Vanadium dioxide (VO$_2$) undergoes a metal-insulator transition (MIT) at 340 K with the structural change between tetragonal and monoclinic crystals as the temperature is lowered. The conductivity $sigma$ drops at MIT by four orders of magnitude. Th
e low-temperature monoclinic phase is known to have a lower ground-state energy. The existence of a $k$-vector ${boldsymbol k}$ is prerequisite for the conduction since the ${boldsymbol k}$ appears in the semiclassical equation of motion for the conduction electron (wave packet). Each wave packet is, by assumption, composed of the plane waves proceeding in the ${boldsymbol k}$ direction perpendicular to the plane. The tetragonal (VO$_2$)$_3$ unit cells are periodic along the crystals $x$-, $y$-, and z-axes, and hence there are three-dimensional $k$-vectors. The periodicity using the non-orthogonal bases does not legitimize the electron dynamics in solids. There are one-dimensional ${boldsymbol k}$ along the c-axis for a monoclinic crystal. We believe this decrease in the dimensionality of the $k$-vectors is the cause of the conductivity drop. Triclinic and trigonal (rhombohedral) crystals have no $k$-vectors, and hence they must be insulators. The majority carriers in graphite are electrons, which is shown by using an orthogonal unit cell for the hexagonal lattice.
