author hackbard Tue, 19 Jun 2012 14:01:50 +0000 (16:01 +0200) committer hackbard Tue, 19 Jun 2012 14:01:50 +0000 (16:01 +0200)

index 7fa936f..e8c61e1 100644 (file)
@@ -283,7 +283,7 @@ To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the
\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
+and if all states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered

\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
@@ -317,11 +317,14 @@ P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
{\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
\frac{2l+1}{4\pi}\\
&=&
--i\hbar(\vec{r'}\times \nabla_{\vec{r'}})
+-i\hbar
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
+&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right)
+\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}
-
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 $\alpha$)

V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
@@ -329,6 +332,15 @@ V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
and the SO projectors are likewise centered on atoms, the SO potential contribution reads

+The $E_l^{\text{SO,KB}}$ are given by
+
+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}}=
+
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots