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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. Additionally, we have a theoretical framework called density functional theory (DFT), which naturally lends itself to computational implementations for modeling their properties. 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 density functional theory is that for ground state properties, we can replace use the total electron density $\rho(\vec{r})$  as the fundamental variable in our equations equations, rather than  the complicated quantum many-body wavefunction by $\Psi$. Kohn and Sham~\cite{Kohn_1965} recast the theory in a form useful for computation, showing that solving for  thetotal  electron density $\rho(\vec{r})$. in the presence of a potential $V_\text{KS}$  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}