\subsection{Born-Oppenheimer approximation}
Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom~\cite{born27}.
-Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
-Thus, on the timescale of electronic motion the ions appear at fixed positions.
-On the other way round, on the timescale of nuclear motion the electrons appear blurred in space adding an extra term to the ion-ion potential.
+Within the Born-Oppenheimer (BO) approximation, the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
+Thus, on the timescale of electronic motion, the ions appear at fixed positions.
+On the other way round, on the timescale of nuclear motion, the electrons appear blurred in space adding an extra term to the ion-ion potential.
The simplified Schr\"odinger equation no longer contains the kinetic energy of the ions.
The momentary positions of the ions enter as fixed parameters and, therefore, the ion-ion interaction may be regarded as a constant added to the electronic energies.
The Schr\"odinger equation describing the remaining electronic problem reads
\end{equation}
where $Z_l$ are the atomic numbers of the nuclei and $\Psi$ is the many-electron wave function, which depends on the positions and spins of the electrons.
Accordingly, there is only a parametric dependence on the ionic coordinates $\vec{R}_l$.
-However, the remaining number of free parameters is still too high and need to be further decreased.
+However, the remaining number of free parameters is still too high and needs to be further decreased.
\subsection{Hohenberg-Kohn theorem and variational principle}
This raised the question how to establish a connection between TF expressed in terms of $n(\vec{r})$ and the exact Schr\"odinger equation expressed in terms of the many-electron wave function $\Psi({\vec{r}})$ and whether a complete description in terms of the charge density is possible in principle.
The answer to this question, whether the charge density completely characterizes a system, became the starting point of modern DFT.
-Considering a system with a nondegenerate ground state there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
-In 1964 Hohenberg and Kohn showed the opposite and far less obvious result~\cite{hohenberg64}.
-Employing no more than the Rayleigh-Ritz minimal principle it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
+Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
+In 1964, Hohenberg and Kohn showed the opposite and far less obvious result~\cite{hohenberg64}.
+Employing no more than the Rayleigh-Ritz minimal principle, it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
Thus, the charge density of the ground state $n_0(\vec{r})$ uniquely determines the potential $V(\vec{r})$ and, consequently, the full Hamiltonian and ground-state energy $E_0$.
-In mathematical terms the full many-electron ground state is a unique functional of the charge density.
+In mathematical terms, the full many-electron ground state is a unique functional of the charge density.
In particular, $E$ is a functional $E[n(\vec{r})]$ of $n(\vec{r})$.
The ground-state charge density $n_0(\vec{r})$ minimizes the energy functional $E[n(\vec{r})]$, its value corresponding to the ground-state energy $E_0$, which enables the formulation of a minimal principle in terms of the charge density~\cite{hohenberg64,levy82}
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}, \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.
+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 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}
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