\subsection{Spin-orbit interaction}
-Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant.
+Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including terms of order $\alpha^2$ \cite{kleinman80,bachelet82} with $\alpha$ being the fine structure constant.
This is advantageous since \ldots
With the solutions of the all-electron Dirac equations, the new pseudopotential reads
\begin{equation}
\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(\vec{r})
\bra{l-\frac{1}{2},m-{\frac{1}{2}}}
\right] \text{ .}
+\label{eq:solid:so_bs1}
\end{equation}
By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
\begin{equation}
the total potential can be expressed as
\begin{equation}
V(\vec{r})=\sum_l
-\ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l}
+\ket{l,m}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l,m}
\text{ ,}
+\label{eq:solid:so_bs2}
\end{equation}
where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling.
+\begin{proof}
+This can be shown by rewriting the $LS$ operator
+\begin{equation}
+J=L+S \Leftrightarrow J^2=L^2+S^2+2LS \Leftrightarrow
+LS=\frac{1}{2}\left(J^2-L^2-S^2\right)
+\end{equation}
+and corresponding eigenvalue
+\begin{eqnarray}
+j(j+1)-l(l+1)-s(s+1)&=&
+(l\pm\frac{1}{2})(l\pm\frac{1}{2}+1)-l^2-l-\frac{3}{4} \nonumber\\
+&=&
+l^2\pm\frac{l}{2}+l\pm\frac{l}{2}+\frac{1}{4}\pm\frac{1}{2}-l^2-l-\frac{3}{4}
+\nonumber\\
+&=&\pm(l+\frac{1}{2})-\frac{1}{2}=\left\{\begin{array}{rl}
+l & \text{for } j=l+\frac{1}{2}\\
+-(l+1) & \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\text{ ,}
+\end{eqnarray}
+which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) matrix elements
+\begin{eqnarray}
+\bra{l\pm\frac{1}{2},m\pm\frac{1}{2}}V(\vec{r})
+\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&=&
+\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})
+\frac{1}{2}\left(l(l+1)-j(j+1)-\frac{3}{4}\right) \nonumber\\
+&=&\bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r})
+\cdot\left\{\begin{array}{cl}
+l & \text{for } j=l+\frac{1}{2}\\
+-(l+1) & \text{for } j=l-\frac{1}{2}
+\end{array}\right. \nonumber\\
+&=&\frac{1}{2l+1}\left(lV_{l,l-\frac{1}{2}}(\vec{r})+
+ (l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)+\nonumber\\
+&&\frac{1}{2l+1}
+\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
+\cdot\left\{\begin{array}{c}
+l \\
+-(l+1)
+\end{array}\right. \nonumber\\
+&=&\left\{\begin{array}{cl}
+V_{l,l+\frac{1}{2}}(\vec{r}) & \text{for } j=l+\frac{1}{2}\\
+V_{l,l-\frac{1}{2}}(\vec{r}) & \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\end{eqnarray}
+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.
+The transformation
+\begin{equation}
+L\cdot S=L_xS_x+L_yS_y+L_zS_z
+\end{equation}
+using the ladder operators
+\begin{equation}
+L_\pm=L_x\pm iL_y \text{ and } S_\pm=S_x\pm iS_y
+\text{ ,}
+\end{equation}
+with properties
+\begin{eqnarray}
+L_+S_- & = & (L_x+iL_y)(S_x-iS_y)=L_xS_x-iL_xS_y+iL_yS_x+L_yS_y \\
+L_-S_+ & = & (L_x-iL_y)(S_x+iS_y)=L_xS_x+iL_xS_y-iL_yS_x+L_yS_y
+\end{eqnarray}
+resulting in
+\begin{equation}
+L_+S_-+L_-S_+=2(L_xS_x+L_yS_y)
+\text{ ,}
+\end{equation}
+reads
+\begin{equation}
+L\cdot S=\frac{1}{2}(L_+S_-+L_-S_+)+L_zS_z
+\text{ .}
+\end{equation}
+The contributions of this operator act differently on $\ket{l,m}$ and --- in fact --- depend on the respectively considered spinor component, which is incorporated by $\ket{l,m,\pm}$.
+\begin{enumerate}
+\item \underline{$L_+S_-$}:
+ Updates spin down component and only acts on spin up 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{-}
+\end{equation}
+\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{-}
+\end{equation}
+\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}
+\end{enumerate}
-\subsubsection{Excursus: real space representation within an iterative treatment}
+\subsubsection{Excursus: Real space representation within an iterative treatment}
In the following, the spin-orbit part is evaluated in real space.
Since spin is treated in another subspace, it can be treated separately.
&=&\int_{r'}
{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
\int_{\Omega_{\vec{r}'}}Y^*_{lm}(\Omega_{\vec{r}'})Y_{lm}(\Omega_{\vec{r}'}) d\Omega_{\vec{r}'}\\
-&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
+&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}\\
+&=&\braket{\delta V_l^{\text{SO}}u_l}{u_l\delta V_l^{\text{SO}}}
\end{eqnarray}
To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the sum of the products of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
\begin{eqnarray}
\end{equation}
In total, the matrix elements of the SO potential can be calculated by
\begin{eqnarray}
--i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
-E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\\
-=-i\hbar\sum_l(\vec{r}'\times \nabla_{\vec{r}'})
+&&-i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\nonumber\\
+&=&-i\hbar\sum_l(\vec{r}'\times \nabla_{\vec{r}'})
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
-P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot\nonumber
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
{\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \cdot
\frac{2l+1}{4\pi}\nonumber\\
-=
+&=&
-i\hbar\sum_l
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
P'_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
\left(\frac{\vec{r}'\times\vec{r}''}{r'r''}\right)\cdot
\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
{\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
-\frac{2l+1}{4\pi}\text{ ,}
+\frac{2l+1}{4\pi}\text{ ,}\nonumber\\
\label{eq:solid:so_fin}
\end{eqnarray}
-where the derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as can be seen from equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}.
+since derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as is illustrated below (equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}).
\begin{eqnarray}
\left(\vec{r}\times\nabla_{\vec{r}}\right)f(r)&=&
\left(\begin{array}{l}
\label{eq:solid:rxp2}
\end{eqnarray}
-If the potential at position $\vec{r}$ is considered a sum of atomic potentials $v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})$ (atom $n$ of species $\alpha$)
-\begin{equation}
-V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
-\end{equation}
-and the SO projectors are likewise centered on atoms, the SO potential contribution reads
+If these projectors are considered to be centered around atom positions $\vec{\tau}_{\alpha n}$ of atoms $n$ of species $\alpha$, the variable $\vec{r}'$ in the previous equations is changed to $\vec{r}'_{\alpha n}=\vec{r}'-\vec{\tau}_{\alpha n}$, which implies
+\begin{eqnarray}
+r'&\rightarrow&r_{\alpha n}=|\vec{r}'-\vec{\tau}_{\alpha n}|\\
+\Omega_{\vec{r}'}&\rightarrow&\Omega_{\vec{r'}-\vec{\tau}_{\alpha n}}\\
+\delta V_l(r')&\rightarrow&\delta V_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+u_l(r')&\rightarrow&u_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+Y_{lm}(\Omega_{\vec{r}'})&\rightarrow&
+Y_{lm}(\Omega_{\vec{r}'-\vec{\tau}_{\alpha n}})
+\text{ .}
+\end{eqnarray}
+Within an iterative treatment on a real space grid consisting of $n_{\text{g}}$ grid points, the sum
\begin{equation}
+\sum_{\vec{r}''_{\alpha n}}
+\sum_{lm}-i\hbar(\vec{r}'_{\alpha n}\times \nabla_{\vec{r}'_{\alpha n}})
+\braket{\vec{r}'_{\alpha n}}{\chi^{\text{SO}}_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi^{\text{SO}}_{lm}}{\vec{r}''_{\alpha n}}
+\braket{\vec{r}''_{\alpha n}}{\Psi}
+\qquad\forall\,\bra{\vec{r}'_{\alpha n}}
\end{equation}
+to obtain all elements $\bra{\vec{r}'_{\alpha n}}$, involves $n_{\text{g}}^2$ evaluations of equation~\eqref{eq:solid:so_fin} for eeach atom, if the projectors are short-ranged, i.e.\ $\delta V_l=0$ outside a certain cut-off radius.
+Thus, this method scales linearly with the number of atoms.
+
The $E_l^{\text{SO,KB}}$ are given by
\begin{equation}
E_l^{\text{SO,KB}}=
{\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l
\ket{r''}\braket{r''}{u_l}}=
\end{equation}
-Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
-\begin{equation}
-\end{equation}
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