total charge from contour integration

usually eq.(76) is unstable. employ contour integration

$$\rho_{ij} = \frac{1}{2\pi i} \int_{C} dz G_{ij}(z)$$ (77)

$$= \frac{1}{2\pi i} \int_{C_1} dz G_{ij}(z) + \frac{1}{2\pi i} \int_{C_2} dz G_{ij}(z)$$ (78)

$$= \frac{1}{\pi} {\rm Im} \int_{C_1} dz G_{ij} (z)$$ (79)

where the path C = ($\mu$,0) $\rightarrow$ ($\mu$,$i \Gamma$) $\rightarrow$ ($E_{min}$, $i \Gamma$) $\rightarrow$ ($E_{min}$, $-i \Gamma$) $\rightarrow$ ($\mu$,$-i \Gamma$) $\rightarrow$ ($\mu$,0), and $C_1$ is the lower half of C and $C_2$ is the upper half of C. i.e. $C_1$ = ($E_{min}$,0) $\rightarrow$ ($E_{min}$, $-i \Gamma$) $\rightarrow$ ($\mu$,$-i \Gamma$) $\rightarrow$ ($\mu$,0),

$E_{min}$ is smaller than the minimum of the eigenvalues.