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 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 A standard exercise in time-dependent perturbation theory shows that the exact many-body W also has the same outer product form 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 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 and ∂F [ G$_{0}$, G$_{0}$ ] ∂ϵ$_{n}$ = ∑ α 〈 n | σ + α | n 〉 ( ξ + α − ϵ$_{n}$ ) 2 + θ ( µ − ϵ$_{n}$ ) 〈 n | d Σ$_{xc}$( ω ) dω | n 〉 ∣ ∣ ω = ϵ$_{n}$ . ∂F [ G$_{0}$, G$_{0}$ ] ∂ϵ$_{n}$ = θ ( ϵ$_{n}$ − µ ) ∑ α 〈 n | σ + α | n 〉 ( ξ + α − ϵ$_{n}$ ) 2 − θ ( µ − ϵ$_{n}$ ) ∑ β 〈 n | σ − β | n 〉 ( ξ − β − ϵ$_{n}$ ) 2 . (25) W $^{RP A}$( ω ) = V + V ∑ p 2 ω$_{p}$ | p 〉〈 p | ω 2 − ω 2 p V . W $^{exact}$( ω ) = V + V ∑ 2( E$_{s}$ − E$_{0}$ ) | n$_{s}$$_{0}$ 〉〈 n$_{s}$$_{0}$ | ω 2 − ( E$_{s}$ − E$_{0}$ ) 2 V . Σ$_{xc}$( x, x $^{′}$, ω ) = i$^{∫}$ ∞ −∞ dω ′ 2 π e − iω $^{′}$0 $^{+}$G ( x, x $^{′}$, ω − ω $^{′}$) W ( x, x $^{′}$, ω$^{′}$) . Σ$_{xc}$( x, x $^{′}$, ω ) = Σ$_{x}$( x, x $^{′}$) + ∑ p 〈 x | V | p 〉〈 p | V | x $^{′}$〉$^{(}$ ∑ v ψ$_{v}$ ( x ) ψ$_{v}$ ( x $^{′}$)∗ ω + ω$_{p}$ − ϵ$_{v}$ + ∑ c ψ$_{c}$ ( x ) ψ$_{c}$ ( x $^{′}$)∗ ω − ω$_{p}$ − ϵ$_{c}$ ) . σ + $_{v,p}$( x, x $^{′}$) = 〈 x | V | p 〉〈 p | V | x $^{′}$〉 ψ$_{v}$ ( x ) ψ$_{v}$ ( x $^{′}$)∗ , ξ + v,p = ϵ$_{v}$ − ω$_{p}$ (26) σ − $_{c,p}$( x, x $^{′}$) = 〈 x | V | p 〉〈 p | V | x $^{′}$〉 ψ$_{c}$ ( x ) ψ$_{c}$ ( x $^{′}$)∗ , ξ − c,p = ϵ$_{c}$ + ω$_{p}$ . (27) 13