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\end{equation}  Thus, we find the self-energy in DFT $\Sigma_\text{KS} = V_\text{xc}$ is in fact frequency independent. In comparing with experiment, we always expect deviations at high frequencies because the non-interacting Kohn-Sham framework should only work well near the Fermi surface, where the quasiparticles of Fermi liquid theory are well-defined. However, at low frequencies (which for this section we take to mean $\omega \lesssim 1$~eV, a typical chemical scale) we can hope the DFT spectrum will resemble observations, and in many cases it does. We call these materials weakly correlated.  Strongly We define strongly  correlated materials are as  those compounds where the difference between the  actual self-energy and DFT reference $\left|\Sigma(\omega) - \Sigma_\text{KS}\right|$ is large at low frequencies. We note that defining correlations by measuringhow much $\Sigma$ deviates from  the reference provided by deviation from a  DFT reference  is not the only route. If we were chemists, rather than solid state physicists, we would use the self-energy produced by the Hartree-Fock approximation $\Sigma_\text{HF}$ as our reference. Our definition for correlations is retrospective, in that it assumes we have access to the exact self-energy and then use it for comparison. In practice, we use methods capable of modeling the effects of correlations on the self-energy, with the specific method selected on the basis of the physics required and computational resources. Most methods begin with a specific approximation to DFT, such as LDA or GGA, as a foundation and augment the theory to capture correlations.  The simplest is LDA+U, which captures magnetic Mott insulators well, requires relatively little additional computational resources and is implemented in mature codes. For Dynamical mean field theory, when combined with density functional theory (LDA+DMFT), provides  a vastly  more complete description of the  Mott physics, as well as transition and the  correlated metals, dynamical mean field theory (DMFT) is metallic state, at  the method expense  of choice. Finally, for accurate description two or more orders  of band gaps magnitude  in semiconductors and momentum-dependent contributions to the self-energy, the GW approximation is used.  We can crudely categorize additional computational resource requirements. While DMFT mainly focuses on capturing  the effects frequency-dependence  ofcorrelations by considering  the dominant term self-energy,  in problems where  the low-order expansion of momentum-dependence (or equivalently  the self-energy deviation  \begin{equation}  \Sigma(\vec{k}, \omega) - \Sigma_\text{KS} \sim \Sigma_0(\vec{k}) + \omega \Sigma_1(\vec{k}) + \ldots  \end{equation}  If $\Sigma_0$ $\vec{r}$-dependence)  is large compared to the linear term for low frequencies, we have a large static shift which typically corresponds to materials dominant, such as band gaps  in semiconductors,  the GW approximation is used. The methods are summarized in Table~\ref{tbl:methods}.  \begin{table}   \label{tbl:methods}  \begin{tabular}{l|c|c}  \hline  Method & $\Sigma-\Sigma_\text{KS}$ & Description \\  \hline  LDA+U & const & good for magnetic  Mott states, over-estimates  insulating regime with magnetic order. Extensions to DFT known as LDA+U describe these systems well. If the linear term dominates, the main effect character \\  LDA+DMFT & $\Sigma(\omega)$ & accurate modeling  of the self-energy is $\omega$-dependence captures Mott transition and correlated metals, omits $\vec{k}$-dependence \\  LDA+GW & $\Sigma(\vec{k}, \omega)$ & captures strong $\vec{k}$-dependence, poor treatment of Mott transition \\  \hline  \end{tabular}   \caption{Summary of methods used  to flatten the bands near the Fermi energy while transferring model correlations}  \end{table}  For  the residual spectral weight to higher energies. These mass-enhanced systems corresponds to correlated metals most widely used starting points, LDA  and require techniques such as GG, what is  the GW approximation or dynamical mean field theory (DMFT) physical  basis  for accurate description. the deviations?  Theoretical chemists have classified the introduced errors into two 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. systems.  Armed with an understanding of methods to treat correlations and their physical  and computational tradeoffs, we proceed to construct a workflow for designing  correlated materials.