under the influence of the second term of eq. (\[eq.crystal\]). Taking account of the multiple scattering from these terms at the origin, we evaluate the resolvent $1/(\hbar\omega-H_{\rm int})$ with respect to the intermediate-state Hamiltonian $H_{\rm int}$: $$\begin{aligned}
&  \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_
{m^ds^d\lambda;m'^ds'^d\lambda'}
\nonumber \\
   = &[G^{5d}_{m^d}(\hbar\omega+i\Gamma-\epsilon_\lambda)^{-1}
     \delta_{\lambda\lambda'}\delta_{m^dm'^d}\delta_{s^ds'^d} 
  \nonumber\\ 
     &- V_{m^ds^d\lambda;m'^ds'^d\lambda'}]^{-1},
\label{eq.matrix1}\end{aligned}$$ where $m^d$ and $s^d$ specify the orbital and spin of the $d$ electron, respectively. The $\epsilon_{\lambda}$ represents the energy of the core hole with $\lambda$ in the $j_p=3/2$ subspace. The scattering potential $V_{m^ds^d\lambda;m'^ds'^d\lambda'}$ includes the second term of eq. (\[eq.crystal\]) and the Coulomb interaction between the $5d$ electron and the $2p$ hole. The latter quantity is expressed in terms of the Slater integrals, which are evaluated within the HF approximation in a Dy$^{3+}$ atom (see Table \[tab.slater\]).[@Com1] The core hole life-time width is set to be $\Gamma=2.5$ eV. Equation (\[eq.matrix1\]) is numerically evaluated.

  --------------------- --------------------- ---------------- ----------------
  $F^{k}(4f,4f)$        $F^{k}(2p,5d)$        $F^{k}(2p,4f)$   $F^{k}(4f,5d)$
  $F^0$ 32.19           $F^0$ 16.13           $F^0$ 44.62      $F^0$ 14.70
  $F^2$ 15.31           $F^2$ 0.489           $F^2$ 1.982      $F^2$ 3.614
  $F^4$ 9.607                                                  $F^4$ 1.741
  $F^6$ 6.911                                                  
                        $G^{k}(2p,5d)$        $G^{k}(2p,4f)$   $G^{k}(4f,5d)$
                        $G^1$ 0.414           $G^2$ 0.207      $G^1$ 1.615
                        $G^3$ 0.245           $G^4$ 0.133      $G^3$ 1.321
                                                               $G^5$ 1.009
  $\zeta_{4f}=$ 0.273   $\zeta_{5d}=$ 0.181                    
  --------------------- --------------------- 