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+%% ****** Start of file auguide.tex ****** %
+%%
+%%   This file is part of the AIP distribution of substyles for REVTeX 4.1
+%%   For version 4.1r of REVTeX, August 2010
+%%
+%%   Copyright (c) 2009,2010 American Institute of Physics
+%%
+%\listfiles
+
+\documentclass[
+ reprint,
+ amssymb, amsmath,
+ aip,jcp
+]{revtex4-1}
+
+\usepackage{comment} 
+\usepackage{dcolumn} 
+%\usepackage{docs}%
+%\usepackage{bm}%
+\usepackage[colorlinks=true,linkcolor=blue]{hyperref}%
+\expandafter\ifx\csname package@font\endcsname\relax\else
+ \expandafter\expandafter
+ \expandafter\usepackage
+ \expandafter\expandafter
+ \expandafter{\csname package@font\endcsname}%
+\fi
+\hyphenation{title}
+\usepackage{xspace}
+
+\usepackage{graphicx}
+%\usepackage{subfig}
+
+\usepackage[version=3]{mhchem}
+\parskip=0.1in
+ 
+\usepackage{amsmath}
+
+\usepackage[normalem]{ulem}
+\usepackage[utf8]{inputenc}
+
+%Macros:
+\newcommand{\basis}[0]{\mathcal{B}}
+\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{srPBE}}^{\basis}[n]}
+\newcommand{\epspbeueg}[0]{\bar{\varepsilon}^{\text{sr},\text{PBE}}_{\text{c,md}}}
+\newcommand{\epspbe}[0]{\varepsilon^{\text{PBE}}_{\text{c}}}
+\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
+\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}
+
+\setcounter{secnumdepth}{4}
+\begin{document}
+
+
+
+\section{PBE-based complementary potential $\potpbeueg$}
+
+
+The PBE-based multideterminant short-range correlation complementary density functional used in this paper is the one presented in Ref.~\onlinecite{LooPraSce} and which is defined with the following equation:
+
+\begin{equation}
+ \label{eq:def_pbe}
+ \efuncbasispbe = \int \, \text{d}{\bf r} \,\,n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}({\bf r})),
+\end{equation}
+
+with,
+
+\begin{equation}
+\label{eq:def_epsipbeueg}
+  \epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3},
+\end{equation}
+
+where $\epspbe$ is the usual PBE correlation functional~\cite{pbe}, $s({\bf r})=\nabla n({\bf r})/n({\bf r})^{4/3}$ is the reduced density gradient,
+
+\begin{equation}
+ \beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
+\end{equation}
+
+ and where 
+ 
+\begin{equation} 
+ \label{eq:uegotop}
+ n_2^{\text{UEG}}(n)=n^2(1-\xi^2)g_0(r_s),
+ \end{equation}
+ 
+is the on-top pair density of the uniform electron gas. In eq.~\ref{eq:uegotop}, $\xi=(n_{\uparrow}-n_{\downarrow})/n$ is the spin polarisation,$r_s=(\frac{4\pi n}{3})^{-1/3}$ the Wigner-Seitz radius and $g_0(r_s)$ is the UEG on-top pair-distribution function. The parametrization used in this paper for this last function is detailed in eq.46 of Ref.~\onlinecite{GorSav2006}.
+
+In the present investigation, we have only studied close shell cases for which $n_{\uparrow} = n_{\downarrow}$, which implies that $\xi = 0$. The on-top pair density of the uniform electron gas can thus be rewritten:
+
+\begin{equation} 
+ \label{eq:uegotop2}
+ n_2^{\text{UEG}}(n)=n^2 g_0(r_s).
+ \end{equation}
+
+The potential of this GGA-based functional has the following form:
+
+\begin{equation}
+ \begin{aligned}
+  & \potpbeueg[n]({\bf r},\mu)  = \frac{\delta \efuncbasispbe}{\delta n({\bf r})} \\
+  & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
+  & =\epspbeueg +n\frac{\partial \epspbeueg }{\partial n}- \nabla . n\frac{\partial  \epspbeueg }{\partial \nabla n}.
+  \end{aligned}
+\end{equation}
+
+So we have to compute two main contributions, the scalar part $\frac{\partial \epspbeueg }{\partial n}$ and the gradient part $\frac{\partial  \epspbeueg }{\partial \nabla n}$. 
+
+
+$\bullet$ For the scalar contribution, we simply derived eq.~\ref{eq:def_epsipbeueg} with respect to the density:
+
+\begin{equation}
+ \frac{\partial \epspbeueg }{\partial n}=\frac{\potpbe(1+\beta \mu^3)-\epspbe  \frac{\partial\beta}{\partial n}\mu^3}{(1+\beta\mu^3)^2},
+\end{equation}
+
+Where 
+
+\begin{equation}
+ \potpbe[n]({\bf r}) =\frac{\partial\epspbe}{\partial n} 
+\end{equation}
+
+and
+
+\begin{equation}
+ \frac{\partial \beta}{\partial n}=\frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\potpbe (n_2^{\text{UEG}}/n)-\epspbe \frac{\partial n_2^{\text{UEG}}/n}{\partial n}}{(n_2^{\text{UEG}}/n)^2}.
+\end{equation}
+
+The only remaining part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:
+
+\begin{equation}
+\frac{\partial n_2^{\text{UEG}}/n}{\partial n} = \frac{\partial n g_0(r_s)}{\partial n} = g_0(r_s)+ n\frac{\partial g_0(r_s)}{\partial n}.
+\end{equation}
+
+
+To compute $\frac{\partial g_0(r_s)}{\partial n}$, we used the chain rule:
+
+\begin{equation}
+\frac{\partial g_0(r_s)}{\partial n} = \frac{\partial g_0(r_s)}{\partial r_s}\frac{\partial r_s}{\partial n}.
+\end{equation}
+
+The derivative with respect to $r_s$ can be express:
+
+\begin{equation}
+\begin{aligned}
+&\frac{\partial g_0(r_s)}{\partial r_s} = \\
+& 0.5e^{-F_{g_0}*r_s} ( (-B_{g_0}+2C_{g_0}r_s+3D_{g_0}*r_s^2+4E_{g_0}r_s^3)  \\
+& -(F_{g_0}(1 - B_{g_0}r_s + C_{g_0}r_s^2 + D_{g_0}rs^3 + E_{g_0}r_s^4))),
+\end{aligned}
+\end{equation}
+ 
+ with
+ 
+ \begin{equation}
+ \begin{aligned}
+  &  C_{g_0} = 0.0819306, \\
+  &  F_{g_0} = 0.752411, \\
+  &  D_{g_0} = -0.0127713,\\
+  &  E_{g_0} =0.00185898,\\
+  &  B_{g_0} = 0.7317 - F_{g_0}.
+  \end{aligned}
+\end{equation}
+
+And finally the derivative of $r_s$ with respect to $n$ is equal to:
+
+\begin{equation}
+\frac{\partial g_0(r_s)}{\partial n} =  -(6^{2/3}n^{4/3}\pi^{1/3})^{-1}.
+\end{equation}
+
+$\bullet$ For the gradient part, we also used the chain rule:
+
+\begin{equation}
+ \frac{\partial \epspbeueg}{\partial \nabla n}=\frac{\partial \epspbeueg}{\partial \epspbe}\frac{\partial \epspbe}{\partial \nabla n}.
+\end{equation}
+
+$\frac{\partial \epspbe}{\partial \nabla n}$ is already known (\textbf{Quelqu'un a une ref pour ça??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is trivial:
+
+\begin{equation}
+ \frac{\partial \epspbeueg}{\partial \epspbe}= \frac{(1+\beta \mu^3)-\epspbe  \frac{\partial\beta}{\partial \epspbe}\mu^3}{(1+\beta\mu^3)^2}
+\end{equation}
+
+where 
+
+\begin{equation}
+ \frac{\partial \beta}{\partial \epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
+\end{equation}
+
+
+\bibliography{paper}
+
+ \end{document}