The ω integral can be turned into a closed contour integral over the upper complex ω plane due to the factor e iω 0 $^{+}$. E$_{H}$ is the Hartree energy stemming from the electron density n ( x ) associated with G ( ω ): The functional Φ$_{xc}$[ G ] is the exchange-correlation energy functional for this approach and is a complicated functional of G . Formally, it is a sum over all diagrams to all orders in the Coulomb interaction obtained by closing all skeleton self-energy diagrams with Green’s functions (with appropriate weight) [22]. Much like in DFT, choosing an approximate form for Φ$_{xc}$ corresponds to including a certain level of treatment of exchange-correlation effects. However, since Φ$_{xc}$ is a functional of the more information-rich, non-local and dynamic G ( x, x $^{′}$, ω ) as opposed to the simpler density n ( x ) in DFT, relatively simple forms for Φ$_{xc}$ will be equivalent to rather complex functionals of the density in DFT. On the other hand, compared to DFT, the price for more information is the increased complexity of the energy functional F and the entire theoretical and computational approach. Within Luttinger-Ward theory, one focuses on the extremum of the functional F . At the extremum, the value of F is the true ground-state total energy, and the extremizing G is the true one-particle Green’s function [10, 22]. Therefore this framework provides, in principle, both exact total energies and one-particle properties such as quasiparticle wave functions, electron densities, and band structures. To find the variation of F versus G , we use a matrix differentiation rule based on the properties of the determinant: where δA is the variation in the matrix A . The variation of F for fixed G$_{0}$ is Again, the situation is analogous to DFT where the exchange-correlation potential is the functional derivative versus the density of the exchange-correlation energy functional. Within Luttinger-Ward theory, there are two separate challenges. The first and most ob- vious is choosing some approximate form for Φ$_{xc}$. Second, there is the additional challenge that, unlike DFT where N -presentability conditions for the electron density n ( x ) have been known for a long time [51, 52], similar conditions for the Green’s function G ( x, x $^{′}$, ω ) are not known to the best of our knowledge. In other words, it is not generally known which subset of functions G ( x, x $^{′}$, ω ) correspond to physically realizable Greens functions for interacting electrons with the standard many-body Hamiltonian. Therefore, one must also decide on some scheme for restricting oneself to physically correct forms of G . From a more pragmatic Setting this to zero for arbitrary δG yields the Dyson Eq. (4) with the self-energy given by the functional derivative Σ$_{xc}$( ω ) = 2 πi δ Φ$_{xc}$ δG ( ω ) . δF ∣ ∣ G$_{0}$ = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr { − G$_{0}$ ( ω ) − $^{1}$δG ( ω ) + G ( ω ) − $^{1}$δG ( ω ) − U$_{0}$ δG ( ω ) } + δE$_{H}$ + δ Φ$_{xc}$ = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr { [ G ( ω ) − 1 − G$_{0}$ ( ω ) − 1 + φ$_{H}$ + 2 πi δ Φ$_{xc}$ δG ( ω ) − U$_{0}$$^{]}$ δG ( ω ) $^{}}$. (10) tr { ln( A ) } = ln { det( A ) } → δ tr { ln( A ) } = tr { A − $^{1}$δA } (9) E$_{H}$ [ n ] = 1 2 ∫ dx$^{∫}$ dx ′ n ( x ) V ( x, x $^{′}$) n ( x $^{′}$) = 1 2 ∫ dx n ( x ) φ$_{H}$ ( x ) . 7