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.
Electronic structure calculations on the low dimensional spin-1/2 compound TiOCl were performed at several pressures in the orthorhombic phase, finding that the structure is quasi-one-dimensional. The Ti3+ (d1) ions have one t2g orbital occupied (dyz
) with a large hopping integral along the b direction of the crystal. The most important magnetic coupling is Ti-Ti along the b axis. The transition temperature (Tc) has a linear evolution with pressure, and at about 10 GPa this Tc is close to room temperature, leading to a room temperature spin-Peierls insulator-insulator transition, with an important reduction of the charge gap in agreement with the experiment. On the high-pressure monoclinic phase, TiOCl presents two possible dimerized structures, with a long or short dimerization. Long dimerized state occurs above 15 GPa, and below this pressure the short dimerized structure is the more stable phase.
On the basis of experimental thermoelectric power results and ab initio calculations, we propose that a metal-insulator transition takes place at high pressure (approximately 6 GPa) in MgV_2O_4.
