the framework of DMFT in Refs.  and , respectively.

Dynamical mean field theory
---------------------------

The matrices of the lattice Green’s function and the local Green’s function are presented on the basis of the non-orthogonal local base set as $$\begin{aligned}
&& [G({\bm k}, {i{\omega}_n})]^{-1}
\nonumber\\
&&\ \ \ \ =[({i{\omega}_n}+\mu)O({\bm k})-\{H^{dc}_{{\mbox{\scriptsize{LDA}}}}({\bm k})+{\Sigma}({i{\omega}_n})\}],
\label{eqn:Gkw}\\
&& G({i{\omega}_n})= {\frac{1}{V}}\int d{\bm k}~G({\bm k},{i{\omega}_n}) ,
\label{eqn:Gw}\end{aligned}$$ where $H^{dc}_{{\mbox{\scriptsize{LDA}}}}({\bm k})$, $O({\bm k})$ and $\mu$ are the Hamiltonian matrix of Eq. (\[eqn:HLDAdc\]), the overlap matrix, both in the ${\bm k}$-space, and the chemical potential, respectively. Here we neglect the ${\bm k}$-dependence of the self-energy ${\Sigma}$ within the framework of DMFT. The chemical potential $\mu$ is determined to satisfy Luttinger’s theorem. [@re:Mul] All the matrices in Eqs. (\[eqn:Gkw\]) and (\[eqn:Gw\]) have suffices $\{jm\sigma, j^\prime m^\prime \sigma^\prime\}$, where $j$, $m$ and $\sigma$ stand for atomic sites in a unit cell, orbitals and spins, respectively. 

We assume that the Green’s function in the effective medium $G^0({i{\omega}_n})$ may be expressed as the self consistent equation of DMFT: $$\begin{aligned}
G^0({i{\omega}_n})^{-1}=G({i{\omega}_n})^{-1}+{\Sigma}({i{\omega}_n})+\tilde{\mu}-\mu,~
\label{eqn:Gw-local}\end{aligned}$$ where $\tilde{\mu}$ is the chemical potential in the effective medium. 

Iterative Perturbation Theory {#sec:IPT}
-----------------------------

In this paper, IPT method is adopted as a solver for the mapped single impurity problem, generalized 