updated data in tabular
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Emmanuel Giner
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@ -73,6 +73,7 @@
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\newcommand{\wbasis}[0]{W_{\psibasis}(\bfr{1},\bfr{2})}
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\newcommand{\fbasis}[0]{f_{\psibasis}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\psibasis}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
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\newcommand{\twodmrpsi}[0]{ n^{(2)}_{\psibasis}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\psibasis}(\rr{1}{2})}
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@ -86,6 +87,7 @@
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% Basis sets
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\newcommand{\basis}[0]{\mathcal{B}}
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\newcommand{\basisval}[0]{\mathcal{B}_\text{val}}
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% densities
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\newcommand{\denfci}[0]{\den_{\psifci}}
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@ -99,6 +101,7 @@
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% operators
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\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\basis}
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\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\basisval}}
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\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
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@ -147,11 +150,11 @@
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The DFT basis-set correction in a nutshell}
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The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing in a finite one-electron basis-set.
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Here, we briefly explain the working equations and notations needed for this work, and we encourage the interested reader to find the detailed formal derivation of the theory in \cite{GinPraFerAssSavTou-JCP-18}.
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The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing from the description of the WFT in a finite basis set.
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Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in \cite{GinPraFerAssSavTou-JCP-18}.
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\subsection{The very basics}
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Consider a basis-set incomplete $\basis$ for which we assume to have accurate approximations of both the FCI density $\denfci$ and energy $\efci$. According to equation (15) of \cite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
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Consider an incomplete basis-set $\basis$ for which we assume to have accurate approximations of both the FCI density $\denfci$ and energy $\efci$. According to equation (15) of \cite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
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\begin{equation}
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\label{eq:e0basis}
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E_0 \approx \efci + \efuncbasisfci
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@ -166,7 +169,7 @@ One of the consequences of the use of an incomplete basis-set $\basis$ is that t
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The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
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\begin{itemize}
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\item is non-divergent at the electron coalescence point as long as a finite-basis set $\basis$ is used
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\item is non-divergent at the electron coalescence point as long as an incomplete basis set $\basis$ is used
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\item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\basis$.
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\end{itemize}
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\subsubsection{General definition of an effective interaction for the basis set $\basis$}
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@ -194,7 +197,7 @@ $\gammamnpq{\psibasis}$ is the two-body density matrix of $\psibasis$
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\begin{equation}
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\gammamnpq{\psibasis} = \elemm{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis},
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\end{equation}
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and $\bfr{}$ collects the space and spin variables.
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and $\bfr{}$ collects the space and spin variables,
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\begin{equation}
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\label{eq:define_x}
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\begin{aligned}
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@ -209,7 +212,7 @@ Then, consider the expectation value of the exact coulomb operator over $\psibas
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\elemm{\psibasis}{\weeop}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \frac{1}{r_{12}} \twodmrdiagpsi\,\, .
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\end{equation}
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where $\twodmrdiagpsi$ is the two-body density associated to $\psibasis$.
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Because $\psibasis$ belongs to $\basis$, such an expectation value coincides with the expectation value of $\weeopbasis$ and therefore one can write:
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Because $\psibasis$ belongs to $\basis$, such an expectation value coincides with the expectation value of $\weeopbasis$
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\begin{equation}
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\elemm{\psibasis}{\weeopbasis}{\psibasis} = \elemm{\psibasis}{\weeop}{\psibasis},
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\end{equation}
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@ -227,6 +230,33 @@ where we introduced $\wbasis$
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\end{equation}
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which is the effective interaction in the basis set $\basis$.
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As already discussed in \cite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\basis$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\basis$.
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Also, as demonstrated in the appendix B of \cite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points in $\rnum^6$ in the limit of a complete basis set $\basis$.
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\subsubsection{Definition of a valence effective interaction}
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As the average inter electronic distances are very different between the valence electrons and the core electrons, it can be advantageous to define an effective interaction taking into account only for the valence electrons which are the most important in most of the chemical processes.
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According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define first the following function $\fbasisval$ satisfying
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\begin{equation}
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\label{eq:expectweeb}
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\elemm{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
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\end{equation}
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where $\weeopbasisval$ is the valence coulomb operator defined as
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\begin{equation}
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\begin{aligned}
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\weeopbasisval = \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basisval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
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\end{aligned}
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\end{equation}
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$\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on,
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and the function $\fbasisval$
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\begin{equation}
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\label{eq:fbasis}
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\begin{aligned}
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\fbasisval = \sum_{ijklmn\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
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\end{aligned}
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\end{equation}
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To define
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -270,10 +300,10 @@ V5Z & 38.0 & 38.7 & 38.8
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& exFCI & 132.0 & 140.3 & 143.6 & 144.3 & \\
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\hline
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& exFCI+LDA & 141.9 & 142.8 & 145.8 & 146.2 & \\
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& exFCI+LDA(FC) & 142.9 & 145.5 & 146.2 & 146.1 & \\
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& exFCI+LDA(FC) & 143.0 & 145.4 & 146.4 & 146.0 & \\
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\hline
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& exFCI+PBE & 146.1 & 143.9 & 145.9 & 145.12 & \\
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& exFCI+PBE (FC) & 147.7 & 146.3 & 146.4 & 146.0 & \\
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& exFCI+PBE (FC) & 147.4 & 146.1 & 146.4 & 145.9 & \\
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\hline
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& exFCI+PBE-on-top& 142.7 & 142.7 & 145.3 & 144.9 & \\
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& exFCI+PBE-on-top(FC) & 143.3 & 144.7 & 145.7 & 145.6 & \\
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@ -281,10 +311,10 @@ V5Z & 38.0 & 38.7 & 38.8
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\ce{N2} & exFCI & 200.9 & 217.1 & 223.5 & 225.7 & 228.5\fnm[2] \\
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\hline
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& exFCI+LDA & 216.3 & 223.1 & 227.9 & 227.9 & \\
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& exFCI+LDA(FC) & 218.2 & 225.8 & 228.8 & 228.4 & \\
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& exFCI+LDA(FC) & 217.8 & 225.9 & 228.1 & 228.5 & \\
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\hline
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& exFCI+PBE & 225.3 & 225.6 & 228.2 & 227.9 & \\
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& exFCI+PBE (FC) & 228.6 & 228.1 & 228.9 & 228.6 & \\
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& exFCI+PBE (FC) & 227.6 & 227.8 & 228.4 & 228.5 & \\
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\hline
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& exFCI+PBE-on-top& 222.3 & 224.6 & 227.7 & 227.7 & \\
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& exFCI+PBE-on-top(FC) & 224.8 & 226.7 & 228.3 & 228.3 & \\
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@ -292,10 +322,10 @@ V5Z & 38.0 & 38.7 & 38.8
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\ce{O2} & exFCI & 105.3 & 114.6 & 118.0 &119.1 & 120.2\fnm[2] \\
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\hline
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& exFCI+LDA & 111.8 & 117.2 & 120.0 &119.9 & \\
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& exFCI+LDA(FC) & 112.5 & 118.5 & 120.2 & 120.2 & \\
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& exFCI+LDA(FC) & 112.4 & 118.5 & 120.2 & 120.3 & \\
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\hline
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& exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\
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& exFCI+PBE (FC) & 117.5 & 119.5 & 120.4 &120.3 & \\
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& exFCI+PBE (FC) & 117.2 & 119.4 & 120.4 &120.3 & \\
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\hline
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& exFCI+PBE-on-top& 115.0 & 118.4 & 120.2 & & \\
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& exFCI+PBE-on-top(FC) & 116.1 & 119.4 & 120.5 & & \\
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@ -306,7 +336,7 @@ V5Z & 38.0 & 38.7 & 38.8
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& exFCI+LDA(FC) & 31.1 & 37.5 & 38.8 & 38.8 & \\
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\hline
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& exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\
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& exFCI+PBE (FC) & 33.9 & 38.2 & 39.0 & 38.8 & \\
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& exFCI+PBE (FC) & 33.7 & 38.2 & 39.0 & 38.8 & \\
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\hline
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& exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\
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& exFCI+PBE-on-top(FC) & 32.4 & 37.8 & 38.8 & 38.8 & \\
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@ -321,6 +351,6 @@ V5Z & 38.0 & 38.7 & 38.8
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%
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\bibliography{G2-srDFT}
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%\bibliography{G2-srDFT}
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\end{document}
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