  We investigate the electronic and structural properties of CuO, which shows
significant deviations from the trends obeyed by other transition-metal
monoxides. Using an extended Hubbard corrective functional, we uncover an
orbitally ordered insulating ground state for the cubic phase of this material,
which was expected but never found before. This insulating state results from a
fine balance between the tendency of Cu to complete its d-shell and Hund's rule
magnetism. Starting from the ground state for the cubic phase, we also study
tetragonal distortions of the unit cell (recently reported in experiments), the
consequent electronic reorganizations and identify the equilibrium structure.
Our calculations reveal an unexpected richness of possible magnetic and orbital
orders, relatively close in energy to the ground state, whose stability depends
on the sign and entity of distortion.
