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The idea behind DFT is that for ground state properties, we can use the total electron density $\rho(\vec{r})$ as the fundamental variable in our equations, rather than the complicated quantum many-body wavefunction $\Psi$. Kohn and Sham~\cite{Kohn_1965} recast the theory in practical form: the proved that there exists a periodic potential $V_\text{KS}(\vec{r}) = V_\text{KS}[\rho](\vec{r})$, which itself is a functional of the density, and that solving the problem of non-interacting electrons in the presence of this potential will give the exact density. The Kohn-Sham potential consists of three contributions: $V_\text{KS} = V_\text{ion} + V_\text{H} + V_\text{xc}$. The first is the one-body attractive potential of the nuclear ions. The other two arise from the electron-electron interaction: the (classical) Hartree component $V_\text{H}$ captures most of the Coulomb interaction, and the remaining contribution is contained in the exchange-correlation term $V_\text{xc}$.  In practice, the exchange-correlation term is difficult to capture, and is generally modeled by approximations known as the local density approximation (LDA) or generalized gradient approximation (GGA). Theoretical chemists have classified the introduced errors into two types\cite{Cohen_2008}. types~\cite{Cohen_2008}.  The delocalization error describes the tendency of LDA/GGA to distribute an electron added to a system over too large a spatial region, which leads to underestimation of band gaps. The static correlation error describes the poor treatment of spin states, which leads to an inability to capture the Mott insulating state ubiquitous in strongly correlated systems, including the superconducting cuprates. Assuming we have the exact $V_\text{xc}$, DFT guarantees the correct ground state density and energy, but makes no claims about the electronic spectrum. For electrons moving in the lattice potential $V_\text{ion}$ of the nuclear ions, the general form of the Green's function is  \begin{equation}