We report a series of electronic structure calculations for CrN using different exchange correlation potentials: PBE, LDA+$U$, the Tran-Blaha modified Becke-Johnson, and hybrid functionals. In every case, our calculations show that the onset of magne
tism in CrN should be accompanied by a gap opening. The experimentally found antiferromagnetic order always leads to an insulating behavior. Our results give further evidence that the Tran-Blaha functional is very useful for treating the electronic structure of correlated semiconductors allowing a parameter free description of the system. Hybrid functionals are also well capable of describing the electronic structure of CrN. The analysis of the system is complemented with our calculations of the thermopower that are in agreement with the experimental data.
The low and high-temperature phases of V$_4$O$_7$ have been studied by textit{ab initio} calculations. At high temperature, all V atoms are electronically equivalent and the material is metallic. Charge and orbital ordering, associated with the disto
rtions in the V pseudo-rutile chains, occur below the metal-insulator transition. Orbital ordering in the low-temperature phase, different in V$^{3+}$ and V$^{4+}$ chains, allows to explain the distortion pattern in the insulating phase of V$_4$O$_7$. The in-chain magnetic couplings in the low-temperature phase turn out to be antiferromagnetic, but very different in the various V$^{4+}$ and V$^{3+}$ bonds. The V$^{4+}$ dimers formed below the transition temperature form spin singlets, but V$^{3+}$ ions, despite dimerization, apparently participate in magnetic ordering.
