Exchange-correlation energy from exact and model Green's functions

Density functional theory (DFT) and Green’s function methods represent two large, and seemingly disparate, approaches to modeling strongly correlated physics. Green’s function based methods, such as Dynamical Mean Field Theory (DMFT), have proven effective in modeling properties of real systems with strong correlation. The intersections of DFT and Green’s function methods are still not well understood and the failures and pathologies of each remains a matter of open research. We derive a formula to determine the exchange-correlation (XC) energy associated to exact and model Green’s functions. Using an exactly solvable interacting system, we show that our formula yields the exact XC energy, and we determine the XC energy associated with the Green’s function of DMFT. A formula connecting DFT quantities with Green’s functions provides a promising step towards better approximate forms of Green’s functions and density functionals.