doped electrons. Figure \[fig:WO3-x\_dos\] shows the density of states for the WO$_{3-x}$ system in the region near the Fermi level. The overall spectra share the same features previously detailed for WO$_3$: the oxygen *2s* band near -18 eV; the broad valence band, comprised mainly of oxygen *2p* orbitals, from -7 to 0 eV; and the conduction band, consisting solely of tungsten *5d* orbitals, lying from roughly 0.5 to 5 eV. Naturally the sub-stoichiometric systems appear more ‘jagged’ than the parent WO$_3$ compound, due in part to the breaking of symmetry, rendering each atom non-equivalent to others within the cell, and causing its contribution to be slightly different.

![\[fig:WO3-x\_dos\] Density of states for the WO$_{3-x}$ system, all with the valence band set at zero. Arrows show the position of the Fermi level. Inset: Position of the Fermi level relative to the top of the valence band.](WO3-x_dos.eps){width="85mm"}

The inset of Figure \[fig:WO3-x\_dos\] shows the progression of the Fermi level into the conduction band. There is a sharp jump between $x=0.167$ and 0.2 as the Fermi level moves up into the conduction band - not a gradual transition as in the case of the sodium bronzes Na$_x$WO$_3$. While the $x=0.167$ compound has the 