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.

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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.

We present periodic "DFT+U" studies of single oxygen vacancies on the CeO 2 (110) surface using a number of different supercells, finding a range of different local minimum structures for the vacancy and its two accompanying Ce(III) ions. We find three different geometrical structures in combination with a variety of different Ce(III) localization patterns, several of which have not been studied before. The desired trapping of electrons was achieved in a two-stage optimization procedure. We find that the surface oxygen nearest to the vacancy either moves within the plane towards the vacancy, or rises out of the surface into either a symmetric or an unsymmetric bridge structure. Results are shown in seven slab geometry supercells, p(2 × 1), p(2 × 2), p(2 × 3), p(3 × 2), p(2 × 4), p(4 × 2), and p(3 × 3), and indicate that the choice of supercell can affect the results qualitatively and quantitatively. An unsymmetric bridge structure with one nearest and one next-nearest neighbour Ce(III) ion (a combination of localizations not previously found) is the ground state in all (but one) of the supercells studied here, and the relative stability of other structures depends strongly on supercell size. Within any one supercell the formation energies of the different vacancy structures differ by up to 0.5 eV, but the same structure can vary by up to ∼1 eV between supercells. Furthermore, finite size scaling suggests that the remaining errors (compared to still larger supercells) can also be ∼1 eV for some vacancy structures.

The errors arising in ab initio density functional theory studies of semiconductor point defects using the supercell approximation are analyzed. It is demonstrated that ͑a͒ the leading finite size errors are inverse linear and inverse cubic in the supercell size and ͑b͒ finite size scaling over a series of supercells gives reliable isolated charged defect formation energies to around ±0.05 eV. The scaled results are used to test three correction methods. The Makov-Payne method is insufficient, but combined with the scaling parameters yields an ab initio dielectric constant of 11.6± 4.1 for InP. ⌫ point corrections for defect level dispersion are completely incorrect, even for shallow levels, but realigning the total potential in real-space between defect and bulk cells actually corrects the electrostatic defect-defect interaction errors as well. Isolated defect energies to ±0.1 eV are then obtained using a 64 atom supercell, though this does not improve for larger cells. Finally, finite size scaling of known dopant levels shows how to treat the band gap problem: in ഛ200 atom supercells with no corrections, continuing to consider levels into the theoretical conduction band ͑extended gap͒ comes closest to experiment. However, for larger cells or when supercell approximation errors are removed, a scissors scheme stretching the theoretical band gap onto the experimental one is in fact correct.

Abstract. Reliable calculations of defect properties may be obtained with density functional theory using the supercell approximation. We systematically review the known sources of error and suggest how to perform calculations of defect properties in order to minimize errors. We argue that any analytical error-correction scheme relying on electrostatic considerations is not appropriate to derive reliable defect formation energies, especially not for relaxed geometries. Instead we propose finite size scaling of the calculated defect formation energies, and compare the use of this with both fully converged and "Gamma" (Γ) point only k-point integration. We give a recipe for practical DFT calculations which will help to obtain reliable defect formation energies and demonstrate it using examples from III-V semiconductors.

In this paper we evaluate the performance of density functional theory with the B3LYP functional for calculations on ceria ͑CeO 2 ͒ and cerium sesquioxide ͑Ce 2 O 3 ͒. We demonstrate that B3LYP is able to describe CeO 2 and Ce 2 O 3 reasonably well. When compared to other functionals, B3LYP performs slightly better than the hybrid functional PBE0 for the electronic properties but slightly worse for the structural properties, although neither performs as well as LDA+ U͑U=6 eV͒ or PBE+ U͑U=5 eV͒. We also make an extensive comparison of atomic basis sets suitable for periodic calculations of these cerium oxides. Here we conclude that there is currently only one type of cerium basis set available in the literature that is able to give a reasonable description of the electronic structure of both CeO 2 and Ce 2 O 3 . These basis sets are based on a 28 electron effective core potential ͑ECP͒ and 30 electrons are attributed to the valence space of cerium. Basis sets based on 46 electron ECPs fail for these materials.