deletions | additions       diff --git a/XC Functionals.tex b/XC Functionals.tex  index 040f65f..8b23015 100644  --- a/XC Functionals.tex  +++ b/XC Functionals.tex 
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families, which have names such as generalized-gradient approximations  (GGAs), meta-GGAs, hybrid functionals, etc. In 2001, John Perdew came  up with a beautiful idea on how to illustrate these families and their  relationship~\cite{perdew:1}. He ordered them the  families as rungs in a ladder that leads to the heaven of ``chemical accuracy'', and that he  christened the Jacob's ladder of density functional approximations for  the xc energy. Every rung adds a dependency on another quantity, 
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functionals in Octopus is provided through the Libxc  library~\cite{Marques_2012}. Libxc started as a spin-off project during  the initial development of Octopus. At that point it became clear that  the task of evaluation of evaluating  the xc functional was completely independent of the main structure of the  code, and could therefore be transformed into a stand-alone library. Over the years, Libxc became more and more independent of  Octopus, and is now used in a variety of DFT codes. There are  currently more than 150 xc functionalscurrently  implemented in Libxc that are available in Octopus, a number that is has been  increasing steadily over the years. All of the standard functionals are included and many  of the less common ones. There is also support for LDAs and GGAs of  systems of reduced dimensionality (1D and 2D), which allow for direct 
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functionals. For example, the method described in section  \ref{sec:mbse} can be used in a straightforward way to obtain reference  data against which to benchmark the performance of a given xc  functional. functional, for example a one-dimensional LDA~\cite{1DLDA}.  In that case, both calculations, exact and approximate, make use of the same real-space grid approach, which makes the  comparison of the results obtained with both straightforward. Despite  the obvious advantage of using exact solutions of the many-body 
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potentials and static polarizabilities of atoms, molecules, and  hydrogen chains.   In this vein, Andrade and Aspuru-Guzik~\cite{Andrade_2011} proposed a method to obtain an asymptotically correct xc potential starting from any approximation. Their method is based on considered the xc potential as an electrostatic potential generated by a fictitious xc charge. In terms of this charge, the asymptotic condition is given as a simple condition formula  that is local in real space that and  can be enforced by a simple procedure. The method, implemented in Octopus, was used to perform test calculations in molecules. Additionally, the with this  correction procedure it is possible to find accurate predictions for the  derivative discontinuity and hence and, hence,  predict the fundamental gap~\cite{Mosquera_2014}.