energy spectrum and the ${\bm k}$-resolved spectrum may be more physically understandable if we present the Green’s function in the orthogonal base, subject to the sum rule of the energy integration of the local Green’s function.  The lattice Green’s function and the local Green’s function in the orthogonal base are defined as $$\begin{aligned}
\!\!\!\!\!\!\!\!\![{\mathscr{G}}({\bm k}, {\omega})]^{-1}\!\!\!\! &=& \!\!\! O^{-\frac{1}{2}}({\bm k})G^{-1}({\bm k}, {\omega})O^{-\frac{1}{2}}({\bm k})
\label{eqn:Gkw-ortho-1}\\
&=&\!\!\! [({\omega}+\mu)
\nonumber\\
&-&\!\!\! O^{-\frac{1}{2}}({\bm k})
\{H^{dc}_{{\mbox{\scriptsize{LDA}}}}({\bm k})+{\Sigma}({\omega})\}O^{-\frac{1}{2}}({\bm k})],
\label{eqn:Gkw-ortho-2}\\
{\mathscr{G}}({\omega})&=& {\frac{1}{V}}\int d{\bm k}~{\mathscr{G}}({\bm k},{\omega}).~
\label{eqn:Gw-ortho-1}\end{aligned}$$ The energy spectrum will be shown with the imaginary part of ${\mathscr{G}}({\omega})$. ${\bm k}$-resolved spectrum will be also presented with an imaginary part of ${\mathscr{G}}({\bm k},{\omega})$, which may be compared with the angle-resolved photoemission spectra (ARPES).

Computational Details {#sec:Comp-Detals}
---------------------

The $8 \times 8 \times 8$ ${\bm k}$-points are used in the ${\bm k}$-integration within the whole Brillouin zone. The ${\bm k}$-integration is carried out by using a generalized tetrahedron method. [@re:Fuji]  The Padé approximation is adopted for analytic continuation of the Green’s function from the Matsubara frequencies ${i{\omega}_n}$ to the real ${\omega}$-axis.  We adopt $2^{11}=2048$ Matsubara frequencies.

Here, all the calculation were carried out at $T=1000~{\mbox{K}}$ to reduce the number of adopted Matsubara frequency. To consider temperature dependence of 