+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 (PW) 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 PW 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 and, as mentioned above, it is simple to check for convergence.
+The biggest advantage, however, is the ability to perform exact calculations by a discrete sum over a numerical grid.
+This is due to the related construction of the grid and the PW basis.
+Ofcourse, exactness is restricted by the fact that the PW basis set is finite.
+The simple form of the PW representation of the KS equations
+\begin{equation}
+\sum_{\vec{G}'} \left[
+ \frac{\hbar^2}{2m}|\vec{k}+\vec{G}|^2 \delta_{\vec{GG}'}
+ + \tilde{V}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{H}}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{xc}}(\vec{G}-\vec{G}')
+\right] c_{i,\vec{k}+\vec{G}} = \epsilon_i c_{i,\vec{k}+\vec{G}}
+\label{eq:basics:pwks}
+\end{equation}
+reveals further advantages.
+The various potentials are described in terms of their Fourier transforms.
+Equation \eqref{eq:basics:pwks} is solved by diagonalization of the Hamiltonian matrix $H_{\vec{k}+\vec{G},\vec{k}+\vec{G}'}$ given by the terms in the brackets.
+The gradient operator is diagonal in reciprocal space whereas the exchange-correlation potential has a diagonal representation in real space.
+This suggests to carry out different operations in real and reciprocal space, which requires frequent Fourier transformations.
+These, however, can be efficiently achieved by the fast Fourier transformation (FFT) algorithm.
+
+There are likewise disadvantages associated with the PW representation.
+By construction, PW calculations require a periodic system.
+This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
+Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
+As a consequence the number of atoms involved in the calculations are increased.
+To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
+Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
+According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
+For the same reason, the description of tightly bound core electrons and the respective, highly localized charge density is hindered.
+However, a much more profound problem exists whenever wave functions for the core as well as the valence electrons need to be calculated within a PW basis set.
+Wave functions of the valence electrons exhibit rapid oscillations in the region occupied by the core electrons near the nuclei.
+The oscillations maintain the orthogonality between the wave functions of the core and valence electrons, which is compulsory due to the exclusion principle.
+Accurately approximating these oscillations demands for an extremely large PW basis set, which is too large for practical use.
+Fortunately, the impossibility to model the core in addition to the valence electrons is eliminated in the pseudopotential approach discussed in the next section.
+