Exchange-correlation energy from model Green's functions: A unified formalism for Green's functions using density functional theory

Kohn-Sham density functional theory (KS-DFT) combined with Green’s function (GF) methods are becoming more routine in the calculation of realistic band-structure and photoemission spectra. The KS orbitals and their energies are typically used as the starting point of a separate GF method, such as one-shot GW or Dynamical Mean Field Theory (DMFT). DFT and the theory of Green’s functions are fundamentally different, and the efficacy of their combination is not understood. In this work we use the exact density-functional formalism to define Green’s functions and characterize their standard spectral features (“quasi-particles” and “satellites”). We derive a Galitskii-Migdal formula to extract density-functional exchange-correlation energies directly from Green’s functions. Using a highly reduced Hubbard model, we isolate the contributions to the exchange-correlation energy from exact and GW spectral features. From a translationally invariant ansatz for the GF, we derive the standard local density approximation (LDA) of DFT and motivate further approximations.