a few changes

Revised_Manuscript/srDFT_SC.tex

@@ -870,24 +870,16 @@ where $\murcipsi$ is the local range-separation function calculated with the CIP
 where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least $10^7$ Slater determinants. In particular, $\murcipsi$ has been calculated from Eqs. \eqref{eq:def_mur_val}-\eqref{eq:twordm_val} with the opposite-spin two-body density matrix $\Gam{pq}{rs}$ of the largest variational CIPSI wave function for a given basis. All quantities in Eqs. \eqref{eq:ontopcas}-\eqref{eq:muaverage} were computed excluding all contributions from the 1s orbitals, \ie, they are ``valence-only'' quantities. 
 }
 
-\manu{
-We report in Table \ref{tab:d1} these quantities for N and \ce{N2} in different basis sets. 
-From this Table    \ref{tab:d1} one can notice that the integral of the on-top pair density at the CIPSI level is systematically lower than that at the CASSCF level, which is expected as the short-range correlation, digging the coulomb hone in a given basis set $\basis$ at near FCI level, is missing from the valence CASSCF wave function. 
-Also, the on-top pair density at the CIPSI level decreases in a monotonous way, roughly by $20\%$ between the aug-cc-pVDZ and aug-cc-pVQZ, whereas the on-top pair density at the CASSCF level is almost constant with respect to the basis set. 
-Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower with respect to their original on-top pair density, which are $\ontopcas$ and $\ontopcipsi$. 
-Nevertheless, the behaviour of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different : $\ontopextrap$ globally increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains qualitatively constant. More precisely, in the case of  \ce{N2} the value $\ontopextrap$ increases by about 50$\%$ between the aug-cc-pVDZ and aug-cc-pVQZ basis sets, whereas $\ontopextrapcipsi$ fluctuates by about 5$\%$ within the same basis sets.  
-The behaviour of $\ontopextrap$ can be understood easily by noticing that (see Eq. \eqref{eq:def_n2extrap}) 
+\alert{
+We report in Table \ref{tab:d1} these quantities for \ce{N2} and N in different basis sets. One can notice that the system-averaged on-top pair density at the CIPSI level $\ontopcipsi$ is systematically lower than that at the CASSCF level $\ontopcas$, which is expected since short-range correlation, digging the Coulomb hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function. 
+Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set increases, leading to roughly a $20\%$ decrease from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas $\ontopcas$ is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower than their non-extrapolated counterparts, $\ontopcas$ and $\ontopcipsi$. Nevertheless, the behaviors of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different: $\ontopextrap$ clearly increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains almost constant. More precisely, in the case of \ce{N2}, the value of $\ontopextrap$ increases by about 30$\%$ from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of $\ontopextrapcipsi$ only fluctuates within 5$\%$ with the same basis sets. The behavior of $\ontopextrap$ can be understood by noticing that i) the value of $\murcas$ globally increases when increasing the size of the basis set (as evidenced by $\muaverage$), and ii) $\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2$ [see Eq. \eqref{eq:def_n2extrap}]. Therefore, in the CBS limit, $\murcas \rightarrow \infty$ and one obtains 
 \begin{equation}
- \lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2, 
+ \lim_{\basis \rightarrow \text{CBS}} \ontopextrap =  \lim_{\basis \rightarrow \text{CBS}} \ontopcas,
 \end{equation}
-that the CASSCF on-top pair density is globally constant with the basis set and that the value of $\murcas$ globally increases (as evidenced by $\muaverage$). 
-Eventually, at the CBS limit, $\murcas \rightarrow \infty$ and therefore one obtains 
-\begin{equation}
- \lim_{\basis \rightarrow \text{CBS}} \ontopextrap = \ontopcas. 
-\end{equation}
-On the other hand, the stability of $\ontopextrapcipsi$ is quite remarkable and must come from i) the fact that the on-top pair density at the CIPSI level already captures the coulomb hole within the basis set $\basis$, and ii) the $\murcipsi$ together with the large-$\mu$ limit extrapolation of the on-top pair density (see Eq. \eqref{eq:def_n2extrap}) are quantitatively correct. 
-Therefore, in order to estimate the integral of exact on-top pair density, we take as reference the value of $\ontopextrapcipsi$ in the aug-cc-pVQZ basis set. 
+\ie, $\ontopextrap$ must increase with the size of the basis set $\basis$ to eventually converge to $\lim_{\basis \rightarrow \text{CBS}} \ontopcas$, the latter limit being essentially reached with the present basis sets.
+On the other hand, the stability of $\ontopextrapcipsi$ with respect to the basis set is quite remarkable and must come from the fact that i) $\ontopcipsi$ is a good approximation to the corresponding FCI value within the considered basis sets, and ii) the extrapolation formula in Eq. \eqref{eq:def_n2extrap} together with the choice of $\murcipsi$ are quantitatively correct. Therefore, we expect the calculated values of $\ontopextrapcipsi$ to be nearly converged with respect to the basis set, and we will take the value of $\ontopextrapcipsi$ in the aug-cc-pVQZ basis set as an estimate of the exact system-averaged on-top pair density.
 }
+
 \manu{
 In the case of the present work, it is important to keep in mind that $\ontopextrap$ is directly linked to the basis set correction in the large $\mu$ limit, and more precisely the correlation energy (in absolute value) is a growing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact (here taken as $\ontopextrapcipsi$ in the aug- cc-pVQZ basis set) provides an indication on the magnitude of the error on the basis set correction for a given system and a given basis set. 
 In the aug-cc-pVQZ, for \ce{N2} $\ontopextrap - \ontopextrapcipsi = 0.120$ whereas $2\times(\ontopextrap - \ontopextrapcipsi) = 0.095$. We can then conclude that the overestimation of the on-top pair density and therefore of the basis set correction is more important on the \ce{N2} molecule at equilibrium distance than on the dissociated molecule, explaining probably the overestimation of the atomization energy.