From 51185022a2f49ea117a33f3442c6840dd4438847 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 10 May 2019 15:35:36 +0200 Subject: [PATCH] titou has modified Manu's part --- JPCL_revision/G2-srDFT.tex | 16 +++++++++++----- 1 file changed, 11 insertions(+), 5 deletions(-) diff --git a/JPCL_revision/G2-srDFT.tex b/JPCL_revision/G2-srDFT.tex index 6c62094..c6a2fbc 100644 --- a/JPCL_revision/G2-srDFT.tex +++ b/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. -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 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. -} +\titou{The fundamental quantity of the present basis set correction is $\rsmu{}{\Bas}(\br{})$. +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 $\br{}$. +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}]. +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.} %%% TABLE II %%% \begin{table}