First-principles determination of charge and orbital interactions in Fe[subscript 3]O[subscript]4

The interactions between charge and orbitally ordered d electrons are important in many transition-metal oxides. We propose an effective energy model for such interactions, parameterized with density-functional theory plus U calculations, so that energy contributions of both electronic and lattice origin can be simultaneously accounted for. The model is applied to the low-temperature phase of magnetite, for which we propose a ground-state structure. The effective interactions on the B lattice of Fe[subscript 3]O[subscript]4 can be interpreted in terms of electrostatics and short-range Kugel-Khomskii exchange coupling. The frustration between optimal charge and orbital orderings leads to a complex energy landscape whereby the supercell for the charge ordering, orbital ordering, and ionic displacements can all be different.