  We have performed {\it ab-initio} electronic structure calculations to
determine the evolution of the electronic structure of WO$_3$ with Na doping.
Na doping introduces an additional electron when introduced into WO$_3$. The
ensuing electronic structure of Na doped WO$_3$, we find, is very similar to
the electronic structure of an electron introduced into WO$_3$, thus clarifying
the role of the impurity potential due to Na. While the electronic structure of
NaWO$_3$ allows a rigid band like description over a certain energy range,
modifications introduced in the electronic structure can be related back to the
electron due to Na and not the impurity potential that one generally believes
to be responsible.
