author hackbard Mon, 18 Jun 2012 19:45:56 +0000 (21:45 +0200) committer hackbard Mon, 18 Jun 2012 19:45:56 +0000 (21:45 +0200)

index c4cf869..1fc7a17 100644 (file)
@@ -200,3 +200,75 @@ V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(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}
+
+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''}
+\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'}
+\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''}
+\text{ .}
+\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 $r$).
+\begin{equation}
+\braket{r'}{\chi_{lm}}=
+\frac{\braket{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}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\text{ .}
+\end{equation}
+In this expression, only the spherical harmonics are complex functions.
+Thus, the complex conjugate with respect to $r''$ is given by
+\begin{equation}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{r''}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
+\text{ .}
+\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
+\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}
+
+Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
+\begin{equation}
+\end{equation}