deletions | additions       diff --git a/XC Functionals.tex b/XC Functionals.tex  index 8b23015..21df25b 100644  --- a/XC Functionals.tex  +++ b/XC Functionals.tex 
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\section{Development of exchange and correlation functionals}  %Micael, Miguel, Xavier  The central quantity of the KS scheme of DFT is the xc energy $E_{\rm xc}[n]$ that xc}[n]$, which  describes all non-trivial many-body effects. Clearly, the exact form of this  quantity is unknown and it must be approximated in any practical  application of DFT. We emphasize that the precision accuracy  of any DFT calculation depends solely on the form of this quantity, as this is  the only real approximation in DFT (neglecting numerical  approximations that are normally controllable). 
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(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 the families as rungs in a  ladder that leads to the heaven of ``chemical accuracy'', and that which  he christened the Jacob's ladder ``Jacob's ladder''  of density functional density-functional  approximations for the xc energy. Every rung adds a dependency on another quantity,  thereby increasing the precision of the functional but also increasing  the numerical complexity and the computational cost.  The first three rungs of this ladder are respectively: :  (i)~the local-density approximation (LDA), where the functional has a local  dependence on the density only; (ii)~the  generalized-gradient approximation (GGA), that which  includes also a local dependence on the gradient of the density \(\nabla n(\vec r)\); and  (iii)~the meta-GGA, that which  adds a local dependence on the Laplacian of the density and on the kinetic energy density; kinetic-energy density.  In the fourth rung we have functionals that depend on the occupied KS orbitals, such as the exact-exchange exact exchange  or hybrid functionals. Finally, the fifth rung adds a dependence on the virtual  KS orbitals.  Support for the first three rungs and for the local part of the hybrid  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 point,  it became clear that the task 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 
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equation for model systems described in section~\ref{sec:mbse}.  Octopus also includes support for other functionals of the fourth  rung, such as exact-exchange exact exchange  or the self-interaction correction of Perdew and Zunger~\cite{Zunger_1980}, through the solution of the optimized effective potential equation . equation.  This can be done exactly~\cite{K_mmel_2003}, or within the Slater~\cite{Slater_1951} or Krieger-Lee-Iafrate approximations~\cite{Krieger_1990}. Besides the functionals that are supported by Octopus, the code has  served as a platform for the testing and development of new 
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problem as reference data, this is often not possible and one usually  needs to resort to the more commonly used experimental or  highly-accurate quantum chemistry data. In this case, the flexibility  of the real-space method that allows method, allowing  for the calculation of many different properties of a wide variety of systems systems,  is again an advantage. Octopus has therefore been used, among others, to benchmark  the performance of xc functionals whose potential has a correct  asymptotic behavior~\cite{Oliveira_2010} when calculating ionization  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 considering  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 formula that is local in real space and can be enforced by a simple procedure. The method, implemented in Octopus, was used to perform test calculations in molecules. Additionally, with this correction procedure it is possible to find accurate predictions for the derivative discontinuity and, hence, predict the fundamental gap~\cite{Mosquera_2014}.