a great deal in describing the bronze systems. A comparison of the density of states for cubic and hexagonal WO$_3$ is shown in Figures \[fig:WO3\_dos\_full\] and \[fig:WO3\_dos\_zoom\]. The lowest band, situated at -18 to -16 eV, corresponds to the oxygen *2s* orbitals. This band is present in all of the tungsten-oxide-based systems studied to date and always occurs at the same energy regardless of the structure or the presence of intercalated atoms or molecules.

![\[fig:WO3\_dos\_full\] The density of states as calculated for cubic (a) and hexagonal (b) WO$_3$.](WO3_dos_full.eps){width="85mm"}

![\[fig:WO3\_dos\_zoom\] A detailed view of the conduction band of cubic (a) and hexagonal (b) WO$_3$, from figure \[fig:WO3\_dos\_full\].](WO3_dos_zoom.eps){width="85mm"}

The broad valence band, from -7 to 0 eV, is comprised mainly of oxygen *2p* orbitals. There is a small tungsten *5d* component but this is negligible above -2 eV. The valence band cut-off is sharp at 0 eV and coincides in both the cubic and hexagonal case with the Fermi level, rendering the oxide materials semiconducting. The conduction band, detailed in Figure 8, lies from 0.5 to 5 eV in both the cubic and hexagonal cases. In the cubic case it is solely comprised of tungsten *5d* orbitals; however in the hexagonal case 