one-electron part. The Coulomb interaction $H_{int}$ is the same as Eq. (\[eqn:int\]). $h^{({\mbox{\scriptsize{LDA}}})}_{mm'{\sigma}}$ is the on-site element of LDA Hamiltonian and therefore $h^{({\mbox{\scriptsize{LDA}}})}_{mm'{\sigma}}$ includes the information of bulk matrix with electron transfer, crystal field and orbital hybridization. $\Delta h^{{\sigma}}_{dc}$ is the double counting term same as in Eq. (\[eqn:HLDAdc\]). Then one electron energy ${\epsilon}_{mm'{\sigma}}$ in Eq. (\[eqn:ep-atom\]) includes the crystal field splitting. Once we obtain ${\epsilon}_{mm'{\sigma}}$ and the Slater integrals, many-electron eigenstates of Eq. (\[eqn:HAM-atom\]) should be obtained within the framework of the ligand field theory (LFT). [@re:LFT] Full configuration interaction (CI) calculation is carried out by using the basis of all the multiplet of d electrons, the number of which is $2^{10}=1024$. Using Full-CI calculation yields an exact solution for local self-energy of Eq. (\[eqn:sig-IPT\]) in the atomic limit. The Green’s function, the self-energy and the second-order self-energy in the atomic limit are then defined as $$\begin{aligned}
G^{at}_{mm'{\sigma}}(i\omega_{n})
&=& {\frac{1}{Z}}\sum_{\eta \nu } 
{\frac{\langle \eta|c_{m{\sigma}}|\nu\rangle \langle \nu|c^{\dag}_{m'{\sigma}}|\eta\rangle}{i{\omega}_n +(E_{\eta}-\mu N_{\eta})-(E_{\nu}-\mu N_{\nu})}}
\nonumber\\
&\:\:& \:\: \:\: \:\: \:\:   \times \{e^{-\beta( E_{\eta}-\mu N_{\eta})}+e^{-\beta(E_{\nu}-\mu N_{\nu})}\}, 
\label{eqn:G-atom}\\
{\Sigma}^{at}_{mm'{\sigma}}(i{\omega})
&=& i{\omega}-({\epsilon}_{mm'{\sigma}}-\mu)-[G^{at}(i{\omega})]^{-1}_{mm'{\sigma}}, 
\label{eqn:Sig-atom}\\
\Sigma^{at(2)}_{m{\sigma}}(i\omega) 
&=& \sum_{m_1m_2m_3{\sigma}'}(U_{mm_2:m_1m_3}-U_{mm_2:m_3m_1}\delta_{{\sigma}{\sigma}'})\nonumber\\
\times && \! \! \! \! \! \! U_{m_1m_3:mm_2}\nonumber\\
\times && \! \! \! \! \! \! {\frac{ f_{m_2{\sigma}'}(1-f_{m_1{\sigma}}-f_{m_3{\sigma}'})+f_{m_1{\sigma}}f_{m_3{\sigma}'}}{i{\omega}- {\epsilon}_{m_1{\sigma}} - 