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.
+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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