\begin{equation}
U^{SOREP} = U_{LJ}^{SOREP}(r) + \sum_{l=0}^{L-1}\sum_{j=|l-1/2|}^{l+1/2}\sum_{m=-j}^{j}
           [U_{ij}^{SOREP}(r)- U_{LJ}^{SOREP}(r)]|ljm\rangle \langle ljm|
\end{equation}