Fermi level at zero (and therefore, still non-conducting), the band structure of the valence band is similar to that of the conducting species. It appears that there is a decrease in the density of states in the valence band, followed by the Fermi level being pushed up into the conduction band. The stoichiometry at which this insulating-conducting transition occurs is in good agreement with the literature value of WO$_{2.76}$, which coincides with the $\beta$-$\gamma$ structural phase transition [@365].

Organic-Inorganic Layered Hybrids
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Following on from the background studies of tungsten bronzes and the oxygen-deficient tungsten oxides, layered organic-inorganic hybrid compounds were studied. These structures consist of WO$_4$ layers (as in H$_2$WO$_4$ [@512]), connected via aliphatic (linear) alkyl diamines. These have been investigated experimentally by us elsewhere [@us1]. Three different length alkyl amines were used in the calculations, with two, four and six carbons. The input structure of the hybrid systems is the most conceptually simple: a single unit formula, WO$_4\cdot$H$_3$N(CH$_2$)$_n$NH$_3$ ($n=2,4,6$ - hereby abbreviated to W-DAn). It is highly conceivable that the calculated structure of the hybrid compounds is in a slightly higher energy state than that of the actual structure, which may be a supercell of the simple input 