Charge and orbital ordering in the low-temperature monoclinic structure of magnetite (Fe3O4) is investigated using the local spin density approximation with Coulomb interaction correction method. While the difference between t(2g) minority occupancies of Fe(2+)(B) and Fe(3+)(B) cations is large and gives direct evidence for charge ordering, the screening is so effective that the total 3d charge disproportion is rather small. The charge order has a pronounced [001] modulation, which is incompatible with the Anderson criterion. The orbital order agrees with the Kugel-Khomskii theory.

The optical and magneto-optical ͑MO͒ spectra of Tm monochalcogenides are investigated theoretically from first principles, using the fully relativistic Dirac linear combination of muffin-tin orbitals band structure method. The electronic structure is obtained with the local-spin-density approximation ͑LSDA͒, as well as with the so-called LSDAϩU approach. In contrast to LSDA, where the stable solution in TmTe is a metal, the LSDAϩU gave an insulating ground state. LSDAϩU theory predicts the thulium ion in TmTe to be in an integer divalent state. It also shows a gradual decreasing of the energy gap with reducing of the lattice constant. LSDAϩU theoretical calculations produce a similar energy band structure in TmS and TmSe, with twelve 4 f bands fully occupied and hybridized with chalcogenide p states. The 14th f hole level was found to be completely unoccupied and well above the Fermi level and a hole 13th f level is partly occupied and pinned at the Fermi level. The occupation number of the 13th f level is equal to 0.12 and 0.27 in TmS and TmSe, respectively ͑valence 2.88ϩ and 2.73ϩ͒. Such an energy band structure of thulium monochalcogenides describes well their measured bremsstrahlung isochromat spectroscopy ͑BIS͒, and x-ray and ultraviolet photoemission spectra as well as the optical and MO spectra. The origin of the Kerr rotation realized in the compounds is examined.

Charge ordering in the low-temperature monoclinic structure of iron oxoborate ͑Fe 2 OBO 3 ͒ is investigated using the local spin density approximation ͑LSDA͒ + U method. While the difference between t 2g minority occupancies of Fe 2+ and Fe 3+ cations is large and gives direct evidence for charge ordering, the static "screening" is so effective that the total 3d charge separation is rather small. The occupied Fe 2+ and Fe 3+ cations are ordered alternately within the chain which is infinite along the a direction. The charge order obtained by LSDA+ U is consistent with observed enlargement of the ␤ angle. An analysis of the exchange interaction parameters demonstrates the predominance of the interribbon exchange interactions which determine the whole L-type ferrimagnetic spin structure.

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Charge ordering in the low-temperature triclinic structure of titanium oxide (Ti4O7) is investigated using the local density approximation (LDA)+U method. Although the total 3d charge separation is rather small, an orbital order parameter defined as the difference between t2g occupancies of Ti 3+ and Ti 4+ cations is large and gives direct evidence for charge ordering. Ti 4s and 4p states make a large contribution to the static "screening" of the total 3d charge difference. This effective charge screening leads to complete loss of the disproportionation between the charges at 3+ and 4+ Ti sites. The occupied t2g states of Ti 3+ cations are predominantly of dxy character and form a spin-singlet molecular orbital via strong direct antiferromagnetic exchange coupling between neighboring Ti(1) and Ti(3) sites, whereas the role of superexchange is found to be negligible.

The electronic structure of the low-temperature (LT) monoclinic magnetite, Fe 3 O 4 , is investigated using the local spin density approximation (LSDA) and the LSDA+U method. The selfconsistent charge ordered LSDA+U solution has a pronounced [001] charge density wave character.In addition, a minor [00 1 2 ] modulation in the phase of the charge order (CO) also occurs. While the existence of CO is evidenced by the large difference between the occupancies of the minority spin t 2g states of "2+" and "3+" Fe B cations, the total 3d charge disproportion is small, in accord with the valence-bond-sum analysis of structural data. Weak Fe orbital moments of ∼0.07µ B are obtained from relativistic calculations for the CO phase which is in good agreement with recent x-ray magnetic circular dichroism measurements. Optical, magneto-optical Kerr effect, and O Kedge x-ray absorption spectra calculated for the charge ordered LSDA+U solution are compared to corresponding LSDA spectra and to available experimental data. The reasonably good agreement between the theoretical and experimental spectra supports the relevance of the CO solution obtained for the monoclinic LT phase. The results of calculations of effective exchange coupling constants between Fe spin magnetic moments are also presented.

The experimental difficulties of observing Fe 2+ /Fe 3+ charge order in Fe 2 OBO 3 and Fe 3 O 4 are different and are considered separately. Charge order in Fe 3 O 4 has a complex arrangement but is long-range coherent, as evidenced by a recent structure determination and two resonant x-ray diffraction studies. Long-range charge order has not been observed crystallographically in Fe 2 OBO 3 , although there is substantial indirect evidence for Fe 2+ /Fe 3+ ordering over shorter distances. Further support is provided by LSDA+ U calculations on a doubled unit cell of Fe 2 OBO 3 , which shows a charge separation similar to that in Fe 3 O 4 , with a large t 2g subshell occupancy difference that is partially screened by Fe-O covalency.

The electronic structure, optical, and magneto-optical spectra of the strongly magnetocaloric Gd5(Si2Ge2) compound are investigated theoretically from first principles, using the fully relativistic Dirac linear muffin tin orbital band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA), as well as with the so-called LSDA+U approach. The Coulomb repulsion Ueff strongly influences the electronic structure of Gd5(Si2Ge2) and for the Gd3+ ions seven spin up 4f bands are fully occupied and situated at the bottom of Si and Ge s states, while the 4f hole levels are completely unoccupied and well above the Fermi level hybridized with Gd 5d states. LSDA+U theory predicts that the magneto-optical spectra are much more sensitive to the α→β phase transition in comparison with the regular nonmagnetic optical properties.