total charge from the retarded Green function

total charge, $n_{tot}$, is calculated from the retarded Green function

$$A_{ij}(\omega ) = -\frac{1}{\pi} {\rm Im} G_{ij}^{R}(w)$$ (70)

$$\rho_{ij} = \int^{\mu} d\omega A_{ij}(w)$$ (71)

$$= -\frac{1}{\pi} {\rm Im} \int^{\mu} d\omega G_{ij}^{R}(w)$$ (72)

$$n_{tot} = {\rm Tr}[ \rho ]$$ (73)

where $G^R(\omega ) = G(\omega +i\delta )$