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The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Since the behavior of the constituent electrons are strongly coupled to one another, studying the behavior of individual particles generally provides little insight into the macroscopic properties of a correlated material. While this conceptual definition is valuable for understanding the fundamental physics, it is of less use to a DFT practitioner, who seeks to make predictions for comparison with experimental observations. To arrive at an operational definition of a correlated material, we examine DFT and how it relates to the observed electronic spectra.  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: solve 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 a periodic this  potential$V_\text{KS}(\vec{r})$ and the sum of their wavefunctions (squared)  will give the exact density. Breaking the The  Kohn-Sham potential apart, we find 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 is challenging to capture, and is generally modeled by approximations known as the local density approximation (LDA) or generalized gradient approximation (GGA).  % Technically, Mott insulators do not exist at T = 0. Magnetism will release 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. The delocalization error describes  the entropy 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 state. 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, given by 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}  G(\omega) = \frac{1}{\omega + \nabla^2/2 + \mu  - V_\text{ion} - V_\text{H} -  \Sigma(\omega)}. \end{equation}  We have written the definition of $G(\omega)$ in used  atomic units for electrons in the lattice units, included a chemical  potential of $\mu$, and separated  the nuclear ions. large Hartree component out from the self-energy $\Sigma$.  The self-energy$\Sigma(\omega)$  is generally frequency-dependent, and we have omitted the argument $\vec{r}$ from all quantities. The eigenenergies that are the result diagonalizing the Kohn-Sham hamiltonian $H_\text{KS} = -\frac{1}{2}\vec{\nabla}^2 - \mu  + V_\text{KS}(\vec{r})$ lack the ability to does not  capture the frequency-dependent effects of many-body interactions, and should not be interpreted as physical eigenvalues. Nevertheless, we often ignore formal justification and compute the Green's function using the Kohn-Sham solution anyway: \begin{equation}  G_\text{KS}(\omega) = \frac{1}{\omega - H_\text{KS}}.  \end{equation}  Thus, we find the self-energy in DFT $\Sigma_\text{KS} =V_\text{H} +  V_\text{xc}$ 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 correlated materials are those compounds where the actual self-energy $\Sigma(\omega)$ deviates strongly from the DFT reference $V_\text{H} + V_\text{xc}$ $V_\text{xc}$  at low frequencies. We note that defining correlations by measuring how much $\Sigma$ deviates from the reference provided by DFT 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.  We Returning to solid state point of view, we  can crudely categorize the effects of correlations by considering the dominant term in the low-order expansion of the self-energy deviation \begin{equation}  \Sigma(\omega) - \Sigma_\text{KS} \sim \Sigma_0 + \omega \Sigma_1 + \ldots  \end{equation}  If $\Sigma_0$ is large compared to the linear term for low frequencies, we have a large static shift which typically corresponds to materials in the Mott insulating regime with magnetic order. Extensions to DFT known as LDA+U describe these systems well. If the linear term dominates, the main effect of the self-energy is to flatten the bands near the Fermi energy while transferring the residual spectral weight to higher energies. These mass-enhanced systems corresponds to correlated metals and require techniques such as the GW approximation or dynamical mean field theory (DMFT) for accurate description.  % Technically, Mott insulators do not exist at T = 0. Magnetism will release the entropy of the Mott state.  % It starts with the Kohn-Sham formulation~\cite{Kohn_1965} of density functional theory. It states the existence of a potential $V_{KS}(r)$, which is itself a functional of the density. One should write $V_{KS}(\vec{r})[\{ \rho(\vec{r}') \}]$ to indicate this dependence, but we omit this in the following. The exact (but unknown) functional is such that the solution of the set of self-consistent equations,  % \begin{equation}  % \left[-\nabla^{2}+V_{KS}\left(\vec{r}\right)\right]\psi_{\vec{k}j}\left(\vec{r}\right)=\epsilon_{\vec{k}j}\psi_{\vec{k}j}\left(\vec{r}\right).