# Why are the bands in unit cell Density Functional Theory (DFT) calculations continuous?

The Bloch theorem states that a non-interacting electronic state of a confined system with periodic boundary conditions (and a periodic potential) is represented as $$\psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) e^{i{\mathbf{k}}r}$$. In the picture of a free-electron confined in a periodic box, the momentum values are quantized as $$k_i=\frac{2\pi n_i}{L}~\mathrm{for}~i\in {x,y,z}$$. So in principle, also in Density Functional Theory (DFT), the $$\mathbf{k}$$-vector should be quantized. However, when we plot the result from DFT band calculations, we always see continuous bands, what is the reason for that?

Edit: I want to stress that I understand that if we make the unit cell very large, there will be a continuous amount of states, but that is not the case in a conventional periodic DFT calculation, where our cell size is finite...

• From your comment to the answer below I see that you now understand that the periodicity of the wave function in the periodic box model differs from the Bloch waves we actually have in infinite crystals. It would be nice if you write an answer to your own question in which you explicitly describe the difference. I guess you are by far not the only person who gets irritated by this. I actually wonder where this periodic box model is useful. Jun 15 '20 at 23:00
• 2 months is long for no upvoted answers, maybe copy and paste into materials.stackexchange.com Jul 23 '20 at 18:18