-\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 states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ 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''}}=
+\sum_{lm}
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}&=&\sum_{lm}
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})\nonumber\\
+&=&\sum_l