\! 
U_{m_1m_2:m_3m_4}c_{jm_1{\sigma}_1}^{\dag}c_{jm_2{\sigma}_2}^{\dag}c_{jm_4{\sigma}_4}c_{jm_3{\sigma}_3}~. 
\nonumber \\
\label{eqn:int}\end{aligned}$$ The Hamiltonian $H$ is divided into two parts: the unperturbed part ($H^{dc}_{{\mbox{\scriptsize{LDA}}}}$) and the interaction part ($H_{int}$). $h^{jj'({\mbox{\scriptsize{LDA}}})}_{mm'{\sigma}}$ in Eq. (\[eqn:HLDAdc\]) is the full LDA Hamiltonian constructed by the Tight-Binding LMTO (TB-LMTO) method, [@re:LMTO] without projecting onto any kind of effective local orbitals. The use of original full LMTO Hamiltonian in LDA+DMFT-IPT(1) enables us to describe the change of hybridization caused by local Coulomb interaction. $\Delta h^{{\sigma}}_{dc}$ in Eq. (\[eqn:HLDAdc\]) is the double-counting term included in LDA Hamiltonian as averaged Coulomb and exchange interactions, because we need to subtract $\Delta h^{{\sigma}}_{dc}$ from $h^{jj'({\mbox{\scriptsize{LDA}}})}_{mm'{\sigma}}$ in order to construct the unperturbed part of the Hamiltonian $H^{dc}_{{\mbox{\scriptsize{LDA}}}}$. Note that an index $j$ in the summation of Eqs. (\[eqn:HLDAdc\]) and (\[eqn:int\]) runs over atomic sites, $\{m\}$ orbital indices and $\{{\sigma}\}$ spins.

$U_{m_1m_2:m_3m_4}$ in Eq. (\[eqn:int\]) is the on-site electron-electron interaction matrix and presented by using the Slater integrals $F^k$ as $$\begin{aligned}
U_{m_1m_2:m_3m_4} &=& \sum_{0\leq k\leq 2l}a_k(m_1m_2:m_3m_4)F^k ,
 \label{eqn:U}\\
a_k(m_1m_2:m_3m_4) &=& {\frac{4\pi}{2k+1}} \sum_{q=-k}^{k} \langle lm_1|Y_{kq}|lm_2\rangle \nonumber\\
&\times & \langle lm_3|Y_{kq}^{*}|lm_4\rangle,
 \label{eqn:ak}\end{aligned}$$ where $\{|lm\rangle\}$ are the basis set of complex spherical harmonics. Note that Eqs. (\[eqn:U\]) and (\[eqn:ak\]) are the general expression and invariant under any rotational operation. [@re:LDA+U2] For d-orbitals 