The electronic band structure gives the relation between crystal momentum $\mathbf{k}$, band index $n$ and the energy $\varepsilon_{n \mathbf{k}}$ of an electron in a crystal, according to Bloch's theorem.

The electronic band structure can be computed using DFT in an approximate way via the Kohn-Sham equations

$$ \left( -\frac{\nabla^2}{2m} + v_\text{ext}(\mathbf{r}) + v_\text{Hartree}(\mathbf{r}) + v_\text{xc}(\mathbf{r}) \right) \psi_{n\mathbf{k}}(\mathbf{r}) = \varepsilon_{n\mathbf{k}} \psi_{n\mathbf{k}}(\mathbf{r}) $$

When using the standard exchange-correlation (xc) functionals like PBE, the band gap between the occupied valence bands and the empty conduction bands is usually underestimated with respect to experiment.

Nevertheless, PBE often gives the correct band ordering, dispersions (i.e., curvature as function of $\mathbf{k}$), and orbital character of the bands.

In this exercise, we compute the band structure of monolayer MoS$_2$, a two-dimensional crystal which has been discovered in 2010 (doi:10.1103/PhysRevLett.105.136805).