deletions | additions       diff --git a/correlations.tex b/correlations.tex  index b58ab7c..a98af9c 100644  --- a/correlations.tex  +++ b/correlations.tex 
...
The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Since the behavior of the constituent electrons are strongly coupled to one another, studying the behavior of individual particles generally provides little insight into the macroscopic properties of a correlated material. While this conceptual definition is valuable for understanding the fundamental physics, it is of less use to a DFT practitioner, who seeks to make predictions for comparison with experimental observations. To arrive at an operational definition of a correlated material, we examine DFT and how it relates to the observed electronic spectra.  The 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: the they  proved that there exists a periodic potential $V_\text{KS}(\vec{r}) = V_\text{KS}[\rho](\vec{r})$, which itself is a functional of the density, and that solving the problem of non-interacting electrons in the presence of this potential will give the exact density. The Kohn-Sham potential consists of three contributions: $V_\text{KS} = V_\text{ion} + V_\text{H} + V_\text{xc}$. The first is the one-body attractive potential of the nuclear ions. The other two arise from the electron-electron interaction: the (classical) Hartree component $V_\text{H}$ captures most of the Coulomb interaction, and the remaining contribution is contained in the exchange-correlation term $V_\text{xc}$. In practice, the exchange-correlation term is difficult to capture, and is generally modeled by approximations known as the local density approximation (LDA) or generalized gradient approximation (GGA). Assuming we have the exact $V_\text{xc}$, DFT guarantees the correct ground state density and energy, but makes no claims about the electronic spectrum. For electrons moving in the lattice potential $V_\text{ion}$ of the nuclear ions, the general form of the Green's function is  \begin{equation}