A modified spin-orbital superexchange theory is developed to derive the effective spin Hamiltonian ${\mathcal{H}}_S$ governing the exchange interactions between pseudospins-$1/2$ on a square lattice, as occurs, for instance, in planar iridium oxide ${\mathrm{Ba}}_2{\mathrm{IrO}}_4$. The procedure begins with the underlying three-orbital Hubbard model for the $t_{2g}$ orbital manifold, which is first unitarily transformed to account for the strong spin-orbit interaction in all orders of the coupling constant $\lambda$, and then an explicit perturbative expansion with respect to the transformed hopping integrals and the Hund's interaction is used. To place this novel analytical approach into a general context, a brief outline is given for the computational superexchange formalism based on the post-Hartree-Fock calculations in the framework of the cluster quantum chemistry methods. A complementary computational approach, based on the density functional theory with strong spin-orbit coupling and aimed at the extraction of a single-bond exchange parameter of the corresponding pseudospin model, is also discussed in some detail.

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• Transition metal oxides
• Electron-correlation calculations
• Hubbard model
• Quantum chemistry methods
• Spin lattice models