Theory OK for T2

Manuscript/G2-srDFT-control.bib

@@ -0,0 +1,15 @@
+%% This BibTeX bibliography file was created using BibDesk.
+%% http://bibdesk.sourceforge.net/
+
+%% Created for Pierre-Francois Loos at 2019-04-06 21:31:18 +0200 
+
+
+%% Saved with string encoding Unicode (UTF-8) 
+
+
+
+@control{achemso-control,
+	Ctrl-Article-Title = {yes},
+	Ctrl-Chapter-Title = {yes},
+	Ctrl-Etal-Firstonly = {yes},
+	Ctrl-Etal-Number = {30}}

Manuscript/G2-srDFT.tex

@@ -1,13 +1,23 @@
 \documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace}
-
 \usepackage{mathpazo,libertine}
-\usepackage[normalem]{ulem}
+
+\usepackage{natbib}
+\bibliographystyle{achemso}
+\AtBeginDocument{\nocite{achemso-control}}
+
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
-\definecolor{darkgreen}{RGB}{0, 180, 0}
+\definecolor{darkgreen}{HTML}{009900}
-\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
+\usepackage[normalem]{ulem}
-\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
+\newcommand{\titou}[1]{\textcolor{red}{#1}}
-\usepackage{xspace}
+\newcommand{\juju}[1]{\textcolor{purple}{#1}}
+\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
+\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
+\newcommand{\trashJL}[1]{\textcolor{purple}{\sout{#1}}}
+\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
+\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
+\newcommand{\JL}[1]{\juju{(\underline{\bf JL}: #1)}}
+\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
 
 \usepackage{hyperref}
 \hypersetup{
@@ -26,11 +36,11 @@
 \newcommand{\SI}{\textcolor{blue}{supporting information}}
 
 % second quantized operators
-\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
+%\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
-\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
+%\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
 \newcommand{\ai}[1]{\hat{a}_{#1}}
 \newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
-\newcommand{\vijkl}[0]{V_{ij}^{kl}}
+\newcommand{\vpqrs}[0]{V_{pq}^{rs}}
 \newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
 \newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
 
@@ -44,16 +54,11 @@
 
 % 
 
-	
 % energies
 \newcommand{\Ec}{E_\text{c}}
-\newcommand{\EPT}{E_\text{PT2}}
-\newcommand{\EsCI}{E_\text{sCI}}
-\newcommand{\EDMC}{E_\text{DMC}}
 \newcommand{\EexFCI}{E_\text{exFCI}}
 \newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}}
 \newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
-\newcommand{\EexDMC}{E_\text{exDMC}}
 \newcommand{\Ead}{\Delta E_\text{ad}}
 \newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
 \newcommand{\emodel}[0]{E_{\model}^{\Bas}}
@@ -80,10 +85,8 @@
 \newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
 
 
-
-
 % numbers
-\newcommand{\rnum}[0]{{\rm I\!R}}
+%\newcommand{\rnum}[0]{{\rm I\!R}}
 \newcommand{\bfr}[1]{{\bf x}_{#1}}
 \newcommand{\bfrb}[1]{{\bf r}_{#1}}
 \newcommand{\dr}[1]{\text{d}\bfr{#1}}
@@ -235,7 +238,7 @@ We report a universal density-based basis set incompleteness correction that can
 \maketitle
 
 %%%%%%%%%%%%%%%%%%%%%%%%
-%\section{Introduction}
+\section{Introduction}
 %%%%%%%%%%%%%%%%%%%%%%%%
 Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
 Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.
@@ -267,18 +270,18 @@ Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, corre
 Unless otherwise stated, atomic used are used.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
-%\section{Theory}
+\section{Theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
-The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set. 
+The present basis set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set. 
 Here, we only provide the main working equations. 
-We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about its formal derivation.  
+We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.  
 
 
 %=================================================================
 %\subsection{Correcting the basis set error of a general WFT model}
 %=================================================================
 Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. 
-According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ {\color{red} are reasonable approximations of the FCI energy and density within $\Bas$ } \sout{\textit{exact} ground state energy }, the exact ground state energy $\E{}{}$ \sout{and density $\n{}{}$, respectively, one may write} may be written as
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \titou{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
 \begin{equation}
 	\label{eq:e0basis}
 	\E{}{} 
@@ -292,20 +295,20 @@ where
 	= \min_{\wf{}{} \to \n{}{}}  \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} 
 	- \min_{\wf{}{\Bas} \to \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
 \end{equation}
-is the basis-dependent complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ are the kinetic and interelectronic repulsion operators, respectively.
+is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
 In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
 Both wave functions yield the same target density $\n{}{}$. 
 
 %\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
 
-An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
+Importantly, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
 \begin{equation}
   \label{eq:limitfunc}
         \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
 \end{equation}
 where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
 In the case $\modX = \FCI$, we have as strict equality as $E_{\FCI}^\infty = E$.
-Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$ {\color{red} not clear to my eyes ... I think that one should say in what sence these are approximations in terms of the density and energy}.
+Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, \manu{the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$}.
 
 %Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$. 
 %As any wave function model is necessary an approximation to the FCI model, one can write 
@@ -400,12 +403,12 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
 However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
 One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points. 
 As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$. 
-Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
+Therefore, as we shall do later on, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
-Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a \textit{range-separated function} $\rsmu{}{}(\br{})$.
+Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, the spatial inhomogeneity of $\Bas$ forces us to define a range-separated \textit{function} $\rsmu{}{}(\br{})$ as the value of $\rsmu{}{}$ must be known at any point in space.
 
-The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set. 
+The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence point as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set. 
 In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$.
-The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
+In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
 %Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. 
 %First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .   
 %Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). 
@@ -423,21 +426,26 @@ The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals
 %=================================================================
 %\subsection{Effective Coulomb operator}
 %=================================================================
-In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that 
+To compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined as
 \begin{equation}
 	\label{eq:int_eq_wee}
 	\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
 \end{equation}
-(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realize that 
+where 
+\begin{equation}
+	\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})
+	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}
+\end{equation}
+is the two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realize that
 \begin{equation}
 	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
 \end{equation}
 which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$ 
 \begin{equation}
 \label{eq:WeeB}
-	\hWee{\Bas} =  \alert{\frac{1}{2} \sum_{ijkl \in \Bas}\vijkl \aic{k} \aic{l} \ai{j} \ai{i}}
+	\hWee{\Bas} =  \frac{1}{2} \sum_{pqrs \in \Bas}\vpqrs \aic{r} \aic{s} \ai{q} \ai{p}
 \end{equation}
-over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$  and $\vijkl$ are the usual two-electron Coulomb integrals. 
+over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals in $\Bas$ and $\vpqrs$ are the usual two-electron Coulomb integrals. 
 Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that  
 \begin{subequations}
 \begin{align}
@@ -453,9 +461,9 @@ where
 	\label{eq:fbasis}
 	\f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) 
 	\\
-	=  \sum_{ijklmn \in \Bas}  \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
+	=  \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \vpqrs \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
 \end{multline}
-and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that 
+it comes naturally that 
 \begin{equation}
 	\label{eq:def_weebasis}
 	\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{}(\bx{1},\bx{2})/\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}).
@@ -477,7 +485,7 @@ Although this choice is not unique, the long-range interaction we have chosen is
 \begin{equation}
 	\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
 \end{equation}
-and ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of same-spin electron pairs yields
+Ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of opposite-spin electron pairs yields
 \begin{equation}
 	\label{eq:mu_of_r}
 	\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\bx{},\Bar{\bx{}}),
@@ -559,7 +567,8 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
 \end{align}
 \end{subequations}
 where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
-These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general WFT method. 
+The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
+This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. 
 
 %--------------------------------------------
 %\subsubsection{Local density approximation}
@@ -576,7 +585,7 @@ where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range co
 %\subsubsection{New PBE functional}
 %--------------------------------------------
 The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. 
-In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
+In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
 \begin{subequations}
 \begin{gather}
 	\label{eq:epsilon_cmdpbe}
@@ -586,7 +595,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
 	\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
 \end{gather}
 \end{subequations}
-The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} =  \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
+The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} =  \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
 Therefore, the PBE complementary functional reads
 \begin{equation}
 	\label{eq:def_lda_tot}
@@ -613,7 +622,7 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
 %\subsection{Valence effective interaction}
 %=================================================================
 As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals. 
-We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
+We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
 
 %According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. 
 Accounting solely for the valence electrons, Eq.~\eqref{eq:expectweeb} becomes
@@ -630,9 +639,9 @@ Following the spirit of Eq.~\eqref{eq:fbasis}, we have
 	\label{eq:fbasisval}
 	\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) 
 	\\
-	=  \sum_{ij \in \Bas} \sum_{klmn \in \Val}  \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
+	=  \sum_{pq \in \Bas} \sum_{rstu \in \Val}  \SO{p}{1} \SO{q}{2} \vpqrs \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
 \end{multline}
-and, the valence part of the effective interaction is
+and the valence part of the effective interaction is
 \begin{subequations}
 \begin{gather}
 	\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})/\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2}),
@@ -640,7 +649,12 @@ and, the valence part of the effective interaction is
 	\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\bx{},\Bar{\bx{}}),
 \end{gather}
 \end{subequations}
-where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
+where 
+\begin{equation}
+	\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})
+	= \sum_{pqrs \in \Val}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}
+\end{equation}
+is the two body density associated to the valence electrons.
 %\begin{equation}
 % \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
 %\end{equation}
@@ -668,10 +682,10 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
 % \label{eq:def_lda_tot}
 %  \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
 %\end{equation}
-Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluate as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
+Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
-%\section{Results}
+\section{Results}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 %%% TABLE I %%%
@@ -777,7 +791,22 @@ Regarding the effect of the basis set correction, both for the CIPSI and CCSD(T)
 First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pv5z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.  
 Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol. 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Supporting information}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+See {\SI} for raw data of the G2 atomization energies.
 
-\bibliography{G2-srDFT}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{acknowledgements}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The authors would like to thank... nobody.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{acknowledgements}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliography{G2-srDFT,G2-srDFT-control}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \end{document}