Band structure of tungsten oxide W$_{20}$O$_{58}$ with ideal octahedra

The band structure, density of states, and the Fermi surface of a tungsten oxide WO$_{2.9}$ with idealized crystal structure (ideal octahedra WO$_6$ creating a "square lattice") is obtained within the density functional theory in the generalized gradient approximation. Because of the oxygen vacancies ordering this system is equivalent to the compound W$_{20}$O$_{58}$ (Magn\'{e}li phase), which has 78 atoms in unit cell. We show that 5$d$-orbitals of tungsten atoms located immediately around the voids in the zigzag chains of edge-sharing octahedra give the dominant contribution near the Fermi level. These particular tungsten atoms are responsible of a low-energy properties of the system.