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\section{Overview} \section{Introduction}
The ability to
discover and control design new materials
with a desired set of properties is crucial to the development of new
technologies. technology. The design of silicon and lithium ion based materials are well known examples which led to the proliferation of consumer hand-held devices today. However, materials discovery has historically proceeded via trial and error, with intuition-guided serendipity being the most fruitful path. For example, all major classes of superconductors--from elemental mercury in 1911, to the heavy fermions, cuprates,
and the iron-based superonductors
and most recently, hydrogen sulfide in
2008 were complete surprises. late 2014--have been discovered by chance.
The dream of materials design is to leverage our theories of electronic structure, rather than ignoring them,
and combine them combined with our increasing computational and storage abilities to discover new materials. Beyond its technological implications, the challenge of materials design is also one of great intellectual depth. In principle, we know all the fundamental equations needed to model the behavior of electrons and nuclei. Solving these equations is another matter.
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
the charge 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,
as a byproduct of these calculations one can obtain
estimates for electronic properties such as
energy 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}.
The It starts with
the Kohn-Sham formulation~\cite{kohn_sham} of density functional
theory theory.
It states the existence of a potential $V_{KS}(r)$, which is itself a functional
of the
density and which added a single particle Hamiltonian can reproduce the exact density of a compound by solving
%One density. One should write $V_{KS}(\vec{r})[\{ \rho(\vec{r}') \}]$ to indicate this dependence, but
%we we omit this in the following. The exact (but unknown) functional is such that the solution
a of the set of self-consistent
equations: 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}
...
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 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}
