\begin{equation}
U^{AREP} = U_{L}^{AREP}(r) + \sum_{l=0}^{L-1}\sum_{m=-l}^{l}
            [U_{l}^{AREP}(r)- U_{L}^{AREP}(r)]|lm\rangle \langle lm|
\end{equation}