however, we need to know explicitly wavefunctions of the intermediate state, which are discussed in the next section.

Absorption coefficient
----------------------

Within the $E1$ transition, the absorption coefficient is given by $$A(\omega) \propto \sum_j\sum_{\alpha}\sum_{\Lambda}
 |\langle\Lambda|x_{\alpha,j}|\psi_0\rangle|^2
 \frac{\Gamma/\pi}{(\hbar\omega-E_{\Lambda}+E_0)^2+\Gamma^2},
\label{eq.absorption}$$ where $|\Lambda\rangle$ with energy $E_{\Lambda}$ represents the final state, which is equivalent to the intermediate state of RXS. A comparison of Eq. (\[eq.absorption\]) with Eq. (\[eq.rxs.amplitude\]) leads to $$A(\omega) \propto -{\rm Im}\, \alpha_0(\omega),$$ where $X$ denotes the imaginary part of $X$.

\[sect.3\]RXS spectra from NO$_{2}$
===================================

Quartet ground state
--------------------

![\[fig.struc1\] Crystal structure of NpO$_{2}$. Gray circles denote Np ions and open circles represent O ions.](Fig.2.eps){width="6.50cm"}

NpO$_{2}$ has the CaF$_2$ type structure ($F_{m\overline{3}m}$) with a lattice constant $a=5.431$ at room temperature, as schematically shown in Fig. \[fig.struc1\].[@Osborne53] Np ions are tetravalent in NpO$_{2}$, as confirmed by the isomer shift in Mössbauer spectra[@Dunlap68] and by the neutron diffraction experiment.[@Delapalme80] In a localized description, each Np ion is in the $(5f)^3$-configuration. The Hamiltonian of Np ions consists of the intra-atomic Coulomb interaction between $5f$ electrons in addition to the SOI of $5f$ electrons. The Slater integrals for the Coulomb interaction and the SOI parameters are evaluated within the Hatree-Fock approximation (HFA),[@Cowan81] and are listed in 