Pseudo-spin model

The interplay between the local Pb polarization and B-site ions may be described by a simple model. As discussed above the magnitude of the local off-centering of Pb " ions is always about 0.5 A, and does not depend on the environment. Thus it could be described as a pseudo-spin, and justifies the pseudo-spin model with the dipolar-dipolar interaction and the local anisotropy energy ... 

We have demonstrated in this contribution that the structural order-disorder transitions in the KDP family are by far more complex than a pseudo-spin model, for example, could describe. Geometrical constraints and couphng to the soft mode, as well as cluster formation of nanometric size, indicate that here the conventional picture of cooperative phenomena has to be revised. 

The three-state extended pseudo spin model is equivalent to the Potts model well-known in solid state physics. Its application to JT structural phase transitions was developed by Hock et al. [63]. Also, as a useful tool, it was mentioned by Kugel and Khomskii [9]. It was applied to hexagonal perovskites by Crama and Maaskant [64]. It has multiple applications to layered manganites (e.g., see [65]). 

Thus starting from the simplest pseudo-spin model of proton dynamics in the hydrogen bond, we have studied a possibility of spontaneous tunnel oscillations of the polarization of a short hydrogen-bonded chain. The phenomenon can be affected by two reasons (a) the coherent motion of protons along the hydrogen bonds and (b) the coherent motion of protons around heavy backbone atoms. 

Furthermore, it was found that it was necessary to consider both the reorientation of the ZFS principal axis system and the changes in magnitude, to describe the fluctuations of the ZFS, These observations were very useful for the improvement of the theoretical models of the electron spin relaxation. In particular the so called pseudo-rotation model is put in serious doubt. 

We start from the model Hamiltonian to describe the strongly correlated electron system with the eg orbital degree of freedom. A system of the present interest is a Mott insulator where one electron occupies one of the doubly degenerate orbitals at each site in a simple cubic lattice. The doubly degenerate orbital degree of freedom is represented by the pseudo-spin operator with an amplitude of 1/2 defined by... 

In this section, we introduce another interesting orbital model with the frustration effect, termed the honeycomb lattice orbital model. We consider the model where the doubly degenerate orbitals, e.g. the d i y7. and d y orbitals, which are described by the pseudo-spin operator T,, are located at each site in a two dimensional honeycomb lattice. The explicit form of the model Hamiltonian is given by... 

The second term is —3JN/16, when Tj is a two-dimensional classical spin, and is —3JN/% in the quantum-spin case. A total number of sites is N. This model is proposed as a orbital state for the layered iron oxide, [5, 6, 26], and is also recently proposed in study of the optical lattice [27-29]. A similar orbital model in a honeycomb lattice termed the Kitaev model is recently well examined. [30,31] Let us introduce the Fourier transformation for the orbital pseudo-spin operator. The Hamiltonian (15) is represented in the momentum space, [5,27]... 

The model proposed by Stasyuk et al. [135] describes the chain of hydrogen bonds connecting by ionic groups. The model is a development of their previous pseudo-spins reduced basis model [138], which took into account only the motion of a proton along the hydrogen bond. Reference 135 discusses the orientational degrees of freedom, which make it possible to include rotations... 

More recently, some states obtained within the above Blume-Capel (pseudo spin 5 = 1) model have been studied using a more realistic treatment of the Hubbard model which takes into account charge transfer and then does not fix an ad hoc value of the magnetic moments (Pinettes and Lacroix 1993b). This model confirms the results obtained with the pseudo spin 5 = 1 model, in particular it shows that the mixed phase can be stable... 

Since Holm and coworkers have previously shown that two different types of structure are found for the [AFe-AS]" " core with different ground state spin multiplicities (63), it is conceivable that an imposed pseudo-three-fold distortion could result in the S = 5/2 or 7/2 ground state found for oxidized P clusters. Ligands such as 15 and 16 will assist in exploring this possibility in synthetic model systems. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

The first coordination sphere of acido-pentamine [Cr(NH3)5X]2+ complexes has C4v symmetry and this leads to a tetragonal resolution of the 2Eg state into 2At and 1Bl components. The classical ligand field model predicts that these components will be virtually degenerate. This is based on a combination of pseudo-spherical and quasi-spin selection rules of the shell and will be discussed later on in Sect. 5.2. At present we welcome this example of a pseudo-degeneracy as another opportunity to observe fine details of the interelectronic repulsion interaction which have the proper anisotropy to induce a splitting of the 2Eg term. 

D. Maynau, P. Durand, J. P. Duadey, and J. P. Malrieu, Phys. Rep. A, 28, 3193 (1983). Direct Determination of Effective-Hamiltonians by Wave-Operator Methods. 2. Application to Effective-Spin Interactions in -Electron Systems. P. Durand and J. P. Malrieu, in Advances in Chemical Physics (Ah Initio Methods in Quantum Chemistry—I), K. P. Lawley, Ed., Wiley, New York, 1987, Vol. 67, pp. 321-412. Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling. 

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