Ionic potentials, which are spherically symmteric, suggest to treat each angular momentum $l,m$ separately leading to $l$-dependent non-local (NL) model potentials $V_l(r)$ and a total potential
\begin{equation}
-V=\sum_{l,m}\ket{lm}V_l(r)\bra{lm} \text{ .}
+V=\sum_{l,m}\ket{lm}V_l(\vec{r})\bra{lm} \text{ .}
\end{equation}
In fact, applied to a function, the potential turns out to be non-local in the angular coordinates but local in the radial variable, which suggests to call it asemilocal (SL) potential.
A local potential can always be separated from the potential \ldots
\begin{equation}
-V=\ldots=V_{\text{local}}(r)+\ldots
+V=\ldots=V_{\text{local}}(\vec{r})+\ldots
\end{equation}
\subsubsection{Norm conserving pseudopotentials}
This is advantageous since \ldots
With the solutions of the all-electron Dirac equations, the new pseudopotential reads
\begin{equation}
-V(r)=\sum_{l,m}\left[
-\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(r)
+V(\vec{r})=\sum_{l,m}\left[
+\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}}} +
-\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(r)
+\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{ .}
\end{equation}
By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
\begin{equation}
\bar{V}_l(r)=\frac{1}{2l+1}\left(
-l V_{l,l-\frac{1}{2}}(r)+(l+1)V_{l,l+\frac{1}{2}}(r)\right)
+l V_{l,l-\frac{1}{2}}(\vec{r})+(l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)
\end{equation}
and a potential describing the difference in the potential with respect to the spin
\begin{equation}
-V^{\text{SO}}_l(r)=\frac{2}{2l+1}\left(
-V_{l,l+\frac{1}{2}}(r)-V_{l,l-\frac{1}{2}}(r)\right)
+V^{\text{SO}}_l(\vec{r})=\frac{2}{2l+1}\left(
+V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
\end{equation}
the total potential can be expressed as
\begin{equation}
-V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(r)LS\right]\bra{l}
+V(\vec{r})=\sum_l
+\ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l}
\text{ ,}
\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.
-\subsubsection{Excursus: real space representation suitable for 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
\begin{equation}
-\bra{r'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{r''}
+\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{r'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{r'} \braket{r'}{\chi_{lm}}
-=-i\hbar\nabla_{r'}\,\chi_{lm}(r') \\
-r\ket{r'} & = & r'\ket{r'}
+\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'}}
\end{eqnarray}
we get
\begin{equation}
--i\hbar(r'\times \nabla_{r'})\braket{r'}{\chi_{lm}}E^{\text{SO,KB}}_l
-\braket{\chi_{lm}}{r''}
+-i\hbar(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}
{\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 $r$).
+must be known in real space (with respect to $\vec{r'}$).
\begin{equation}
-\braket{r'}{\chi_{lm}}=
-\frac{\braket{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{r'}{\delta V_l^{\text{SO}}\Phi_{lm}}=
+\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'})
\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 $r''$ 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}}}{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_{r''})
\text{ .}
+\label{eq:solid:so_r2}
\end{equation}
Using the orthonormality property
\begin{equation}
\int Y^*_{l'm'}(\Omega_r)Y_{lm}(\Omega_r) d\Omega_r = \delta_{ll'}\delta_{mm'}
\label{eq:solid:y_ortho}
\end{equation}
-of the spherical harmonics, the norm of the $\chi_{lm}$ reduces to
+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'})
\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{ .}
\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.
+\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}
+All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$.
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
\begin{equation}
\end{equation}
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