+Finally, a set of basis functions is required to represent the one-electron KS wave functions.
+With respect to the numerical treatment it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions.
+Covergence of the basis set, i.e. convergence of the wave functions with respect to the amount of basis functions, is most crucial for the accuracy of the numerical calulations.
+Two classes of basis sets, the plane-wave and local basis sets, exist.
+
+Local basis set functions usually are atomic orbitals, i.e. mathematical functions that describe the wave-like behavior of electrons, which are localized, i.e. centered on atoms or bonds.
+Molecular orbitals can be represented by linear combinations of atomic orbitals (LCAO).
+By construction, only a small number of basis functions is required to represent all of the electrons of each atom within reasonable accuracy.
+Thus, local basis sets enable the implementation of methods that scale linearly with the number of atoms.
+However, these methods rely on the fact that the wave functions are localized and exhibit an exponential decay resulting in a sparse Hamiltonian.
+
+Another approach is to represent the KS wave functions by plane waves.
+In fact, the employed {\textsc vasp} software is solving the KS equations within a plane-wave basis set.
+The idea is based on the Bloch theorem \cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice.
+The latter one can be expressed by a Fourier series, i.e. a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal.
+Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete plane-wave basis set
+\begin{equation}
+\Phi_i(\vec{r})=\sum_{\vec{G}
+%, |\vec{G}+\vec{k}|<G_{\text{cut}}}
+}c_{i,\vec{k}+\vec{G}} \exp\left(i(\vec{k}+\vec{G})\vec{r}\right)
+\text{ .}
+%E_{\text{cut}}=\frac{\hbar^2 G^2_{\text{cut}}}{2m}
+%\text{, }
+\end{equation}
+The basis set, which in principle should be infinite, can be truncated to include only plane waves that have kinetic energies $\hbar^2|\vec{k}+\vec{G}|^2/2m$ less than a particular cut-off energy $E_{\text{cut}}$.
+Although coefficients $c_{i,\vec{k}+\vec{G}}$ corresponding to small kinetic energies are typically more important, convergence with respect to the cut-off energy is crucial for the accuracy of the calculations.
+Convergence with respect to the basis set, however, is easily achieved by increasing $E_{\text{cut}}$ until the respective differences in total energy approximate zero.
+Next to their simplicity, plane waves have several advantages.
+The basis set is orthonormal by construction.
+matrix elements of the Hamiltonian have a simple form (pw rep of ks equations)
+As mentioned above ... simple to check for convergence.
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