hexagonal bronzes (apart from hydrogen) there is a steady downward trend in the energy of formation as the size of the intercalated alkali metal ion increases. Therefore the larger ions form more stable bronzes than the smaller ones, which is also indicated in the literature [@15]. In the cubic system however, the energy drops to a point and then, beyond Na, increases in the case of the larger intercalates. This point coincides with the stability predicted by the Goldschmidt perovskite tolerance factor. For the stable compounds (WO$_3$, LiWO$_3$, and NaWO$_3$) there is a progressive decrease in the energy of formation. This supports experimental evidence that sodium may well be the most stable of the cubic bronzes, as it is the only one for which a fully intercalated compound has been reported [@hobbs].

Density of states
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For both the cubic and hexagonal systems other than hydrogen, the basic band structures of the bronzes are essentially identical to the parent oxide of the same phase. The only difference amongst them is the position of the Fermi level relative to the valence and conduction bands, which will be discussed later. Hence comparing the band structure of hexagonal and cubic WO$_3$ will aid 