of the density of states contribution from each atom reveals that the intercalated atom contributes very little if at all to the overall density of states. This is expected due to its complete ionisation, as the single valence electron of the intercalated atom is contributed to the W$-$O framework [@186]. All compounds have the Fermi level located well into the conduction band, rendering them metallic. The shape of the band structure does not change as atoms are intercalated. This is also noted in other literature [@56; @111]. Also, the magnitude of the band gap and the position of the Fermi level is relatively constant - even for those compounds which are known to be unstable (cubic potassium, rubidium and caesium bronzes).

![\[fig:dos\_cub\_MWO3\] Calculated density of states for cubic tungsten bronzes, MWO$_3$, near the Fermi level: (a) WO$_3$, (b) HWO$_3$, (c) LiWO$_3$, (d) NaWO$_3$, (e) KWO$_3$, (f) RbWO$_3$, (g) CsWO$_3$. The Fermi level is indicated in each case.](dos_cub_MWO3.eps){width="85mm"}

We have also derived the band structure curves along lines of symmetry in cubic WO$_3$ and cubic NaWO$_3$. These are given in Figure \[fig:Na-WO3-band-comp\]. These results agree extremely well with the calculated band structures of both Bullett [@186] and Cora et. al. [@111], 