Electronic structure and quadrupole interactions in promising cathode materials Na$_{x}M_{y}$(MoO$_{4}$)$_{3}$, $M$ = Mn, Fe, Co, and Ni

The electronic structure and the magnetic properties of molybdates Na$_{x}M_{y}$(MoO$_{4}$)$_{3}$, ($M$ = Mn, Fe, Co, and Ni) which are promising materials for sodium batteries have been studied in the framework of the density functional theory with the GGA and GGA+ U approximations for the first time. The calculations show that all the compounds are insulators. An important role of the correlation effects, provided by the on-site Coulomb interactions, was established in the formation of the band gap in these compounds. The quadrupole constants of $^{23}$Na nuclei are calculated in the nonmagnetic and ferromagnetic states within the GGA and GGA+ U approaches. It is shown that the quadrupole frequencies for nonequivalent crystallographic positions of sodium are in different frequency ranges, which allows to study the diffusion of sodium in these compounds by the Nuclear Magnetic Resonance method.