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JPCL_revision/G2-srDFT.tex

@@ -471,7 +471,7 @@ In most cases, the basis-set corrected triple-$\zeta$ atomization energies are o
 \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. 
 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 $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$. 
 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 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 must have a strong differential effect for the atomization energy; 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. 
+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. 
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