We arrive at the desired derivative ∂F $^{GW}$[ G$_{0}$, G$_{0}$ ] ∂ϵ$_{n}$ = θ ( ϵ$_{n}$ − µ ) ∑ v,p |〈 nv | V | p 〉| 2 ( ϵ$_{n}$ − ϵ$_{v}$ + ω$_{p}$ ) 2 − θ ( µ − ϵ$_{n}$ ) ∑ c,p |〈 nc | V | p 〉| 2 ( ϵ$_{c}$ − ϵ$_{n}$ + ω$_{p}$ ) 2 . (28) It is clear that for unoccupied states the derivative is always positive and for filled states it is always negative. Repeating the derivation with the exact W produces the same conclusion which is thus inherent to the GW form of the self-energy when G is non-interacting. We discuss the implications of this result in Section 8. For completeness, we provide the derivatives of F versus the wave functions ψ$_{n}$ ( x ) and versus the potential U$_{0}$ . For the wave function derivatives we have The functional derivative of G$_{0}$ is We insert this into the integral and perform the contour integral. For the general case, and expressions for the GW -RPA approximation are found by performing the substitutions of Eqs. (26,27). For the derivative versus U$_{0}$ , it is convenient to express the potential in the one-particle eigenbasis so U$_{0}$$_{j,k}$ = 〈 j | U$_{0}$ | k 〉 . The variation δU$_{0}$ is then Inserting this into Eq. (17) and performing the contour integral and some algebraic manip- ulations yields δF [ G$_{0}$, G$_{0}$ ] δψ$_{n}$ ( x ) ∗ = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ T + U$_{ion}$ + φ$_{H}$ + Σ$_{xc}$( ω )] δG$_{0}$ ( ω ) δψ$_{n}$ ( x ) ∗ $^{}}$. δG$_{0}$ ( x $^{′}$, x$^{′′}$, ω ) δψ$_{n}$ ( x ) ∗ = ψ$_{n}$ ( x $^{′}$) δ ( x ′′ − x ) ω − ϵ$_{n}$ − i 0 $^{+}$sgn( µ − ϵ$_{n}$ ) . δF [ G$_{0}$, G$_{0}$ ] δψ$_{n}$ ( x ) ∗ = θ ( µ − ϵ$_{n}$ ) 〈 x | T + U$_{ion}$ + φ$_{H}$ + Σ$_{xc}$( ϵ$_{n}$ ) | n 〉 + ∑ α 〈 x | σ + α | n 〉 ξ + α − ϵ$_{n}$ (29) = θ ( µ − ϵ$_{n}$ ) 〈 x | T + U$_{ion}$ + φ$_{H}$ + Σ$_{x}$ | n 〉 + θ ( ϵ$_{n}$ − µ ) ∑ α 〈 x | σ + α | n 〉 ξ + α − ϵ$_{n}$ − θ ( µ − ϵ$_{n}$ ) ∑ α 〈 x | σ − α | n 〉 ξ − α − ϵ$_{n}$ , (30) δU$_{0}$ = ∑ j,k | j 〉 δU$_{0}$$_{j,k}$ 〈 k | . ∂F [ G$_{0}$, G$_{0}$ ] ∂U$_{0}$$_{v,v}$ ′ = − ∑ α 〈 v $^{′}$| σ − α | v 〉 ( ξ − α − ϵ$_{v}$ ′ )( ξ − α − ϵ$_{v}$ ) (31) ∂F [ G$_{0}$, G$_{0}$ ] ∂U$_{0}$$_{c,v}$ = 〈 v | φ$_{H}$ + Σ$_{xc}$( ϵ$_{v}$ ) − U$_{0}$ | c 〉 ϵ$_{v}$ − ϵ$_{c}$ + ∑ α 〈 v | σ + α | c 〉 ( ξ + α − ϵ$_{v}$ )( ξ + α − ϵ$_{c}$ ) (32) ∂F [ G$_{0}$, G$_{0}$ ] ∂U$_{0}$$_{v,c}$ = 〈 c | φ$_{H}$ + Σ$_{xc}$( ϵ$_{v}$ ) − U$_{0}$ | v 〉 ϵ$_{v}$ − ϵ$_{c}$ + ∑ α 〈 c | σ + α | v 〉 ( ξ + α − ϵ$_{c}$ )( ξ + α − ϵ$_{v}$ ) (33) ∂F [ G$_{0}$, G$_{0}$ ] ∂U$_{0}$$_{c,c}$ ′ = ∑ α 〈 c $^{′}$| σ + α | c 〉 ( ξ + α − ϵ$_{c}$ ′ )( ξ + α − ϵ$_{c}$ ) . (34) 14