moment. We calculate the energy profiles with taking full account of multiplet structures in the intermediate state. We obtain spikes-like curves at Np $M_{4}$ edges for smaller values of the core-level width $\Gamma$ as a reflection of multiplet structures. They are found to merge into a single peak with $\Gamma >1$ eV. The energy profile with $\Gamma\sim 2$ eV seems to agree with the experiment. The azimuthal-angle dependence of the RXS spectra is obtained in agreement with the previous analysis.[@Paixao02; @Caciuffo03] The present analysis provides a sound basis to the previous phenomenological analysis.

The present paper is organized as follows. In Sec. \[sect.2\], we present an expression for the RXS amplitude, which is useful to calculate the energy profiles. In Sec. \[sect.3\], we analyze the RXS spectra in the triple-**k** octupole ordering of NpO$_2$ on the basis of a localized electron model. Section \[sect.4\] is devoted to concluding remarks. In Appendix, we derive the general expression of RXS characterizing energy profiles.

\[sect.2\]Theoretical Framework of RXS
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Second-order optical process
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In the resonant process, an incident photon with energy $\hbar \omega$, wavevector ${\textbf k}$, and polarization vector $\mbox{\boldmath{$\epsilon$}}$ excites a core electron to an empty valence shell of the intermediate 