Pseudo-spin operator

As it follows from (12), ground-state averages of orbital pseudo spin operators determine the numeric values of Q (i) and, thereby, the respective symmetry-adapted distortion of every elementary cell of the crystal. Instead of ferromagnetic , antiferromagnetic , etc., the appropriate name to use is ferrodistortive , antiferrodistortive , etc. 

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]... 

Let the chain obey the periodical boundary conditions and let protons be strongly connected with heavy atoms, so that tunnel proton transitions along the hydrogen bond are negligible in comparison with the reorientation motion of A—H groups. This means that only one proton is localized near each heavy atom in the chain that is, equality + /z/ / i = 1 holds, which makes it possible to introduce the following pseudo-spin operators ... 

Section II.D above. The pseudo-spin operator Sf [see formulas (68) and (72)] corresponded to two possible projections of the coordinate of the ith proton onto the z axis. The order parameter obeyed the equation... 

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 NMR characterisation of polymeric systems requires first the search for the existence of networks. The observation of a time reversal effect, specific to residual spin-spin interactions, gives evidence for the presence of polymeric networks [3]. This property is reflected by so-called pseudo-solid spin-echoes formed by applying a suitable radiofrequency pulse sequence that results in a rotation of the spin operators (Figure 8.3). 

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

The OOA was not designed for and does not apply to temperature dependencies of any kind in JT crystals. In particular, one cannot expect a reasonable estimate of the temperature of phase transitions in crystal lattice (structural), electron orbital, and/or spin system. This follows from the partitioning procedure that includes averaging over vibrational degrees of freedom. One can see the same reason from another perspective. The pseudo spin of a JT site, as the basic concept used in the OOA, operates in the basis of degenerate ground state wave functions. Excited vibronic states are beyond the pseudo spin setup. Therefore, in the OOA, by its very definition, temperature population of excited states does not make sense. 

This Hamiltonian is invariant under the following two-symmetry operations [16, 23] (1) The global four-fold symmetry the pseudo-spins at all sites and the crystal lattice are rotated by njl, simultaneously, with respect to the y axis. (2) The local symmetry at each column and row the z (x)-component of all pseudo-spins at each... 

H Hamiltonian operator S pseudo-spin Bloch vector... 

Note that this has identical spin operators to those involving in Equation [31]. This means that the spectra of anisotropic systems depend on (2D y + / ), and that these two interactions cannot be determined separately from the spectra. For this reason, Jfj is often referred to a pseudo-dipolar coupling. 

Here, S is the pseudo-spin of the system, and F is the KS Fock operator up to first order in the external field or the nuclear spin magnetic perturbation. LWA is suitable for Kramers doublets. Assuming no spatial degeneracy, a pair of degenerate Kramers orbitals is initially calculated from SO DFT by assigning equal occupations of 0.5 to two frontier orbitals. One then chooses one of them, (p, and constructs from its real and imaginary parts of the spin a and l3 components the Kramers pair d>i, 2 as... 

It is obvious that the projection operators for the different species have different numbers of terms in them. The HON species have 12 terms (3 x 2 ) while the A2B-type species have four terms, and the HDT+ isotopomer has only two terms. This results in different sizes of the spin-projected basis sets, and for this reason the properties obtained in this work are not precisely comparable between the A3, A2B, and ABC systems, although a very good idea of the trends may be obtained from the data in Table XVI. While all of the above are given in terms of the original particles, it should be noted that the permutations used in the internal particle basis functions are pseudo -permutations induced by the permutations on real particles. 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

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. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Teichteil et al.41 fit a spin-orbit pseudo-operator such that its action on a pseudo-orbital optimally reproduces the effect of the true spin-orbit operator on the corresponding all-electron orbital. Ermler, Ross, Christiansen, and co-workers42 9 and Titov and Mosyagin50 define a spin-orbit operator as the difference between the and j dependent relativistic effective pseudopotentials (REPs)51... 

A detailed consideration of all these steps in the Overhauser triplet mechanism operating in the photoreduction of qulnone systems has been given by Adrian et al. (10). With some reasonable approximations (a pseudo-first-order reaction of the polarized radicals and steady-state conditions), it is possible to show that the rate of production of nuclear spin polarized products is... 

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