From 4c648fb53116d8525ae0f815963259968b9e0e59 Mon Sep 17 00:00:00 2001
From: eginer <giner.emmanuel@gmail.com>
Date: Tue, 18 Jun 2019 16:37:25 +0200
Subject: [PATCH] modifs

---
 Manuscript/Ex-srDFT.tex | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex
index 18d30d7..6bdc44b 100644
--- a/Manuscript/Ex-srDFT.tex
+++ b/Manuscript/Ex-srDFT.tex
@@ -334,10 +334,10 @@ The PBE correlation functional has clearly shown to improve energetics for small
 Therefore, based on the proposition of Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general form of the PBE complementary functional: 
 \begin{multline}
 	\label{eq:def_pbe_tot}
-	\bE{\PBE}{}[\n{}{},\tn{2}{},\rsmu{}{}] =
+	\bE{\PBE}{\manu{\Bas}}[\n{}{},\tn{2}{},\rsmu{}{\manu{\Bas}}] =
 	\int \n{}{}(\br{}) 
 	\\
-	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
+	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
 \end{multline}
 with
 \begin{subequations}
@@ -353,10 +353,10 @@ with
 In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a new PBE-based functional, here-referred as \titou{PBE-UEG}, 
 \begin{multline}
 	\label{eq:def_pbe_tot}
-	\bE{\titou{\PBE\text{-}\UEG}}{}[\n{}{},\n{2}{\UEG},\rsmu{}{}] =
+	\bE{\titou{\PBE\text{-}\UEG}}{\manu{\Bas}}[\n{}{},\n{2}{\UEG},\rsmu{}{\manu{\Bas}}] =
 	\int \n{}{}(\br{}) 
 	\\
-	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
+	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
 \end{multline}
 in which the on-top pair density was approximated by its UEG version,
 \begin{equation}
@@ -369,14 +369,14 @@ However, the underlying UEG on-top pair density might not be suited for the trea
 Therefore, we propose here the "PBE-ontop" (PBEot) functional, 
 \begin{multline}
 	\label{eq:def_pbe_tot}
-	\bE{\PBEot}{}[\n{}{},\ttn{2}{},\rsmu{}{}] =
+	\bE{\PBEot}{\manu{\Bas}}[\n{}{},\ttn{2}{\manu{\Bas}},\rsmu{}{\manu{\Bas}}] =
 	\int \n{}{}(\br{}) 
 	\\
-	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
+	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{\manu{\Bas}}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
 \end{multline}
-a variant inspired by the work of Ref.~\onlinecite{FerGinTou-JCP-18} where the exact on-top pair density is approximated by the extrapolation formula proposed by Gori-Giorgi and Savin:\cite{GorSav-PRA-06} 
+a variant inspired by the work of Ref.~\onlinecite{FerGinTou-JCP-18} where the exact on-top pair density is approximated by the extrapolation formula proposed by Gori-Giorgi and Savin:\cite{GorSav-PRA-06}\manu{. In the present context, we use the on-top pair density in the basis set $\Bas$ $ \n{2}{\manu{\Bas}}(\br{})$ together with the associated range separation function $\rsmu{}{\Bas}(\br{})$, which leads to the following approximated on-top pair density: } 
 \begin{equation}
-  \ttn{2}{}(\br{}) = \n{2}{}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}. 
+  \ttn{2}{\manu{\Bas}}(\br{}) = \n{2}{\manu{\Bas}}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\manu{\Bas}}(\br{})})^{-1}. 
 \end{equation}
 %Such formula relies on the association between $\n{2}{\Bas}(\br{ },\br{ })$, which is the on-top pair density computed in the basis set $\Bas$ in a given point, and the local value of the range-separation parameter $\rsmu{}{\Bas}$ at the same point. 
 The sole distinction between \titou{PBE-UEG} and PBEot is the level of approximation of the exact on-top pair density.