Orbital and charge ordering in ${\mathrm{Pr}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Ca}}_{\mathrm{x}}$${\mathrm{MnO}}_3$ (x=0 and 0.5) from the ab initio calculations

The electronic structure of the doped manganites ${\mathrm{Pr}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Ca}}_{\mathrm{x}}$ ${\mathrm{MnO}}_3$ was calculated by the LSDA+U method which takes into account the local Coulomb interaction between d electrons of transition-metal ions. In contrast to the standard local-spin-density approximation, no Jahn-Teller lattice distortions are needed to reproduce the experimentally observed orbital ordering in the undoped ${\mathrm{PrMnO}}_3$. The correct ground state, a charge and orbital ordered antiferromagnetic insulator, was obtained for ${\mathrm{Pr}}_{1\mathrm{/}2}$ ${\mathrm{Ca}}_{1\mathrm{/}2}$ ${\mathrm{MnO}}_3$. The results are in a good agreement with the neutron-diffraction data.