discussed in Section \[sec:LF\]. 

The self-energy in Eq. (\[eqn:sig-IPT\]) also includes spin-flip and pair-hopping terms of the exchange interactions since the second order self-energy in Eq. (\[eqn:2nd-pertu-Sig\]) includes those terms. In addition, the IPT can be achieved within efficient CPU time since the self-energy in Eq. (\[eqn:sig-IPT\]) is obtained by only matrix calculation and hence the IPT is applicable to realistic materials with larger size by using full-LDA Hamiltonian. Form the discussion in this subsection, present IPT is much appropriate impurity solver and present IPT appropriately treats on-site dynamical correlation effects including both hole-hole and electron-electron scattering as well as electron-hole scattering.

In the following Sections (Secs. \[sec:result-Fe-Ni\] and \[sec:result-NiO\]), we assume the cubic symmetry around transition metal ions and, therefore, the self-energy can be assumed as $$\begin{aligned}
   \Sigma_{jm\sigma j'm'\sigma '}(i\omega_n)=
   \Sigma_{jm\sigma}(i\omega_n)\delta_{mm'}\delta_{jj'}\delta_{\sigma\sigma '}.
\label{eqn:sig-scalar}   \end{aligned}$$ For a cubic symmetry, the local Green’s function becomes diagonal after the ${\bm k}$-integration.

Calculation in the atomic limit based on the ligand field theory {#sec:LF}
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The Hamiltonian mapped onto an isolated atom is defined as $$\begin{aligned}
 H_{atom}      &=& H^{dc}_{atom}+H_{int} \ ,
\label{eqn:HAM-atom}\\
 H^{dc}_{atom} &=& \sum_{(m{\sigma})}{\epsilon}_{mm'{\sigma}}c_{m{\sigma}}^{\dag}c_{m'{\sigma}} \ ,
\label{eqn:HAM-atom-dc}\\
 {\epsilon}_{mm'{\sigma}}&=&h^{({\mbox{\scriptsize{LDA}}})}_{mm'{\sigma}}-\Delta h^{{\sigma}}_{dc}\delta_{mm'} \ .
\label{eqn:ep-atom}\end{aligned}$$ The first term $H^{dc}_{atom}$ in Eq. (\[eqn:HAM-atom\]) is an unperturbed 