General superexchange Hamiltonians for magnetic and orbital physics in eg and t2g systems

Material-specific super-exchange Hamiltonians are the key to studying spin and orbital physics in strongly correlated materials. Recently, via an irreducible-tensor operator representation, we derived the orbital superexchange Hamiltonian for ${t}_{2g}^{1}$ perovskites and successfully used it, in combination with many-body approaches, to explain orbital physics in these systems. Here, we generalize our method to ${e}_{g}^{n}$ and ${t}_{2g}^{n}$ systems at arbitrary integer filling $n$, including both spin and orbital interactions. The approach is suitable for numerical implementations based on ab initio hopping parameters and realistic screened Coulomb interactions and allows for a systematic exploration of superexchange energy surfaces in a realistic context.