author hackbard Tue, 19 Jun 2012 09:46:45 +0000 (11:46 +0200) committer hackbard Tue, 19 Jun 2012 09:46:45 +0000 (11:46 +0200)

index 507de12..7fa936f 100644 (file)
@@ -220,7 +220,7 @@ r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
\end{eqnarray}
we get

--i\hbar(r'\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
+-i\hbar(\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}
@@ -310,14 +310,26 @@ In total, the matrix elements of the potential for angular momentum $l$ can be c
\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'}})
+-i\hbar(\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\\
+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}
+\frac{2l+1}{4\pi}\\
+&=&
+-i\hbar(\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\\
\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})
+
+and the SO projectors are likewise centered on atoms, the SO potential contribution reads
+
+
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots

+