F[n(\vec{r})] + \int n(\vec{r}) V(\vec{r}) d\vec{r}
\right\}
\text{ ,}
+\label{eq:basics:hkm}
\end{equation}
where $F[n(\vec{r})]$ is a universal functional of the charge density $n(\vec{r})$, which is composed of the kinetic energy functional $T[n(\vec{r})]$ and the interaction energy functional $U[n(\vec{r})]$.
The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional.
\subsection{Kohn-Sham system}
-Now find $F[n]$ ...
-
-As in the last section, the complex many-electron effects are relocated, this time into the exchange-correlation functional.
+Inspired by the Hartree equations, i.e. a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}.
+However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$.
+The respective Kohn-Sham equations for the effective single-particle wave functions $\Phi_i(\vec{r})$ take the form
+\begin{equation}
+\left[
+ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(\vec{r})
+\right] \Phi_i(\vec{r})=\epsilon_i\Phi_i(\vec{r})
+\label{eq:basics:kse1}
+\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{xc}(\vec{r})}
+\text{ ,}
+\label{eq:basics:kse2}
+\end{equation}
+\begin{equation}
+n(\vec{r})=\sum_{i=1}^N |\Phi_i(\vec{r})|^2
+\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.
+%\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})}
+%\end{equation}
+
+The system of interacting electrons is mapped to an auxiliary system, the Kohn-Sham (KS) system, of non-interacting electrons in an effective potential.
+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.
+
+The self-consistent KS equations \eqref{eq:basics:kse1,eq:basics:kse2,eq:basics:kse3} 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.
+
+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.
+Clearly, this directs attention to the functional, which now contains the costs involved with the many-electron problem.
\subsection{Approximations for exchange and correlation}
+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
+\begin{equation}
+E_{\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}
+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).
+
+\subsection{Plane-wave basis set}
+
\subsection{Pseudopotentials}
+\subsection{Brillouin zone sampling}
+
+\subsection{Hellmann-Feynman forces}
+
\section{Modeling of defects}
\label{section:basics:defects}
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