As discussed above, the HK and KS formulations are exact and so far no approximations except the adiabatic approximation entered the theory.
However, to make concrete use of the theory, effective approximations for the exchange and correlation energy functional $E_{\text{xc}}[n(\vec{r})]$ are required.
-Most simple and at the same time remarkably useful is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65}
+Most simple and at the same time remarkably useful, is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65},
\begin{equation}
E^{\text{LDA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
\text{ ,}
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}
-These functionals constitute the simplest extensions of 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.
+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}
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.
+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.
Convergence 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 calculations.
Two classes of basis sets, the plane-wave and local basis sets, exist.
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
+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}}}
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.
+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.
\label{section:basics:defects}
Point defects are defects that affect a single lattice site.
-At this site the crystalline periodicity is interrupted.
+At this site, the crystalline periodicity is interrupted.
An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
A substitutional defect exists, if an atom belonging to the perfect crystal is replaced with an atom of another species.
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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