E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
\text{ .}
\end{equation}
-This functional constitutes the simplest extension of LDA for inhomogeneous systems.
+This functional constitutes the simplest extension of the LDA for inhomogeneous systems.
At modest computational costs, gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.
\subsection{Plane-wave basis set}
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.
+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.
%To guarantee transferability of the PP the logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives need to agree outside of the core region.
%A simple procedure was proposed to extract norm-conserving PPs obyeing the above-mentioned conditions from {\em ab initio} atomic calculations~\cite{hamann79}.
-In order to achieve these properties different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
+In order to achieve these properties, different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
Mathematically, a non-local PP, which depends on the angular momentum, has the form
\begin{equation}
V_{\text{nl}}(\vec{r}) = \sum_{lm} | lm \rangle V_l(\vec{r}) \langle lm |
\subsection{Brillouin zone sampling}
\label{subsection:basics:bzs}
-Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
+Following Bloch's theorem, only a finite number of electronic wave functions need to be calculated for a periodic system.
However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
-Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $k$ points, each representing the respective region of the wave function in $\vec{k}$ space.
+Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $\vec{k}$ points, each representing the respective region of the wave function in $\vec{k}$ space.
Methods have been derived for obtaining very accurate approximations by a summation over special sets of $\vec{k}$ points with distinct, associated weights~\cite{baldereschi73,chadi73,monkhorst76}.
If present, symmetries in reciprocal space may further reduce the number of calculations.
For supercells, i.e.\ repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurate results.
If not by construction, the system should be fully relaxed.
The substitutional or vacancy defect is realized by replacing or removing one atom contained in the supercell.
Interstitial defects are created by adding an atom at positions located in the space between regular lattice sites.
-Once the intuitively created defect structure is generated structural relaxation methods will yield the respective local minimum configuration.
+Once the intuitively created defect structure is generated, structural relaxation methods will yield the respective local minimum configuration.
Since the supercell approach applies periodic boundary conditions, enough bulk material surrounding the defect is required to exclude possible interaction of the defect with its periodic image.
\begin{figure}[t]
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