In order to use DFT for practical calculations on real systems, we need to solve the Kohn-Sham equations numerically with a computer, which means that the problem must be cast in a finite manner. Furthermore, it is advantageous to cast the problem in a way that is computationally efficient, and that allows the numerical accuracy to be controlled in a sensible way. In all of the calculations in this work we will use the plane wave pseudopotential approach to solving the Kohn-Sham equations. This involves using a plane wave basis set to represent the orbitals, and pseudopotentials to represent the nuclei and core electrons. In this section, we will describe this plane wave pseudopotential approach. Alternative approaches to the plane wave pseudopotential exist. These involve using basis functions that are localised around individual atoms. While cheaper computationally, they suffer from the problem that the basis set is incomplete and so it is often unclear whether or not a given calculation is truly converged with respect to the basis.