The average cubic cell parameter was found to increase linearly with $x$ as shown in figure \[fig:NaxWO3\_cell\]. Also shown in this figure is a series of reported experimental results. For higher $x$ contents the two data sets are very close, with the calculated parameters being about 0.8$\%$ larger than the experimental; again, a good result given the approximations made using this method. However below $x=0.3$, the experimental results differ markedly from the calculated values, due to a phase change to a tetragonal form at low $x$ values in the experimental system [@11]. The structure of the tetragonal phase is not related to the cubic phase [@15]. It contains seven WO$_3$ units per cell (28 atoms), and when one takes the fractional doping into account, the resulting system quickly becomes much too large to attempt a calculation with the computational resources available to us. In the work of Walkingshaw et. al. [@521] the volume versus $x$ deviates from this linear behaviour for $x$ as large as 0.5.

![\[fig:NaxWO3\_cell\] Calculated and experimental values of cell parameter for cubic Na$_x$WO$_3$ with variable *x*.](NaxWO3_cell.eps){width="85mm"}

There are no obvious changes in the appearance of the density of states as $x$ increases from zero to one. 