titou has modified Manu's part

JPCL_revision/G2-srDFT.tex

@@ -468,11 +468,17 @@ In most cases, the basis-set corrected triple-$\zeta$ atomization energies are o
 	\label{fig:N2}}
 \end{figure}
 
-\manu{A fundamental quantity for the present basis set correction is the function $\rsmu{}{\Bas}(\br{})$ in space. As $\rsmu{}{\Bas}(\br{})$ should tend to infinity in any points in space when reaching the CBS, the local value of $\rsmu{}{\Bas}(\br{})$ can be used to quantify quality of a given basis set in a given point in space. Indeed, the larger the value of $\rsmu{}{\Bas}(\br{})$, the closer it is to the CBS limit, and therefore the smaller (in absolute value) will be the energetic correction. 
+\titou{The fundamental quantity of the present basis set correction is $\rsmu{}{\Bas}(\br{})$. 
-In order to qualitatively illustrate how the basis set correction operates, we report in Figure \ref{fig:N2} $\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for \ce{N2} and $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$. 
+As it grows when one gets closer to the CBS limit, the value of $\rsmu{}{\Bas}(\br{})$ quantifies the quality of a given basis set at a given $\br{}$. 
-This figure illustrates several general trends: i) the global value of $\rsmu{}{\Bas}(z)$ is much larger than 0.5 which is the standard value used in RS-DFT ii) the local value of $\rsmu{}{\Bas}(z)$ systematically grows when improving the basis set $\Bas$, which means that the total DFT correction will diminish while improving the basis set, iii) the value of $\rsmu{}{\Bas}(z)$ are highly non uniform in space, illustrating the non homogeneity of quality of the basis sets used in quantum chemistry, iv) the value of $\rsmu{}{\Bas}(z)$ are signigicantly larger close to the nucleis, a signature that atom-centered basis sets describe better these regions than the bonding region. 
+Another important quantity closely related to $\rsmu{}{\Bas}(\br{})$ is the local energetic correction, $\n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}))$, which integrates to the total basis set correction $\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}]$ [see Eq.~\eqref{eq:def_pbe_tot}]. 
-Another important aspect closely related to $\rsmu{}{\Bas}(\br{})$  is the local energetic correction at a given point in space $\br{}$, which integrated gives the total basis set correction (see equation \eqref{eq:def_pbe_tot}). Such quantity essentially depends on the local value of $\rsmu{}{\Bas}(\br{})$ together with the local value of the density at a given point $\br{}$. We report in Figure \ref{fig:N2} the value of $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) for \ce{N2} and $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$. This figure illustrates that several things: i) for all basis sets used, except for the high density regions (\textit{i.e.} close to the nuclei), the largest contribution is the bonding region which highlights that the correlation effects are poorly described in this region; ii) the global value of the energy correction get smaller as one improves the basis set quality, and the reduction is spectacular close to the nuclei, a sign that atom-centered basis sets give a better description of these regions. 
+Such quantity essentially depends on the local values of both $\rsmu{}{\Bas}(\br{})$ and $\n{}{}(\br{})$.
-}
+In order to qualitatively illustrate how the basis set correction operates, we report, in Figure \ref{fig:N2}, $\rsmu{}{\Bas}$ and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) of \ce{N2} for $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$. 
+This figure illustrates several general trends: 
+i) the global value of $\rsmu{}{\Bas}(z)$ is much larger than 0.5 which is the standard value used in RS-DFT, 
+ii) $\rsmu{}{\Bas}(z)$ is highly non uniform in space, illustrating the non-homogeneity of basis set quality in quantum chemistry, 
+iii) $\rsmu{}{\Bas}(z)$ is significantly larger close to the nuclei, a signature that atom-centered basis sets better describe these high-density regions than the bonding regions,
+v) the global value of the energy correction get smaller as one improves the basis set quality, and the reduction is spectacular close to the nuclei, and
+iv) a large energetic contribution comes from the bonding regions highlighting the poor description of correlation effects in these region with Gaussian basis sets.}
 
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 \begin{table}