are shown in Figure \[fig:WO3-x\_cell\] and are in good agreement with experimental values, despite the absence of the phase change in the calculated system.

![\[fig:WO3-x\_cell\] The volume-averaged cubic cell parameter for calculated and experimental sub-stoichiometric ‘cubic’ WO$_{3-x}$ systems. The curve is given as a guide.](WO3-x_cell.eps){width="85mm"}

The energies of formation for the different species studied are given in Table \[table:enform\_WO3-x\]. We note that a slight deficit of oxygen ($x=\frac{1}{6}$) is more favourable energetically than stoichiometric WO$_3$. This is observed experimentally, as commercial WO$_3$ powder exhibits a loss of oxygen over 1-2 days in atmospheric conditions. Further loss of oxygen renders WO$_{3-x}$ less energetically favourable than its parent oxide, and once again, as was the case with sodium tungsten bronzes, the presence of a phase change in the experimental system may explain any discrepancies seen.

              Formula               Energy of formation (eV/unit formula)
  -------------------------------- ---------------------------------------
           WO$_3$ ($x=0$)                             0
   WO$_{2.833}$ ($x=\frac{1}{6}$)                  -0.103
    WO$_{2.8}$ ($x=\frac{1}{5}$)                    0.447
   WO$_{2.75}$ ($x=\frac{1}{4}$)                    1.007
   WO$_{2.667}$ ($x=\frac{1}{3}$)                   1.661

It is also of interest to look at the changes in the density of states as oxygen is removed from WO$_3$. As mentioned in the introduction, oxygen-deficient WO$_3$ exhibits an increased conductivity, but not as great as that due solely to the presence of 