where the last form is the Green’s function in the time domain evaluated for infinitesimal negative times. Note that the relations of Eq. (5) among n , ρ , G and G also hold for the non-interacting n$_{0}$ , ρ$_{0}$ , G$_{0}$ and G$_{0}$ . The frequency dependent dielectric matrix ε ( ω ) is related to the irreducible polarizability matrix P ( ω ) via Within the RPA approximation, P ( ω ) is a sum over transitions between valence and con- duction states, The pair states | cv 〉 are defined in coordinate space via the single-particle wave functions ψ$_{n}$ ( x ) through For a more compact notation, we label each interband transition ( c, v ) by t with energy Δ$_{t}$ ≡ ϵ$_{c}$ − ϵ$_{v}$ − i 0 + and can write where Π$_{t}$( ω ) is the polarizability contribution of transition t . We can write P ( ω ) in a more general form to handle possible degeneracies in the transition energies Δ$_{t}$. Namely, we break the sum over all transitions t into a first sum over the distinct energies Δ and then a sum over all degenerate transitions δ with energy Δ: Finally, the time domain RPA polarizability is One of the central points of the overall approach of Luttinger and Ward [22] is that one can obtain both the ground-state total energy and interacting G ( ω ) from the extremum of an energy functional of G . In this work we concentrate on the specific case of the Klein functional [50], a functional of both G and an auxiliary non-interacting G$_{0}$ that is meant to be an initial guess for G . It is given by ε ( ω ) = I − V P ( ω ) . P ( ω ) = ∑ c,v 2( ϵ$_{c}$ − ϵ$_{v}$ ) | cv 〉〈 cv | ω 2 − ( ϵ$_{c}$ − ϵ$_{v}$ − i 0 $^{+}$)2 . 〈 x | cv 〉 ≡ 〈 x | c 〉〈 x | v 〉 ∗ = ψ$_{c}$ ( x ) ψ$_{v}$ ( x ) ∗ . P ( ω ) = ∑ t Π$_{t}$( ω ) , Π$_{t}$( ω ) = 2Δ$_{t}$ | t 〉〈 t | ω 2 − Δ 2 t , (6) P ( ω ) = ∑ Δ Π$_{Δ}$( ω ) , Π$_{Δ}$( ω ) = 2Δ ω 2 − Δ 2 ∑ δ | Δ$_{δ}$=Δ | δ 〉〈 δ | . (7) P ( x, x $^{′}$, t ) = − i G$_{0}$ ( x, x $^{′}$, t ) G$_{0}$ ( x $^{′}$, x, − t ) = ∫ ∞ −∞ dω 2 π e − $^{iωt}$P ( x, x $^{′}$, ω ) . (8) F [ G, G$_{0}$ ] = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$H$_{0}$ G$_{0}$ ( ω ) + I − G$_{0}$ ( ω ) − $^{1}$G ( ω ) + ln[ G$_{0}$ ( ω ) − $^{1}$G ( ω )] − U$_{0}$G ( ω ) } + E$_{H}$ [ n ] + Φ$_{xc}$[ G ] . 3. LUTTINGER-WARD APPROACH 6