content_image/338805.jpg content_image/338806.jpg content_image/338807.jpg content_image/338808.jpg content_image/338809.jpg content_image/338810.jpg content_image/338811.jpg content_image/338812.jpg
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 

<img src='content_image/338805.jpg'>

Within the RPA approximation, P ( ω ) is a sum over transitions between valence and con- duction states, 

<img src='content_image/338806.jpg'>

The pair states | cv 〉 are defined in coordinate space via the single-particle wave functions ψ$_{n}$ ( x ) through 

<img src='content_image/338807.jpg'>

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 

<img src='content_image/338808.jpg'>

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: 

<img src='content_image/338809.jpg'>

Finally, the time domain RPA polarizability is 

<img src='content_image/338810.jpg'>

## 3. LUTTINGER-WARD APPROACH 

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 

<img src='content_image/338811.jpg'>

<img src='content_image/338812.jpg'>

