Merge branch 'master' of https://github.com/pfloos/srDFT_G2

Manuscript/G2-srDFT.bib

@@ -12233,3 +12233,15 @@
 	Year = {1998},
 	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261498001110},
 	Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
+
+@article{DasHer-JCC-17,
+author = {Dasgupta, Saswata and Herbert, John M.},
+title = {Standard grids for high-precision integration of modern density functionals: SG-2 and SG-3},
+journal = {Journal of Computational Chemistry},
+volume = {38},
+number = {12},
+pages = {869-882},
+doi = {10.1002/jcc.24761},
+eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24761},
+year = {2017}
+}

Manuscript/G2-srDFT.tex

@@ -207,7 +207,7 @@ Importantly, in the limit of a complete basis set $\Bas$ (which we refer to as $
 \end{equation}
 where  $\E{\modX}{\infty}$ 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 $\modY$ for the FCI energy and density within $\Bas$.
+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 $\modY$ for the FCI energy and density within $\Bas$, respectively.
 
 %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 
@@ -300,12 +300,16 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
 %\end{equation}
 
 Rigorously speaking,  the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
-Nevertheless, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct for the lack of electronic cups in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points, which is universal. 
+Nevertheless, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct 
+for the lack of electronic cups in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the electron-electron (e-e) 
+coalescence points. 
+Therefore, the physical role of $\bE{}{\Bas}[\n{}{}]$ is to account for a universal condition of exact wave functions. 
 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, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth non divergent two-electron interaction.
-Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{}{}(\br{})$ which quantifies the incompleteness of a basis set $\Bas$ for each point in ${\rm I\!R}^3$.  
+Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{}{}(\br{})$ which automatically adapts to quantify the incompleteness of a basis set $\Bas$ for each point in ${\rm I\!R}^3$.  
 
-The first step of the basis set correction consists in obtaining an effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ 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 interaction in the limit of a complete basis set. 
+The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which represents the effect of the projection in an incomplete basis set $\Bas$ of the Coulomb operator. 
+We use a definition for which $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$  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 interaction in the limit of a complete basis set. 
 In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
 In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} with $\rsmu{}{}(\br{})$ as the range separation. 
 %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}]. 
@@ -325,20 +329,20 @@ In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-
 %=================================================================
 %\subsection{Effective Coulomb operator}
 %=================================================================
-We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
+We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as 
 \begin{equation}
 	\label{eq:def_weebasis}
  \W{\wf{}{\Bas}}{}(\br{1},\br{2})   = \left\{
     \begin{array}{ll}
-       \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) & \mbox{if } \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \ne 0\\
+       \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
-        \,\,\,\,+\infty & \mbox{if not.}
+        \,\,\,\,+\infty & \mbox{otherwise.}
     \end{array}
  \right.
 \end{equation}
-where  $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
+where  $\n{2}{\wf{}{\Bas}}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
 \begin{equation}
 	\label{eq:n2basis}
-	\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
+	\n{2}{\wf{}{\Bas}}(\br{1},\br{2})
 	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}, 
 \end{equation}
 $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\wf{}{\Bas}}{}(\br{1},\br{2})$ is defined as
@@ -349,31 +353,32 @@ $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\w
 	=  \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
 \end{multline}
 and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. 
+The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction. 
 With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
 \begin{equation}
 	\label{eq:int_eq_wee}
-	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
+	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
-where the $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
+where here $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
 \begin{equation}
-  \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
+  \iint r_{12}^{-1} \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
 which intuitively motivates $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
 As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries. 
-An important quantity to define is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
+An important quantity to define in the present context is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
 \begin{equation}
   \label{eq:wcoal}
   \W{\wf{}{\Bas}}{}(\br{}) =  \W{\wf{}{\Bas}}{}(\br{},{\br{}})
 \end{equation}
-and which is necessarily \textit{finite} at for an \textit{incomplete} basis set. 
+and which is necessarily \textit{finite} at for an \textit{incomplete} basis set as long as the on-top two-body density is non vanishing. 
 
 Of course, there exists \textit{a priori} an infinite set of functions in ${\rm I\!R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that 
 \begin{equation}
 	\label{eq:lim_W}
 	\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\   
 \end{equation}
-for all points $(\br{1},\br{2})$ such that $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. 
+for all points $(\br{1},\br{2})$ such that $\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. 
-An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  
+An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  
 As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems. 
 
 %=================================================================
@@ -472,7 +477,7 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
 \end{subequations}
 where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
 The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}]. 
-Indeed, provided that $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) \approx \W{\wf{}{\Bas}}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\wf{}{\Bas}}{}(\br{})}[\n{}{}(\br{})]$ coincides with $\wf{}{\Bas}$. 
+Indeed, provided that $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) \approx \W{\wf{}{\Bas}}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\wf{}{\Bas}}{}(\br{})}$ coincides with $\wf{}{\Bas}$. 
 %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. 
 
@@ -484,25 +489,25 @@ Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Ba
 	\label{eq:def_lda_tot}
         \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
 \end{equation}
-where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
+where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range ECMD per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
 %In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
 
 %--------------------------------------------
 %\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 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, PazMorGorBac-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 $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
 \begin{subequations}
 \begin{gather}
 	\label{eq:epsilon_cmdpbe}
 	\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} },
 	\\
 	\label{eq:beta_cmdpbe}
-	\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
+	\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
 \end{gather}
 \end{subequations}
-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}. 
+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 $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{})  \approx \n{2}{\UEG}(\br{}) =  \left(\n{}{}(\br{})\right)^2  g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
-This represents a major computational saving without loss of performance as we eschew the computation of $\n{}{(2)}$.
+This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}$.
 Therefore, the PBE complementary functional reads
 \begin{equation}
 	\label{eq:def_pbe_tot}
@@ -536,8 +541,8 @@ We therefore define the valence-only effective interaction
 %       \label{eq:Wval}
  \W{\wf{}{\Bas}}{\Val}(\br{1},\br{2})  = \left\{
     \begin{array}{ll}
-     \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})  & \mbox{if } \n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})\ne 0\\
+     \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2})  & \mbox{if } \n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2})\ne 0\\
-        \,\,\,\,+\infty & \mbox{if not.}
+        \,\,\,\,+\infty & \mbox{otherwise. }
     \end{array}
   \right.
  \end{equation}
@@ -548,14 +553,14 @@ with
 	\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
 	=  \sum_{pq \in \Bas} \sum_{rstu \in \Val}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
 	\\
-	\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
+	\n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2})
 	= \sum_{pqrs \in \Val}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}, 
 \end{gather}
 \end{subequations}
 and the corresponding valence range separation function $\rsmu{\wf{}{\Bas}}{\Val}(\br{})$
 \begin{equation}
 	\label{eq:muval}
-	\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}).
+	\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\br{}).
 \end{equation}
 %\begin{equation}
 % \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
@@ -586,7 +591,11 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
 %\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 evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
 
-Regarding now the main computational source of the present approach, it consists in the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of 
+Regarding now the main computational source of the present approach, it consists in the evaluation 
+of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
+All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) 
+for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation 
+at each quadrature grid point of 
 \begin{equation}
 	\label{eq:fcoal}
         f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
@@ -595,8 +604,9 @@ which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly paralle
 When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$  but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly spped up the calculations.  
 
 
-To conclude this theory session, it is important to notice that in the limit of a complete basis set, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part. 
+To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any wave function models that provide an energy and density, ii) it vanishes for one-electron systems, 
-Thefore in the limit of a complete basis set one recovers the correct limit of the WFT model whatever approximations are made in the DFT part, just like in equation \eqref{eq:limitfunc}. 
+iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model. 
+%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}
@@ -693,28 +703,27 @@ Thefore in the limit of a complete basis set one recovers the correct limit of t
 We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
 In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.
 \ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set, whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11} 
-In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family.
+In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family, except for Li, Be and Na for which we use the cc-pCVXZ basis sets.
 This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08,Gro-JCP-09,FelPet-JCP-09,NemTowNee-JCP-10,FelPetHil-JCP-11,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
 %The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
 As a method $\modX$ we employ either CCSD(T) or exFCI.
 Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
 We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
 In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the Restricted Open Shel Hartree-Fock (ROHF) one-electron density to compute the complementary energy, and for the CIPSI calculations we use the density of a converged variational wave function. 
-For the definition of the interaction, we use a single Slater determinant in the ROHF basis. 
+For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculations, and built with the natural orbitals of the converged variational wave function for the exFCI calculations. 
 The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
 RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
-\titou{For the quadrature grid, we employ ... radial and angular points.}
+For the quadrature grid, we employ the radial and angular points of the SG2 grid\cite{DasHer-JCC-17}.
 Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
 Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{B} to \ce{Mg}, while a \ce{Ne} core is frozen from \ce{Al} to \ce{Xe}.
-In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the effective interaction [see Eq.~\eqref{eq:Wval}] refers to the non-frozen spinorbitals.
+In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals.
 The ``valence'' correction was used consistently when the FC approximation was applied.
 In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies. 
 We refer to these atomization energies as $\CBS$. 
 
 %\subsection{Convergence of the atomization energies with the WFT models }
 As the exFCI calculations were converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values.
-They will be our references for a given system in a given basis. 
+They will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} in a given basis, and the results for these diatomics are reported in Table \ref{tab:diatomics}. 
-The results for four diatomics mentioned above are reported in Table \ref{tab:diatomics}. 
 As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with cc-pV5Z. 
 Also, the atomization energies are consistently underestimated, reflecting that, in a given basis, the atom is always better described than the molecule due to the larger number of interacting electron pairs in the molecular system. 
 A similar trend holds for CCSD(T).
@@ -728,6 +737,7 @@ Nevertheless, the deviation observed for the largest basis set is typically with
 %Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. 
 Importantly, the sensitivity with respect to the SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. 
 However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}. 
+Such weak sensitivity to the approximated functionals in the DFT part when reaching large basis sets shows the robustness of the approach. 
 
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 \section*{Supporting information}