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The pyrochlore oxides $A_2B_2$O$_7$ exhibit a complex interplay between geometrical frustration, electronic correlations, and spin-orbit coupling, due to the lattice structure and active charge, spin, and orbital degrees of freedom. Understanding the properties of these materials is a theoretical chalenge, because their intricate nature depends on material-specific details and quantum many-body effects. Here we review our recent studies based on first-principles calculations and quantum many-body theories for 4$d$ and 5$d$ pyrochlore oxides with $B$=Mo, Os, and Ir. In these studies, the spin-orbit coupling and local electron correlations are treated within the LDA+$U$ and LDA+dynamical mean-field theory formalisms. We also discuss the technical aspects of these calculations.
The self-interaction-corrected local-spin-density approximation is used to describe the electronic structure of dioxides, REO$_2$, and sesquioxides, RE$_2$O$_3$, for the rare earths, RE=Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy and Ho. The valencies of the
The interplay of spin-orbit interactions and Coulomb correlations has become a hot topic in condensed matter theory. Here, we review recent advances in dynamical mean-field theory-based electronic structure calculations for iridates and rhodates. We
Present work demonstrates the formation of spin-orbital polarons in electron doped copper oxides, that arise due to doping-induced polarisation of the oxygen orbitals in the CuO$_2$ planes. The concept of such polarons is fundamentally different from
The ground state electronic structures of the actinide oxides AO, A2O3 and AO2 (A=U, Np, Pu, Am, Cm, Bk, Cf) are determined from first-principles calculations, using the self-interaction corrected local spin-density (SIC-LSD) approximation. Emphasis
Fully relativistic first-principles electronic structure calculations based on a noncollinear local spin density approximation (LSDA) are performed for pyrochlore iridates Y$_2$Ir$_2$O$_7$ and Pr$_2$Ir$_2$O$_7$. The all-in, all-out antiferromagnetic