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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.  The conceptual 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, showing that one could compute the exact electronic density by solving form: solve  the problem ofan auxiliary system of  non-interacting electrons in the presence of a periodic potential $V_\text{KS}(\vec{r})$. $V_\text{KS}(\vec{r})$ and the sum of their wavefunctions (squared) will give  the exact density. In this Kohn-Sham formulation, their potential $V_\text{KS}$  functionally depends on the total density $\rho$, which is not a problem as  there are standard algorithms for iterative solution. The problem lies in the  lack of known methods for constructing $V_\text{KS}$ exactly.  % Technically, Mott insulators do not exist at T = 0. Magnetism will release  % the entropy of the Mott state.  Breaking the Kohn-Sham potential apart, we find three contributions:  $V_\text{KS} = V_\text{ion} + V_\text{H} + V_\text{xc}$. The first two are the  attractive potential of the nuclear ions and the (classical) Hartree component  of the Coulomb interaction, and both are known. The last exchange-correlation  term is challenging to compute, and is generally modeled by approximations  known as the local density approximation (LDA) or generalized gradient  approximation (GGA).  DFT guarantees the correct ground state density and energy, and makes no  statements about the electronic spectrum. The eigenenergies that are the result  diagonalizing the Kohn-Sham hamiltonian $H_\text{KS} =  -\frac{1}{2}\vec{\nabla}^2 + V_\text{KS}(\vec{r})$ are not the eigenenergies of  the full many-body problem. Nevertheless, we often ignore theoretical  guarantees and compute the Green's function using the Kohn-Sham solution,  schematically written as  \begin{equation}  G(\omega) = \frac{1}{\omega - H_\text{KS}} = \frac{1}{\omega + \vec{\nabla}^2/2 - V_\text{ion} - \Sigma(\omega)}  \end{equation}  where the self-energy $\Sigma(\omega) = V_\text{H} + V_\text{xc}$. We have not  written the $\vec{r}$ dependence in all quantities and the self-energy $\Sigma$  is in fact frequency-independent within DFT. In many cases, the Kohn-Sham  solution matches well with experiment, and we call these materials weakly  correlated.  However, there is an entire class of materials where using the DFT form of the  self-energy gives a poor description of the spectrum. We arrive at the  following operational definition: a material is correlated if $\Sigma(\omega) -  V_\text{H} - V_\text{xc}$ is large at low frequencies.  % 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} 
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% \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 $ 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} $ is large at %%  small energies. Here $\Sigma_{HF} $ is the self energy computed in the Hartree Fock approximation. At infinite frequency, %%  $\Sigma $ is given by the Hartree Fock graph evaluated 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 as the LDA. Hence, for condensed matter scientists, by definition, a strongly correlated material is one where $\Sigma - V_\text{xc} $ is large 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. 
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% \item they do interesting things and pose special challenges for the material,  % and we want to summarize outsanding problems in this area and possible  % directions to overcome them.  % \end{itemize}