-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.
-...
+This fact is exploited in the pseudopotential (PP) approach \cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions.
+This way, the pseudo wave functions become smooth near the nuclei.
+
+Most PPs statisfy four general conditions.
+The pseudo wave functions generated by the PP should not contain nodes, i.e. the pseudo wave functions should be smooth and free of wiggles in the core region.
+Outside the core region, the pseudo and real valence wave functions as well as the generated charge densities need to be identical.
+The charge enclosed within the core region must be equal for both wave functions.
+Last, almost redundantly, the valence all-electron and pseudopotential eigenvalues must be equal.
+Pseudopotentials that meet the conditions outlined above are referred to as norm-conserving pseudopotentials \cite{hamann79}.
+
+%Certain properties need to be fulfilled by PPs and the resulting pseudo wave functions.
+%The pseudo wave functions should yield the same energy eigenvalues than the true valence wave functions.
+%The PP is called norm-conserving if the pseudo and real charge contained within the core region matches.
+%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.
+Mathematically, a non-local PP, which depends on the angular momentum, has the form
+\begin{equation}
+V_{\text{nl}}(\vec{r}) = \sum_{lm} \mid lm \rangle V_l(\vec{r}) \langle lm \mid
+\text{ .}
+\end{equation}
+Applying of the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $\mid lm \rangle$, i.e. the orbitals with azimuthal angular momentum $l$ and magnetic number $m$, which are then multiplied by the respective psuedopotential $V_l(\vec{r})$ for angular momentum $l$.
+The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are eual to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius detrmining the core region.
+Modified methods to generate ultra-soft pseudopotentials were proposed, which address the rapid convergence with respect to the size of the plane wave basis set \cite{vanderbilt90,troullier91}.
+
+Using PPs the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
+More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.