Electronic structure of NaWO$_3$: Role of the impurity potential

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.