content_image/369821.jpg content_image/369822.jpg content_image/369823.jpg content_image/369824.jpg content_image/369825.jpg content_image/369826.jpg content_image/369827.jpg
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 

<img src='content_image/369821.jpg'>

<img src='content_image/369822.jpg'>

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 

<img src='content_image/369823.jpg'>

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$_{0}$ ]: 

<img src='content_image/369824.jpg'>

<img src='content_image/369825.jpg'>

<img src='content_image/369826.jpg'>

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 

<img src='content_image/369827.jpg'>

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 

