Superexchange interaction in insulating manganites $R_{1-x}{A}_x{\mathrm{MnO}}_3$ $(x=0,0.5)$

Manganites with general formulas $R{\mathrm{MnO}}_3$ and $R_{0.5}{A}_{0.5}{\mathrm{MnO}}_3,$ where $R^{3+}$ is a rare-earth ion, and $A^{2+}$ is an alkaline-earth ion, are known as insulating pure or charge ordered compounds at low temperatures with some R and A. In the present work, it is shown how the charge ordering and crystal structure form the orbital structure, which strongly effects the superexchange interaction. An explicit space group and the magnitudes of the distortions of each crystal were used in the calculations. We assume an ideal insulating state of the crystals. The model presented in this work allows one to explain the $\mathrm{CE}$ and A magnetic structures in terms of charge and orbital structures using quantitative estimations of the exchange parameters. The model also allows one to find easy axis and noncollinear components of a multisublattice magnetic structure of manganites.