular momentum of Co(II), (with fictious L=1). The best fit to the experimental data is achieved for the exchange parameters JFe-Fe=−87.5 cm−1, JFe-Co=−17 cm−1 and orbital reduction factor for Co(II) ion κ=0.93. The ground state and the first excited one are found to possess total angular momentum J=1/2, Effective g-factors for the ground- and first-excited states calculated using the hybrid spin-orbital functions prove to be 4.3 and 0.75, that are distinctly different from the pure spin value 2 peculiar for the Heisenberg model. The comparison of the magnetic measurements with the theory clearly demonstrates the insufficiency of a simple Heisenberg spin-coupling model as evidenced by the discovered strong orbital magnetic contribution.

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

The irreducible tensor operator technique in R 3 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 D 4h 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.

High-Spin Heteronuclear Cyanometalates: g-Tensors and Magnetic Properties of CrIIINiII n (n=2,3,4,5,6) and Heptanuclear CrIIICuII 6 Compounds

Using the irreducible tensor operator technique we develop a computational approach to the problem of g-tensors for the heteronuclear polymetallic systems consisting of the arbitrary number of coupled paramagnetic ions. We deduce the effective g-values for the family of the heterometallic CrIIINiII n cyanometalates (n=2,3,4,5,6) possessing different topologies. For CrIIINiII 6 complexes a good fit to experimental μ(T) is achieved. For CrIIICuII 6 the local anisotropy of g for CuII ions is taken into account.

Magnetic properties of new Fe 6(triethanolaminate(3-)) 6 spin-clusters

We report the magnetic properties of three cyclic hexanuclear iron(III) spin-clusters ([Fe 6(tea) 6] · 6MeOH ( 1), [NaFe 6(tea) 6]ClO 4 · 2MeOH ( 2) and [LiFe 6(tea) 6]ClO 4 · 2MeOH ( 3), (tea: triethanolaminate(3-)). The intra-molecular exchange coupling between the Fe III-spins is determined by measurements of the magnetic susceptibility in the range 2–300 K. The zero-field splitting of the lowest cluster spin states ( E/k B < 200K) is studied by ESR (electron-spin resonance) at frequencies between 9.5 and 350 GHz. The level splitting is analyzed by the irreducible-tensor operator technique.

Heisenberg model of tetrameric manganese cluster of S 2 center of Photosystem II

The Heisenberg exchange model of the mixed-valence [Mn 3(III)Mn(IV)] and [Mn 3(IV)Mn(III)] clusters — the prototypes of the manganese active center of Photosystem II in the S 2 state — is developed. The proposed approach gives us the generalized solutions of the exchange problem. The exchange model H A = −2[ J 12 s 1 s 2+ J a( s 1 s 3+ s 2 s 3)+ J b( s 1 s 4+ s 2 s 4)+ J 34 s 3 s 4] describes the system with fixed intermediate spin S 12, the states with different S 34 are mixed. The Heisenberg model H B=−2[ J 12 s 1 s 2+ J α( s 1 s 3+ s 1 s 4)+ J β( s 2 s 3+ s 2 s 4) + J 34 s 3 s 4] describes the manganese clusters with mixing of the spin states with different intermediate spin S 12 ( S and S 34 are fixed). The two Heisenberg models H A and H B have analytical solutions for the exchange energy for the states with S= 1 2 . These analytical solutions give the results of different Kambe models with fixed intermediate spins in the particular cases. The exchange matrices ( S= 1 2 ) for the general exchange model H =−2( J 12 s 1 s 2+ J 13 s 1 s 3+ J 23 s 2 s 3+ J 14 s 2 s 4+ J 34 s 3 s 4) were obtained by using the technique of irreducible tensor operators. The general model explains the splittings and the transformation of spin levels of complex manganese clusters: trapezoid, T-type, bent structures, and strongly distorted tetrahedron models, which do not allow the Kambe solutions of the exchange problem and also the solutions in the generalized models H A and H B. The states of the general model are characterized by the net spin S and by the exchange mixture of the intermediate spins. Double exchange leads to the formation of the exchange-resonance states.

Transition metal ions in crystals: A refined treatment and deduction of coulomb and exchange interaction constants

Making use of Racah's irreducible tensor operator technique a refined treatment for the 3 d-transition metal ions in crystals has been presented. The refined theory has been applied to analyse the recently observed fine optical spectra of MgF 2:Co 2+ and MgF 2:Mn 2+. Important differences have been noticed in the assignments of the optical lines obtained from the present analysis and from the conventional three-parameter theory. First numerical values of the exchange and Coulomb integrals in these systems have also been given.

Coherent active spectroscopy of optically isotropic media formed by randomly oriented elements

The technique of irreducible tensor operators is used to obtain an accurate expression for the macroscopic cubic susceptibility tensor of a homogeneous isotropic medium formed by randomly oriented molecules (liquids and gases). The result is expressed in terms of the invariants of the molecular hyperpolarizability tensor. Purely rotational and vibrational-rotational molecular transitions are considered. The conditions for maximum intensity of the coherent active Raman (anti-Stokes) spectroscopy (CARS) signal are determined for a finely dispersed medium (crystalline powder). A separate analysis is made of CARS of a noncentrosymmetric crystal powder in which suppression of the cascade process by the quadratic nonlinearity is possible. The first experimental observation of an ARS signal from a finely dispersed sample is reported.