as expected and --- in fact --- obtained from equation~\eqref{eq:solid:so_bs1}.
\end{proof}
-In order to include the spin-orbit interaction into the scalar-relativistic formalism of a normconserving, non-local pseudopotential, scalar-relativistic in contrast to fully relativistic pseudopotential wavefunctions are needed as a basis for the projectors of the spin-orbit potential.
+\subsubsection{Scalar relativistic basis}
+
+In order to include the spin-orbit interaction into the scalar relativistic formalism of a normconserving, non-local pseudopotential, scalar relativistic in contrast to fully relativistic pseudopotential wavefunctions are needed as a basis for the projectors of the spin-orbit potential.
The transformation
\begin{equation}
L\cdot S=L_xS_x+L_yS_y+L_zS_z
L_+S_-\ket{l,m,+}=L_+\ket{l,m}S_-\ket{+}=
\sqrt{(l-m)(l+m+1)}\hbar\ket{l,m+1}\hbar\ket{-}
\end{equation}
+ Moreover, this part only acts on magnetic quantum numbers
+ $m=-l,\ldots,l-1$ and updates quantum numbers $m=-l+1,\ldots,l$.
\item \underline{$L_-S_+$}:
Updates spin up component and only acts on spin down component
\begin{equation}
-L_+S_-\ket{l,m,-}=L_+\ket{l,m}S_-\ket{+}=
-\sqrt{(l-m)(l+m+1)}\hbar\ket{l,m+1}\hbar\ket{-}
+L_-S_+\ket{l,m,-}=L_+\ket{l,m}S_+\ket{-}=
+\sqrt{(l+m)(l-m+1)}\hbar\ket{l,m-1}\hbar\ket{+}
\end{equation}
+ Moreover, this part only acts on magnetic quantum numbers
+ $m=-l+1,\ldots,l$ and updates quantum numbers $m=-l,\ldots,l-1$.
\item \underline{$L_zS_z$}: Acts on both and updates both spinor components
\begin{equation}
L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
\pm\frac{1}{2}m\hbar^2\ket{l,m,\pm}
\end{equation}
+ It acts on all magnetic quantum numbers and updates all of them.
\end{enumerate}
+Please note that the $\ket{l,m,\pm}$ are not eigenfunctions of the two combinations of ladder operators, i.e.\ the $\ket{l,m,\pm}$ do not diagonalize the spin-orbit part of the Hamiltonian.
+
+These equations can be simplified to read
+\begin{eqnarray}
+\ldots
+\text{ .}
+\end{eqnarray}
+
+\subsubsection{A different basis set}
+
+The above basis is composed of eigenfunctions
+\begin{equation}
+\ket{l,m} \text{, } \ket{\pm} \text{ of operators }
+L^2\text{, } L_z \text{ and } S_z
+\text{.}
+\end{equation}
+These eigenfunctions diagonalize the scalar relativistic Hamiltonian.
+Introducing spin-orbit interaction, however, it is a good idea to chose eigenfunctions that diagonalize the perturbation
+\begin{equation}
+L\cdot S=\frac{1}{2}(J^2-L^2-S^2)
+\text{ ,}
+\end{equation}
+i.e.\ simultaneous eigenfunctions of $J^2$, $L^2$ and $S^2$.
\subsubsection{Excursus: Real space representation within an iterative treatment}