+In total, the matrix elements of the SO potential can be calculated by
+\begin{eqnarray}
+&&-i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\nonumber\\
+&=&-i\hbar\sum_l(\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)\cdot
+\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} \cdot
+\frac{2l+1}{4\pi}\nonumber\\
+&=&
+-i\hbar\sum_l
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P'_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
+\left(\frac{\vec{r}'\times\vec{r}''}{r'r''}\right)\cdot
+\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}\text{ ,}\nonumber\\
+\label{eq:solid:so_fin}
+\end{eqnarray}
+since derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as is illustrated below (equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}).
+\begin{eqnarray}
+\left(\vec{r}\times\nabla_{\vec{r}}\right)f(r)&=&
+\left(\begin{array}{l}
+r_y\frac{\partial}{\partial r_z}f(r)-r_z\frac{\partial}{\partial r_y}f(r)\\
+r_z\frac{\partial}{\partial r_x}f(r)-r_x\frac{\partial}{\partial r_z}f(r)\\
+r_x\frac{\partial}{\partial r_y}f(r)-r_y\frac{\partial}{\partial r_x}f(r)
+\end{array}\right)
+\label{eq:solid:rxp1}
+\end{eqnarray}
+\begin{eqnarray}
+r_i\frac{\partial}{\partial r_j}f(r)-r_j\frac{\partial}{\partial r_i}f(r)&=&
+r_if'(r)\frac{\partial}{\partial r_j}(r_x^2+r_y^2+r_z^2)^{1/2}-
+r_jf'(r)\frac{\partial}{\partial r_i}(r_x^2+r_y^2+r_z^2)^{1/2}\nonumber\\
+&=&
+r_if'(r)\frac{1}{2r}2r_j-r_jf'(r)\frac{1}{2r}2r_i=0
+\label{eq:solid:rxp2}
+\end{eqnarray}
+
+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
+\begin{equation}
+\sum_{\vec{r}''_{\alpha n}}
+\sum_{lm}-i\hbar(\vec{r}'_{\alpha n}\times \nabla_{\vec{r}'_{\alpha n}})
+\braket{\vec{r}'_{\alpha n}}{\chi^{\text{SO}}_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi^{\text{SO}}_{lm}}{\vec{r}''_{\alpha n}}
+\braket{\vec{r}''_{\alpha n}}{\Psi}
+\qquad\forall\,\bra{\vec{r}'_{\alpha n}}
+\end{equation}
+to obtain all elements $\bra{\vec{r}'_{\alpha n}}$, involves $n_{\text{g}}^2$ evaluations of equation~\eqref{eq:solid:so_fin} for eeach atom, if the projectors are short-ranged, i.e.\ $\delta V_l=0$ outside a certain cut-off radius.
+Thus, this method scales linearly with the number of atoms.
+
+The $E_l^{\text{SO,KB}}$ are given by
+\begin{equation}
+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}}=
+\end{equation}
+