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abstract: |
    Tungsten trioxide adopts a variety of structures which can be intercalated with charged species to alter the electronic properties, thus forming ‘tungsten bronzes’. Similar optical effects are observed upon removing oxygen from WO$_3$, although the electronic properties are slightly different. Here we present a computational study of cubic and hexagonal alkali bronzes and examine the effects on cell size and band structure as the size of the intercalated ion is increased. With the exception of hydrogen (which is predicted to be unstable as an intercalate), the behaviour of the bronzes are relatively consistent. NaWO$_3$ is the most stable of the cubic systems, although in the hexagonal system the larger ions are more stable. The band structures are identical, with the intercalated atom donating its single electron to the tungsten *5d* valence band. Next, this was extended to a study of fractional doping in the Na$_x$WO$_3$ system ($0
    \leq x \leq 1$). A linear variation in cell parameter, and a systematic change in the position of the Fermi level up into the valence band was observed with increasing $x$. In the underdoped WO$_{3-x}$ system however, the Fermi level undergoes a sudden jump into the conduction band at around 