by Fujiwara [*et al.*]{} for multi-orbital bands on arbitrary electron occupation with different Coulomb $U$. [@re:Fuji; @re:miura] In the IPT scheme, the self-energy is determined with the interpolation scheme between the high frequency limit and the strong interaction limit (the atomic limit) by using the second order self-energy.

The second order self-energy is calculated as $$\begin{aligned}
\!\!\!\!\!\!\!\!\Sigma^{(2)}_{m{\sigma}}(\tau) \!\!\!\! &=& \!\!\!\! - \!\!\!\!\!\!\!\!\!\!\sum_{m_1m_2m_3{\sigma}'}\!\!\!\!\!\!\!(U_{m_1m_2:mm_3}-U_{mm_2:m_3m_1}\delta_{{\sigma}{\sigma}'})\nonumber\\
&\times& \!\!\!\! U_{m_1m_3:mm_2}G^0_{m_1{\sigma}}(\tau)G^0_{m_3{\sigma}'}(\tau)G^0_{m_2{\sigma}'}(-\tau)
\label{eqn:2nd-pertu-Sig}\\
\!\!\!\!\!\!\!\!\Sigma^{(2)}(i\omega_{n}) \!\!\!\! &=& \!\!\!\!\! \int_{0}^{\beta} \!\!\!\! d\tau 
e^{i\omega_{n}\tau}\Sigma^{(2)}(\tau) \ . \label{eqn:2nd-pertu-Sig-fourier}  \end{aligned}$$ By using the second order self-energy $\Sigma^{(2)}({i{\omega}_n})$,  the self-energy is then expressed in a matrix form as $$\begin{aligned}
\! \! \Sigma(i\omega_n) \! = \! \Sigma^{{\mbox{\scriptsize{HF}}}} \! \! \! 
+\! A\Sigma^{(2)}(i\omega_n)[1\! -\! B(i\omega_n)\Sigma^{(2)}(i\omega_n)]^{-1},
\label{eqn:sig-IPT}   \end{aligned}$$ where $\Sigma^{{\mbox{\scriptsize{HF}}}}$ is the Hartree-Fock energy. The matrices $A$ and $B(i\omega_n)$ are determined by requiring the self-energy ${\Sigma}({i{\omega}_n})$ to be exact in the high-frequency limit $({i{\omega}_n}\to \infty)$,  and in the atomic limit $(U\to \infty)$ [@re:Fuji] as $$\begin{aligned}
\!\!\!\!\!\! A \!\! &=& \!\! \lim_{i{\omega}_n\to\infty}\{\Sigma(i\omega_n)-\Sigma^{HF}\}\Sigma^{(2)}(i\omega_n)^{-1},
\label{eqn:Sig-IPT-A}\\ 
\!\!\!\!\!\! B(i{\omega}_n) \!\! &=& \!\! \Sigma^{at(2)}(i\omega_n)^{-1}-\{\Sigma^{at}(i\omega_n)-\Sigma^{HF}\}^{-1}A ,
\label{eqn:Sig-IPT-B} \end{aligned}$$ where $\Sigma^{at}(i\omega_n)$ and $\Sigma^{at(2)}(i\omega_n)$ in Eq. (\[eqn:Sig-IPT-B\]) are the self-energy and the second order self-energy in the atomic limit.  These values are calculated exactly within the framework of the ligand field theory (LFT), which is 