intermediate state is evaluated by the following steps. Let the $E_1$ transition take place at the origin. The complex of $4f$ electrons and the $2p$ hole is assigned as the eigenstate $|\nu\rangle$ with energy $E_{\nu}$, which is calculated by diagonalizing the matrix of the intra-atomic Coulomb interaction. To keep the matrix size manageable, the space of $4f$ states is restricted within the space of $J=15/2$. This restriction causes only minor errors in the RXS spectra, since the RXS amplitude contains the overlap between the $4f$ states in the intermediate state and that in the initial state, which becomes very small for the $4f$ states outside the $J=15/2$ subspace. Since the photoexcited $5d$ electron interacts with the $2p$ hole and with $4f$ electrons, the complex of $4f$ electrons and the $2p$ hole serves as a scatterer to the $5d$ electron. Thus we have the resolvent at the origin, $$\begin{aligned}
 &  \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_
{m^ds^d\nu;m'^ds'^d\nu'}
\nonumber \\
 &  = [G^{5d}_{m^d}(\hbar\omega+i\Gamma-E_\nu)^{-1}
     \delta_{\nu\nu'}\delta_{m^dm'^d}\delta_{s^ds'^d} 
  \nonumber\\ 
     &  - U_{m^ds^d\nu;m'^ds'^d\nu'}]^{-1},
\label{eq.matrix2}\end{aligned}$$ where $m^d$ and $s^d$ specify $5d$ states. The potential $U_{m^ds^d\nu;m'^ds'^d\nu'}$ includes the $5d$-$4f$ Coulomb interaction as well as the $5d$-$2p$ Coulomb interaction. They are expressed in terms of the Slater integrals, which are 