content_image/329171.jpg content_image/329172.jpg content_image/329173.jpg content_image/329174.jpg content_image/329175.jpg content_image/329176.jpg content_image/329177.jpg content_image/329178.jpg
The ξ + α are the poles of Σ$_{xc}$( ω ) that are above the real axis, Im ξ + α > 0, and σ + α are their residues, while σ − β and ξ − β are the residues and poles below the real axis, Im ξ − β < 0. This allows us to write the derivative as 

A quick manipulation yields 

<img src='content_image/329171.jpg'>

<img src='content_image/329172.jpg'>

If the residue matrices σ ± are positive definite, then the derivatives for empty states are always positive while those of filled states are always negative. For the GW -RPA approxi- mation, we can demonstrate this explicitly: as we will discuss in more detail in in Section 7, the RPA screened Coulomb interaction W ( ω ) = ε − $^{1}$( ω ) has the outer product form 

<img src='content_image/329173.jpg'>

A standard exercise in time-dependent perturbation theory shows that the exact many-body W also has the same outer product form 

<img src='content_image/329174.jpg'>

where E$_{0}$ is the exact ground-state energy, E$_{s}$ are the exact excited-state energies, 〈 x | n$_{s}$$_{0}$ 〉 = 〈 0 | ˆ n ( x ) | s 〉 where ˆ n ( x ) is the fermion density operator and | 0 〉 and | s 〉 are the many-body eigenstates. Thus the exact and RPA W are related via the replacements p, ω$_{p}$, | p 〉 → s, E$_{s}$ − E$_{0}$, | n$_{s}$$_{0}$ 〉 . 

Whether we have the exact or RPA W , the GW self-energy is 

For G = G$_{0}$ and the RPA W , the integral yields 

<img src='content_image/329175.jpg'>

<img src='content_image/329176.jpg'>

Thus the poles and residues of the GW Σ$_{xc}$ are indexed by an eigenvalue ϵ$_{n}$ and a plasmon ω$_{p}$ : the labels α or β in Eq. (24) correspond a pair index ( n, p ). The precise correspondence for the GW -RPA approximation is 

<img src='content_image/329177.jpg'>

and 

<img src='content_image/329178.jpg'>

