intermediate state. We first evaluate the Slater integrals for the Coulomb interaction and the SOI parameters within the HFA, which are shown in Table \[table.4\]. These values are reduced by taking account of screening effects. The reduction factors are set the same as in the ground state. The Hamiltonian of the intermediate state, consisting of the full intra-atomic Coulomb interactions between $5f$-$5f$, $5f$-$3d$ and $3d$-$3d$ electrons as well as the SOI of $5f$ and $3d$ electrons, is represented by $1001 \times (2 j_d+1)$ microscopic states with the total angular momentum of the core hole $j_d=3/2$ and $5/2$ corresponding to the $M_4$ and the $M_5$ edges, respectively. Diagonalizing the Hamiltonian matrix, we obtain multiplet structures in the intermediate state. The $\alpha_2(\omega)$ is calculated by using Eq. (A.8).

  ---------------------- --------------------- ------------- -------------
  $F^0$ 181.0            $F^0$ 29.13           $F^0$ 20.54   $G^1$ 2.158
  $F^2$ 92.62            $F^2$ 2.749           $F^2$ 10.39   $G^3$ 1.306
  $F^4$ 59.67            $F^4$ 1.281           $F^4$ 6.943   $G^5$ 0.914
                                               $F^6$ 5.017   
  $\zeta_{3d}=$ 77.278   $\zeta_{5f}=$ 0.339                 
  ---------------------- --------------------- ------------- -------------

  : \[table.4\] Slater integrals and spin-orbit interaction parameters in the $(3d)^{9}(5f)^4$ configuration within the HFA (in units of eV).[@Cowan81]

The energy profile is proportional to $|\alpha_2(\omega)|^2$ in the octupole ordering phase. The calculated spectra around $M_4$ 