-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 $\alpha$)
-\begin{equation}
-V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
-\end{equation}
-and the SO projectors are likewise centered on atoms, the SO potential contribution reads
+If these projectors are considered to be centered around atom positions $\vec{\tau}_{\alpha n}$ of atoms $n$ of species $\alpha$, the variable $\vec{r}'$ in the previous equations is changed to $\vec{r}'_{\alpha n}=\vec{r}'-\vec{\tau}_{\alpha n}$, which implies
+\begin{eqnarray}
+r'&\rightarrow&r_{\alpha n}=|\vec{r}'-\vec{\tau}_{\alpha n}|\\
+\Omega_{\vec{r}'}&\rightarrow&\Omega_{\vec{r'}-\vec{\tau}_{\alpha n}}\\
+\delta V_l(r')&\rightarrow&\delta V_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+u_l(r')&\rightarrow&u_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+Y_{lm}(\Omega_{\vec{r}'})&\rightarrow&
+Y_{lm}(\Omega_{\vec{r}'-\vec{\tau}_{\alpha n}})
+\text{ .}
+\end{eqnarray}
+Within an iterative treatment on a real space grid consisting of $n_{\text{g}}$ grid points, the sum