\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})
+\left\{\begin{array}{rl}
+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\{\begin{array}{rl}
+l\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
+ \text{for } j=l+\frac{1}{2}\\
+-(l+1)\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
+ \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\end{eqnarray}
+as equation~\eqref{eq:solid:so_bs1}
+\begin{equation}
+\text{ .}
+\end{equation}
+\end{proof}
-\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.
-The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read
+The matrix elements of the orbital angular momentum part of the potential in KB form read
\begin{equation}
-\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
+\sum_{lm}
+\bra{\vec{r}'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{\vec{r}''}
\text{ .}
\end{equation}
With
\begin{eqnarray}
-\bra{\vec{r'}}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r'}} \braket{\vec{r'}}{\chi_{lm}}
-=-i\hbar\nabla_{\vec{r'}}\,\chi_{lm}(\vec{r'}) \\
-r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
+\bra{\vec{r}'}r\ket{\chi_{lm}} & = & \vec{r}'\braket{\vec{r}'}{\chi_{lm}}\\
+\bra{\vec{r}'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r}'}
+\braket{\vec{r}'}{\chi_{lm}}
\end{eqnarray}
we get
\begin{equation}
--i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
-E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\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}''}
\text{ .}
\label{eq:solid:so_me}
\end{equation}
To further evaluate this expression, the KB projectors
\begin{equation}
-\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
+\ket{\chi_{lm}}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
{\braket{\delta V_l^{\text{SO}}\Phi_{lm}}
{\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}}
\end{equation}
-must be known in real space (with respect to $\vec{r'}$).
+must be known in real space (with respect to $\vec{r}'$).
\begin{equation}
-\braket{\vec{r'}}{\chi_{lm}}=
-\frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
+\braket{\vec{r}'}{\chi_{lm}}=
+\frac{\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}}
^{1/2}}
\end{equation}
and
\begin{equation}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}=
-\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
\text{ .}
\label{eq:solid:so_r1}
\end{equation}
In this expression, only the spherical harmonics are complex functions.
-Thus, the complex conjugate with respect to $\vec{r''}$ is given by
+Thus, the complex conjugate with respect to $\vec{r}''$ is given by
\begin{equation}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})
\text{ .}
\label{eq:solid:so_r2}
\end{equation}
of the spherical harmonics, the squared norm of the $\chi_{lm}$ reduces to
\begin{eqnarray}
\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=&
-\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r'}}
-\braket{\vec{r'}}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}\\
+\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r}'}
+\braket{\vec{r}'}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r}'\\
&=&\int
-{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'})
-Y_{lm}(\Omega_{r'})
-r'^2 dr' d\Omega_{r'} \\
+{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{\vec{r}'})
+Y_{lm}(\Omega_{\vec{r}'})
+r'^2 dr' d\Omega_{\vec{r}'} \\
&=&\int_{r'}
{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
-\int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
-&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
+\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{ .}\\
+&=&\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 product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+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}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
-\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
-&=&
-\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
-Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
-\end{eqnarray}
-and if all megnetic states $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
-\begin{equation}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\sum_{lm}
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}&=&\sum_{lm}
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})\nonumber\\
+&=&\sum_l
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
-\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,}
-\end{equation}
-which can be rewritten as
-\begin{equation}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}\sum_m
+Y^*_{lm}(\Omega_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})\nonumber\\
+&=&\sum_l
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
-\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)
-\end{equation}
-using the vector addition theorem
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\frac{2l+1}{4\pi}\nonumber\\
+\end{eqnarray}
+due to the vector addition theorem
\begin{equation}
-P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)=
-\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)=
+\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})
+\text{ .}
\end{equation}
-In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
+In total, the matrix elements of the SO potential can be calculated by
\begin{eqnarray}
-\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=&
-\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
-\braket{\chi_{lm}}{\vec{r''}}\\
-&=&
--i\hbar(\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)\times\nonumber\\
-&&\times\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}\\
+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(\vec{r'}\times \nabla_{\vec{r'}})
+-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)\times\nonumber\\
+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{ ,}\nonumber\\
+\label{eq:solid:so_fin}
+\end{eqnarray}
+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}
+r_y\frac{\partial}{\partial r_z}f(r)-r_z\frac{\partial}{\partial r_y}f(r)\\
+r_z\frac{\partial}{\partial r_x}f(r)-r_x\frac{\partial}{\partial r_z}f(r)\\
+r_x\frac{\partial}{\partial r_y}f(r)-r_y\frac{\partial}{\partial r_x}f(r)
+\end{array}\right)
+\label{eq:solid:rxp1}
+\end{eqnarray}
+\begin{eqnarray}
+r_i\frac{\partial}{\partial r_j}f(r)-r_j\frac{\partial}{\partial r_i}f(r)&=&
+r_if'(r)\frac{\partial}{\partial r_j}(r_x^2+r_y^2+r_z^2)^{1/2}-
+r_jf'(r)\frac{\partial}{\partial r_i}(r_x^2+r_y^2+r_z^2)^{1/2}\nonumber\\
+&=&
+r_if'(r)\frac{1}{2r}2r_j-r_jf'(r)\frac{1}{2r}2r_i=0
+\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}
-Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
+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}}=
+\frac{\braket{\delta V_lu_l}{u_l\delta V_l}}
+ {\bra{u_l}\delta V_l\ket{u_l}}=
+\frac{\int_{r}\delta V^2_l(r)u^2_l(r)}r^2dr
+ {\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l
+\ket{r''}\braket{r''}{u_l}}=
\end{equation}
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