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The dream of materials design is to leverage, rather than ignore, our theories of electronic structure and combine them with our increasing computational ability to discover new materials. Beyond its technological implications, the challenge of materials design is also one of great intellectual depth. In principle, we know the fundamental equation needed to model the behavior of a material: it is the Schr\"odinger equation describing electrons moving in the potential of a periodic lattice, mutually interacting via the Coulomb repulsion. Solving this equation is another matter.  In practice, we can classify materials by how well we can solve their corresponding Schr\"odinger equation. For the class of compounds encompassing simple metals, insulators and semiconductors, termed weakly correlated materials, we have a well-developed theory of their excitation spectra called Fermi liquid theory. From a practical viewpoint, the theoretical framework of density functional theory (DFT) naturally lends itself to computational implementations for modeling properties. Materials which are not well-described by DFT are colloquially termed strongly correlated materials. In order to understand the challenges particular to correlations in materials design, we need to better define what we mean by a correlated material, which we do in Sec.~\ref{sec:correlations}.  %,  and we start by examining DFT. 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: solve the  problem of non-interacting electrons in the presence of a periodic potential  $V_\text{KS}(\vec{r})$ 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}  % \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 $ 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.  In quantum chemistry~\cite{Mori_S_nchez_2008}, there is a classification of the errors introduced by 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 energy (i.e. a self energy which varies weakly with frequency at low energies) corresponds to the concept of dynamical correlations. For weakly correlated electron systems, the tools for predicting the properties of solids have advanced to the point that one can contemplate materials design using these tools and their extensions. [ Materials Project AFLOW, what else ] , and these methods are being currently used to predict new materials, as for example in Feenie 2008, Gautier 2015 and Fredeman 2011. A clear example that theoretical approaches in the field of weakly correlated electron materials is coming of age, is the recent prediction of superconductivity in HS3 under pressure, which was recently found to superconduct near 200 K, the highest temperature superconductor discovered so far. Strongly-correlated compounds have many unusual properties. For example, there are very sensitive 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.  % TODO: Define the problem: What IS design of correlated materials? Describe the intersection of materials design with correlated materials. Also describe the need for large computable databases.  % [Chuck: cite examples where LDA+U doesn't quite construct the correct convex hull -- maybe Gutzwiller could do better.]  % Materials design also necessarily involves handling and organizing large bodies of data since the process of check => machine learning. To be precise, we can phrase the question of materials design concretely as follows: given a chemical system, determine the crystal structures and electronic properties of all stable compounds formed by the constituent elements. For example, if the chemical system of interest is Li-Fe-P-O, determine all binaries, ternaries and quaternaries and compute their properties (turns out LiFePO$_4$ is a promising battery material). This problem involves coordinating many moving pieces, including structural prediction, determination of thermodynamic stability against competing phases and computation of electronic properties. In this article, we seek to summarize outstanding challenges in the area, especially as it pertains to correlated materials, and propose strategies to solve them.% The ability to design new materials with desired properties is a key challenge.  % Its solution would have far-reaching implications in both fundamental science  % and technological applications. Whether it is a new class of semiconductors for  % the next generation of integrated circuits, superconductors for dissipationless  % transport of electricity, or thermoelectrics for efficient recovery of waste  % heat, advances in underlying materials results in advances in technology.  % However, the road to saying "I want a material that has these mechanical  % properties combined with these optical properties with these specific thermal  % switching characteristics" and being able to design a new material satisfying  % those properties from nothing more than the constituent elements is a long one.  % Why is it so difficult?  % Ingredients of materials design.  % The underlying workhorse for all materials design is a box which takes as input  % the coordinates of the atoms within a unit cell and produces the total energy  % of the configuration. For materials without partially-filled $d$ or $f$ shells,  % density functional theory performs quite well, providing total energies that  % are accurate to within 50meV.  % \begin{itemize}  % \item define the material design challenge  % \item say that in weakly 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 the material,  % and we want to summarize outsanding problems in this area and possible  % directions to overcome them.  % \end{itemize}