The numerators have no ω dependence. The standard contour integral for analytic f ( z ) applied to this case with k = 2 gives a zero derivative and null result. So we have our desired result Hence, for a fixed G , we can evaluate F at any valid G$_{0}$ without changing its value. Physically, F doesn’t depend on the initial choice of non-interacting Hamiltonian. Since we are restricting G to be noninteracting, we may as well set G = G$_{0}$ . Then F simplifies to We have the noninteracting kinetic, electron-ion, Hartree and exchange-correlation energies. Operationally, the first three terms are identical to their counterparts in DFT. Due to the extremal nature of F about the extremizing G , F [ G$_{0}$, G$_{0}$ ] provides a variational estimate of the ground-state energy with the error being smallest for the “best” G$_{0}$ . Except for Φ$_{xc}$, the other energy terms depend only on the density matrix ρ$_{0}$ . Only Φ$_{xc}$ depends on the energy eigenvalues ϵ$_{n}$ . Before we end this section, we make brief comments on the functional derivative of F [ G$_{0}$, G$_{0}$ ] versus G$_{0}$ or equivalently versus the potential U$_{0}$ . Starting from Eq. (10) with G = G$_{0}$ or Eq. (14) and using the relation of δG$_{0}$ to δU$_{0}$ in Eq. (11), we have at least three equivalent ways of writing the variation of F [ G$_{0}$, G$_{0}$ ]: Eq. (15), written in terms of the variation δG$_{0}$ , is used below when computing derivatives of F versus the ψ$_{n}$ ( x ) and ϵ$_{n}$ that characterize G$_{0}$ . Eq. (17) is written in terms of the variation of the potential U$_{0}$ that generates G$_{0}$ . Setting δF = 0 in Eq. (17) for arbitrary δU$_{0}$ yields the matrix equation which is the linearized Sham-Schl¨ uter (LSS) equation for the exchange-correlation potential operator V$_{xc}$ = U$_{0}$ − φ$_{H}$ [47, 48]. This is the condition determining the static and Hermitian V$_{xc}$ that most closely resembles the dynamic and non-Hermitian Σ$_{xc}$( ω ). As Casida has noted, this is equivalent to saying that V$_{xc}$ is optimal in a variational sense [53]. The LSS ∮ dz 2 πi · f ( z ) ( z − a ) k = 1 ( k − 1)! d k − $^{1}$f ( z ) dz k − 1 ∣ ∣ z = a (12) δF [ G, G$_{0}$ ] δU$_{0}$ ∣ ∣ G = 0 . (13) F [ G$_{0}$, G$_{0}$ ] = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ T + U$_{ion}$ ] G$_{0}$ ( ω ) } + E$_{H}$ [ n$_{0}$ ] + Φ$_{xc}$[ G$_{0}$ ] = tr $^{{}$[ T + U$_{ion}$ ] ρ$_{0}$ } + E$_{H}$ [ n$_{0}$ ] + Φ$_{xc}$[ G$_{0}$ ] (14) δF [ G$_{0}$, G$_{0}$ ] = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ φ$_{H}$ + Σ$_{xc}$( ω ) − U$_{0}$ ] δG$_{0}$ ( ω ) $^{}}$. (15) = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ T + U$_{ion}$ + φ$_{H}$ + Σ$_{xc}$( ω )] δG$_{0}$ ( ω ) $^{}}$. (16) = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$G$_{0}$ ( ω ) [ φ$_{H}$ + Σ$_{xc}$( ω ) − U$_{0}$ ] G$_{0}$ ( ω ) δU$_{0}$ $^{}}$. (17) ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$G$_{0}$ ( ω ) [ φ$_{H}$ + Σ$_{xc}$( ω )] G$_{0}$ ( ω ) = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$G$_{0}$ ( ω ) U$_{0}$G$_{0}$ ( ω ) (18) 9