\text{ ,}
\end{equation}
\begin{equation}
-V_{\text{eff}}=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'
+V_{\text{eff}}(\vec{r})=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'
+ V_{\text{xc}(\vec{r})}
\text{ ,}
\label{eq:basics:kse2}
\text{ ,}
\label{eq:basics:kse3}
\end{equation}
-where the local exchange-correlation potential $V_{\text{xc}}(\vec{r})$ is the partial derivative of the exchange-correlation functional $E_{\text{xc}}[n(vec{r})]$ with respect to the charge density $n(\vec{r})$ for the ground-state $n_0(\vec{r})$.
-The first term in equation \eqref{eq:basics:kse1} corresponds to the kinetic energy of non-interacting electrons and the second term of equation \eqref{eq:basics:kse2} is just the Hartree contribution to the interaction energy.
+where the local exchange-correlation potential $V_{\text{xc}}(\vec{r})$ is the partial derivative of the exchange-correlation functional $E_{\text{xc}}[n(\vec{r})]$ with respect to the charge density $n(\vec{r})$ for the ground-state $n_0(\vec{r})$.
+The first term in equation \eqref{eq:basics:kse1} corresponds to the kinetic energy of non-interacting electrons and the second term of equation \eqref{eq:basics:kse2} is just the Hartree contribution $V_{\text{H}}(\vec{r})$ to the interaction energy.
%\begin{equation}
%V_{\text{xc}}(\vec{r})=\frac{\partial}{\partial n(\vec{r})}
% E_{\text{xc}}[n(\vec{r})] |_{n(\vec{r})=n_0(\vec{r})}
The exact effective potential $V_{\text{eff}}(\vec{r})$ may be regarded as a fictious external potential yielding a gound-state density for non-interacting electrons, which is equal to that for interacting electrons in the external potential $V(\vec{r})$.
The one-electron KS orbitals $\Phi_i(\vec{r})$ as well as the respective KS energies $\epsilon_i$ are not directly attached to any physical observable except for the ground-state density, which is determined by equation \eqref{eq:basics:kse3} and the ionization energy, which is equal to the highest occupied relative to the vacuum level.
The KS equations may be considered the formal exactification of the Hartree theory, which it is reduced to if the exchange-correlation potential and functional are neglected.
-In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that is not part of the HF correlation.
+In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that are not part of the HF correlation.
-The self-consistent KS equations \eqref{eq:basics:kse1,eq:basics:kse2,eq:basics:kse3} may be solved numerically by an iterative process.
+The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} are non-linear partial differential equations, which may be solved numerically by an iterative process.
Starting from a first approximation for $n(\vec{r})$ the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential.
The $\Phi_i(\vec{r})$ are used to obtain a new expression for $n(\vec{r})$.
-These steps are repeated until the initial and new density are equal or vary only slightly.
+These steps are repeated until the initial and new density are equal or reasonably converged.
Again, it is worth to note that the KS equations are formally exact.
-Assuming exact functionals $E_{\text{xc}}[n(vec{r})]$ and potentials $V_{\text{xc}}(\vec{r})$ all many-body effects are included.
+Assuming exact functionals $E_{\text{xc}}[n(\vec{r})]$ and potentials $V_{\text{xc}}(\vec{r})$ all many-body effects are included.
Clearly, this directs attention to the functional, which now contains the costs involved with the many-electron problem.
\subsection{Approximations for exchange and correlation}
+\label{subsection:ldagga}
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.
+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
+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{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
-\text{ .}
+E^{\text{LDA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
+\text{ ,}
\label{eq:basics:xca}
\end{equation}
+which is, thus, called local density approximation (LDA).
Here, the exchange-correlation energy per particle of the uniform electron gas of constant density $n$ is used for $\epsilon_{\text{xc}}(n(\vec{r}))$.
-This is called the local density approximation (LDA).
+Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods \cite{ceperley80} and corresponding paramterizations exist \cite{perdew81}.
+Obviously exact for the homogeneous electron gas, the LDA was {\em a priori} expected to be useful only for densities varying slowly on scales of the local Fermi or TF wavelength.
+Nevertheless, LDA turned out to be extremely successful in describing some properties of highly inhomogeneous systems accurately within a few percent.
+Although LDA is known to overestimate the cohesive energy in solids by \unit[10-20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
+
+More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density \cite{kohn65}.
+Respective considerations are based on the concept of an exchange-correlation hole density describing the depletion of the electron density in the vicinity of an electron.
+The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(\vec{r})]$ and an equivalent expression \cite{kohn96,kohn98}, which makes use of the electron density distribution $n(\tilde{\vec{r}})$ at positions $\tilde{\vec{r}}$ near $\vec{r}$, yielding the form
+\begin{equation}
+E_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})])n(\vec{r}) d\vec{r}
+\end{equation}
+for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})])$ becomes a nearsighted functional of $n(\tilde{\vec{r}})$.
+Expressing $n(\tilde{\vec{r}})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion.
+Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r})$ results in the functional equal to LDA, which requires the function of variable $n$.
+Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$.
+This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
+\begin{equation}
+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.
\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.
+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.
+
\subsection{Pseudopotentials}
+As discussed in the last part of the previous section, an extremely large basis set of plane waves would be required to perform an all-electron calculation and a vast amount of computational time would be required to calculate the electronic wave functions.
+It is worth to stress out one more time, that this is due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
+Thus, existing core states practically prevent the use of a PW basis set.
+However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid.
+This fact is exploited in the pseudopotential approach \cite{} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker pseudopotential that acts on a set of pseudo wave functions rather than the true valance wave functions.
+Certain conditions need to be fulfilled by the constructed pseudopotentials and the resulting pseudo wave functions.
+Outside the core region, the pseudo and real wafe functions as well as the generated charge densities need to be identical.
+...
+A pseudopotential is called norm-conserving if the pseudo and real charge contained within the core region match.
+...
+
\subsection{Brillouin zone sampling}
+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.
+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 accurat results.
+In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space.
+In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictious anyway.
+
\subsection{Hellmann-Feynman forces}
\section{Modeling of defects}
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