or in matrix form The direction of approach of the limit, t → 0 + versus t → − 0 $^{+}$, is immaterial as both yield the same answer for real valued Green’s functions G$_{0}$ ( ω ) due to time reversal symmetry. We then have our desired result: Φ GW xc written in terms of the time-ordered ε ( ω ), This expression is our main workhorse. In Sections 7 and 9, we evaluate this integral exactly and approximately to generate alternative forms for Φ GW xc . In this section, we provide expressions for the derivatives of the exchange-correlation functional Φ GW xc [ G$_{0}$ ] and the total energy functional F [ G$_{0}$, G$_{0}$ ] versus the wave functions ψ$_{n}$ and eigenenergies ϵ$_{n}$ or equivalently the potential U$_{0}$ that determine G$_{0}$ . The ϵ$_{n}$ derivatives are used in Section 8 to prove the unboundedness of the energy functional. Separately, these derivative expressions can prove useful to investigators contemplating variational approaches that extremize F [ G$_{0}$, G$_{0}$ ] which require analytical derivatives. We begin with variations of the eigenenergies ϵ$_{n}$ . The derivative of G$_{0}$ is As per Section 4, the only non-zero contribution to the variation of F [ G$_{0}$, G$_{0}$ ] when changing ϵ$_{n}$ comes from the exchange-correlation functional Φ GW xc so We turn this into a contour integral over the upper complex ω half plane. Using Eq. (12), we get contributions from the poles of Σ$_{xc}$( ω ) that are above the real axis and a possible contribution from the pole of ∂G$_{0}$/∂ϵ$_{n}$ if n is occupied (if ϵ$_{n}$ < µ ). To organize the process, we write Σ$_{xc}$( ω ) in the general form of a sum over poles Here, Σ$_{x}$ is the Fock (bare) exchange operator P ( t → 0) = − i ∑ c,v | cv 〉〈 cv | = − i ∑ t | t 〉〈 t | . Φ GW xc [ G$_{0}$ ] = E$_{X}$ [ ρ$_{0}$ ] − 1 2 ∑ t 〈 t | V | t 〉 + 1 2 ∫ ∞ −∞ dω 2 πi tr $^{{}$ln ε ( ω ) $^{}}$. (23) ∂G$_{0}$ ( ω ) ∂ϵ$_{n}$ = | n 〉〈 n | ( ω − ϵ$_{n}$ ) 2 . ∂F [ G$_{0}$, G$_{0}$ ] ∂ϵ$_{n}$ = ∂ Φ GW xc [ G$_{0}$ ] ∂ϵ$_{n}$ = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$Σ$_{xc}$ ( ω ) ∂G$_{0}$ ( ω ) ∂ϵ$_{n}$ $^{}}$. Σ$_{xc}$( ω ) = Σ$_{x}$ + ∑ α σ + α ω − ξ + α + ∑ β σ − β ω − ξ − α . (24) Σ$_{x}$( x, x $^{′}$) = − ∑ v ψ$_{v}$ ( x ) ψ$_{v}$ ( x $^{′}$)$^{∗}$V ( x, x $^{′}$) 12 6. DERIVATIVES OF F AND Φ GW xc