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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: 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}. 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} 
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Strongly correlated materials are those compounds where the difference between actual self-energy and DFT reference $\left|\Sigma(\omega) - \Sigma_\text{KS}\right|$ is large 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.  Returning Our definition for correlations is retrospective, in that it assumes we have  access  to solid state point of view, 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. The simplest is LDA+U, which captures magnetic Mott  insulators well, requires relatively little additional computational resources  and is implemented in mature codes. For a more complete description of Mott  physics, as well as correlated metals, dynamical mean field theory (DMFT) is  the method of choice. Finally, for accurate description of band gaps in  semiconductors and momentum-dependent contributions to the self-energy, the GW approximation is used.  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(\vec{k}, \omega)  - \Sigma_\text{KS} \sim \Sigma_0 \Sigma_0(\vec{k})  + \omega \Sigma_1 \Sigma_1(\vec{k})  + \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).  % \label{Kohn-Sham}  % \end{equation}  % \begin{equation}  % \sum_{\vec{k}j} |\phi_{\vec{k}j}(\vec{r})|^2f(\epsilon_{\vec{k}j}) = \rho(\vec{r})  % \label{KS2}  % \end{equation}  % reproduces the density of the solid. It is useful to divide the Kohn-Sham potential into several parts: $ V_{KS} = V_{Hartree}+V_{cryst} +V_{xc}$, where one lumps into $V_{xc}$ exchange and correlation effects beyond Hartree.  %% The eigenvalues $\epsilon_{\vec{k}j} $ of the solution of the self-consistent set of Eq.~\ref{Kohn-Sham} and~\ref{KS2} are not to be interpreted as excitation energies. Instead the excitation spectra should be extracted from the poles of the one particle Green's function:  %% \begin{equation}  %% G\left( \omega \right) = \frac{1}{ \left[ \omega+\nabla^2+\mu-V_{Hartree}-V_{cryst} \right] - \Sigma \left( \omega \right) }. \label{eq:gwk}  %% \end{equation}  %% Here $\mu $ If $\Sigma_0$  is large compared to  the chemical potential and linear term for low frequencies,  we havesingled out in Eq.~\ref{eq:gwk} the Hartree potential expressed in terms of the exact density and the crystal potential, and lumped the rest of the effects of the correlation in the self energy operator which depends on frequency as well as on two space variables.  %% In chemistry,  aquantum mechanical system is strongly correlated when $\Sigma( i \omega) - \Sigma_{HF} $ is  large at  %% small energies. Here $\Sigma_{HF} $ is the self energy computed static shift which typically corresponds to materials  in the Hartree Fock approximation. At infinite frequency,  %% $\Sigma $ is given by the Hartree Fock graph evaluated Mott insulating regime  with the exact Greens function, hence the Hartree Fock approximation is not exact even at infinite frequency, but it is a good starting point for the treatment of atoms and molecules.  %% Solid state physicists adopt a very different definition of strong correlations. Here, a good reference system is the Kohn-Sham Greens function evaluated in some implementation of the density functional theory such magnetic order. Extensions to DFT known  as LDA+U describe these systems well. If the linear term dominates, the main effect of  the LDA. Hence, for condensed matter scientists, by definition, a strongly correlated material is one where $\Sigma - V_\text{xc} $ self-energy  islarge at low frequencies. Strongly correlated materials, are those for which this is not the case, a famous example are materials such as La2CuO4 which are predicted  to be metals in LDA but which are experimentally antiferromagnetic Mott insulators.  %% In quantum chemistry~\cite{Mori_S_nchez_2008}, there is a classification of flatten  the errors introduced by bands near  the use of approximate density functionals, as being of two types, static correlation and dynamic correlation\cite{Yang_2012}. Similar ideas, also appear in the solid state context, but the nomenclature is exchanged. A \emph{static} self Fermi  energy (i.e. a self energy which varies weakly with frequency at low energies) while transferring the residual spectral weight to higher energies. These mass-enhanced systems  corresponds to correlated metals and require techniques such as  the concept of GW approximation or  dynamical correlations.  % [Chuck: cite examples where LDA+U doesn't quite construct the correct convex hull -- maybe Gutzwiller could do better.]  % Ingredients of materials design. mean field theory (DMFT) for accurate description.  % \begin{itemize}  % \item define Theoretical chemists have classified  the material design challenge  % \item say that 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  inweakly correlated systems it has been fulfilled within DFT. Can mention Zunger’s Heussler compound.  % \item define  strongly correlated materials  % \item they do interesting things and pose special challenges for systems, including  the material,  % and we want to summarize outsanding problems in this area and possible  % directions to overcome them.  % \end{itemize} superconducting cuprates.