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 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{\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'}}
+\end{eqnarray}
+we get
+\begin{equation}
+-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}
+\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 $\vec{r'}$).
+\begin{equation}
+\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'})
+\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
+\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''})
+\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 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
+{\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'} \\
+&=&\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{ .}
+\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}
+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''}}=
+\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'}
+\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
+\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'})
+\end{equation}
+In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
+\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(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\\
+&&\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}
+\end{eqnarray}
+
+Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
+\begin{equation}
+\end{equation}
projects / lectures / latex.git / blobdiff