The potential is often given in position space. It is then convenient to establish the connection between \( V_{lm, l'm'}(k,k') \) and \( V_{lm, l'm'}(r,r') \). Inserting the completeness relation for the position quantum numbers in equation (4) results in $$ \begin{equation} V =\int d\mathbf{r}\int d\mathbf{r}'\left\{\int d{\hat{k}}Y_{lm}^*(\hat{k})\langle \mathbf{k}\vert \mathbf{r}\rangle\right\}\langle\mathbf{r}\vert V\vert\mathbf{r}'\rangle\left\{\int d\hat{k}'Y_{lm}(\hat{k}')\langle\mathbf{r'}\vert\mathbf{k}'\rangle\right\} \tag{5} \end{equation} $$