{\epsilon}_{m_3{\sigma}'} + {\epsilon}_{m_2{\sigma}'} +{\tilde{\mu}}}} \ ,
\label{eqn:sig2-atom}\end{aligned}$$ where $|\eta\rangle$, $E_{\eta}$, $N_{\eta}$, $f_{m{\sigma}}$ are the many-electron eigenstates in the Hilbert space of $2^{10}$ multiplets, the energy eigenvalue, the total electron number of the eigenstate $|\eta\rangle$ and the Fermi distribution function obtained from full-CI calculation within the framework of LFT.

One can assign, with a help of the atomic spectrum in LFT, the origin of peaks in LDA+DMFT spectra, e.g. the initial and final multiplets of the corresponding transitions. Generally, peak structures in LFT spectra would suggest the existence of strong scattering channels in LDA+DMFT spectra at around these particular energies.

From discussion in Sections \[sec:IPT\] and \[sec:LF\], the IPT is a appropriate approximation method and is worth being extended to the application of realistic materials in following four reasons: (1) Efficient computational cost (2) Applicability of realistic materials with larger size and more complicated hybridization by using full-LDA Hamiltonian \[This is followed by (1)\] (3) Care of spin-flip and pair-hopping terms of the exchange interactions in multi-orbital systems (4) Direct comparison of LFT spectra within the IPT method with the LDA+DMFT spectra and assignment of the origin of peaks in LDA+DMFT spectra

Energy spectrum and ${\bm k}$-resolved spectrum {#sec:Eq-Gw-Gkw}
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The 