Antisymmetric Exchange in Antiferromagnetic Materials of Rhombohedral Structures

Carriers transferrings, determined by wave functions and energy levels of i j magnetic and oxygen ions, which are determined by rhombohedral oxygen crystal fields and their particularities, are in discussions for identification of antisymmetric, Dzyloshinskii-Moria exchange, \textbf{D}$_{z}${\{}\textbf{S}$_{ix}$\textbf{S}$_{jy}-$\textbf{S}$_{iy}$\textbf{S}$_{jx }${\}}, taking into account Hubbard Hamiltonians, including spin-orbit interactions. Wave functions symmetry dependence are described by, depending on trigonal symmetry $\alpha $, $\beta $ coefficients in wave functions of energy levels of magnetic ions. Particularities of i j oxygen crystal fields are concerned with rotations of j fields at angles 60 degrees with respect to i fields. Taking spin-orbit, transferrings as perturbations, exchange symmetric,$_{ }$antisymmetric parts of spin Hamiltonians are H$_{ex}=\Sigma _{i,j}$J$_{i,j}$(\textbf{S}$_{i}\cdot $\textbf{S}$_{j})+\Sigma _{i,j}$\textbf{D}$_{i,j}$[\textbf{S}$_{i}\times $\textbf{S}$_{j}$], where J$_{i,j}$ and \textbf{D}$_{i,j}$ are determined by carriers transferrings, kinetic energies, Coulomb interactions, magnetic and oxygen energy levels. As examples, after some assumptions \textbf{D}$_{ij }$= J$\cdot (-\lambda )\cdot $i$\cdot ${\{}$\Sigma _{m}\langle \psi _{im }$/\textbf{L}$_{i}$/$\psi _{i0}\rangle ^{\ast }$/($\varepsilon _{im}-\varepsilon _{i0})\cdot $(t$_{im,kn}$/t$_{i0,kn})$ - $\Sigma _{m}\langle \psi _{jm}$/\textbf{L}$_{j}$/$\psi _{j0}\rangle ^{\ast }$/($\varepsilon _{jm}-\varepsilon _{j0})\cdot $(t$_{kn,jm}$/t$_{kn,j0})${\}}, \textbf{D}$_{z}\sim $J$\cdot \alpha \beta \cdot ${\{}$\lambda $/($\varepsilon _{m}-\varepsilon _{0})${\}}$\cdot ${\{}(t$_{im,kn}$/t$_{i0,kn}-$t$_{kn,jm}$/t$_{kn,j0})${\}}. Energy levels and Pauli requirements, determine depending on spin-orbit interactions, carriers transferrings between magnetic -oxygen -magnetic ions, which determine vectors \textbf{D}$_{z}$, oriented, according to trigonal symmetry, along trigonal - z axis.