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For a large class of materials called weakly-correlated compounds, encompassing simple metals, insulators and semiconductors, implementations of density functional theory (DFT) performs extremely well. DFT is a workhorse of the materials science community, providing efficient and accurate computations of the total energy and distribution of electrons of a compound, requiring only the coordinates of the atoms in its crystal lattice as input. From the total energy, one can obtain lattice constants, equations of state and the spectrum of lattice vibrations. Furthermore, one can obtain electronic properties such as band gaps, electric polarization and topological numbers, which are by no means trivial for these "simple" compounds. Using DFT, several groups have been constructing databases spanning large swaths of the space of known compounds, containing these computed properties which can then be data mined for specific properties. Successes include the prediction of multiferroics~\cite{Fennie_2008}, intermetallic semiconductors\cite{Gautier_2015} and even heavy fermion materials\cite{Fredeman_2011}.  It starts with  the Kohn-Sham formulation~\cite{kohn_sham} 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.  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 $ is the chemical potential and we have singled 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, a quantum mechanical system is strongly correlated when $\Sigma( i \omega) - \Sigma_{HF} (i \omega) $ is large at  small energies. Here $\Sigma_{HF} (i omega) $ is the self energy computed in the Hartree Fock approximation. NOTE At infinite frequency,  $\Sigma $ is given by the Hartree Fock graph evaluated with the exact Greens function.   Solid state physicists adopt a very different definition of strong correlations. Here, the reference system is the Kohn Sham Greens function  evaluated in some implementation of the density functional theory such as the LDA. Hence, for condensed matter scientists, by definition, a  strongly correlated material is one where $\Sigma - V_KS $ is large at low energy.   In quantum chemistry   Aron J Cohen, Paula Mori-Sánchez, Weitao Yang  Publication date  2008/8/8  Journal  Science  Volume  321  Issue  5890  Pages  792-794  there is a classification of the errors introduced by approximate density functionals, as being of two types, static correlation and dynamic  correlation. These ideas, have also been reformulated in the solid state context. Unfortunately the terms are exchanged. A {\it static } self energy   (i.e. a self energy which varies weakly with frequency at low energies) correspondes to the concept of dynamical correlations.  However, there is another large class of materials called strongly-correlated compounds, whose distinguishing phenomenological feature is sensitivity to external perturbation, which makes them technologically useful. Small changes in pressure, temperature or chemical doping often drives large changes in electronic or structural behavior. For example, changing the temperature by only several degrees Kelvin can drive a transition between a metallic and insulating state, behavior not observed in weakly-correlated compounds. In addition to metal-insulator transitions, these compounds display unusual magnetic properties, high-temperature superconductivity and strange-metal behavior.  The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Often, this occurs because the material contains atoms with partially-filled $d$ or $f$ orbitals. The electrons occupying these orbitals retain a strong atomic-like character to their behavior, while the remaining electrons form bands; their interplay poses special challenges for theory. Consequently current implementations of DFT cannot describe their properties accurately. This led to the development of extensions to DFT such as LDA+U, and entirely more sophisticated approaches such as dynamical mean field theory (DMFT) and the GW approximation.