We examine the real space structure and the electronic structure ͑particularly Ce4f electron localization͒ of oxygen vacancies in CeO 2 ͑ceria͒ as a function of U in density functional theory studies with the rotationally invariant forms of the LDA+ U and GGA+ U functionals. The four nearest neighbor Ce ions always relax outwards, with those not carrying localized Ce4f charge moving furthest. Several quantification schemes show that the charge starts to become localized at U Ϸ 3 eV and that the degree of localization reaches a maximum at ϳ6 eV for LDA+ U or at ϳ5.5 eV for GGA+ U. For higher U it decreases rapidly as charge is transferred onto second neighbor O ions and beyond. The localization is never into atomic corelike states; at maximum localization about 80-90% of the Ce4f charge is located on the two nearest neighboring Ce ions. However, if we look at the total atomic charge we find that the two ions only make a net gain of ͑0.2-0.4͒e each, so localization is actually very incomplete, with localization of Ce4f electrons coming at the expense of moving other electrons off the Ce ions. We have also revisited some properties of defect-free ceria and find that with LDA+ U the crystal structure is actually best described with U =3-4 eV, while the experimental band structure is obtained with U =7-8 eV. ͑For GGA+ U the lattice parameters worsen for U Ͼ 0 eV, but the band structure is similar to LDA + U.͒ The best overall choice is U Ϸ 6 eV with LDA+ U and Ϸ5.5 eV for GGA+ U, since the localization is most important, but a consistent choice for both CeO 2 and Ce 2 O 3 , with and without vacancies, is hard to find.
The relaxed and unrelaxed formation energies of neutral antisites and interstitial defects in InP are calculated using ab initio density functional theory and simple cubic supercells of up to 512 atoms. The finite-size errors in the formation energies of all the neutral defects arising from the supercell approximation are examined and corrected for using finite-size scaling methods, which are shown to be a very promising approach to the problem. Elastic errors scale linearly, while the errors arising from charge multipole interactions between the defect and its images in the periodic boundary conditions have a linear plus a higher order term, for which a cubic provides the best fit. These latter errors are shown to be significant even for neutral defects. Instances are also presented where even the 512 atom supercell is not sufficiently converged. Instead, physically relevant results can be obtained only by finite-size scaling the results of calculations in several supercells, up to and including the 512 atom cell and in extreme cases possibly even including the 1000 atom supercell.
It is proposed that the observation of orbital ordering in manganite materials should be possible at the L II and L III edges of manganese using x-ray resonant scattering. If performed, dipole selection rules would make the measurements much more direct than the disputed observations at the manganese K edge. They would yield specific information about the type and mechanism of the ordering not available at the K edge, as well as permitting the effects of orbital ordering and Jahn-Teller ordering to be detected and distinguished from one another. Predictions are presented based on atomic multiplet calculations, indicating distinctive dependence on energy, as well as on polarization and on the azimuthal angle around the scattering vector.
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