working on the theory section
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@ -18,27 +18,41 @@
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\providecommand\HyField@AuxAddToCoFields[2]{}
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\citation{alex_thom,piotr}
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\citation{scuseria}
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\bibdata{srDFT_SCNotes,srDFT_SC}
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\bibstyle{aipnum4-1}
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\citation{REVTEX41Control}
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\citation{aip41Control}
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\citation{GinPraFerAssSavTou-JCP-18}
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\newlabel{FirstPage}{{}{1}{}{section*.1}{}}
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\@writefile{toc}{\contentsline {title}{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}{1}{section*.2}}
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\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}}
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\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}}
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\@writefile{toc}{\contentsline {section}{\numberline {II}Theory}{1}{section*.4}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{1}{section*.5}}
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\newlabel{LastBibItem}{{0}{1}{}{figure.6}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{2}{figure.1}}
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\newlabel{fig:N2_avdz}{{1}{2}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{3}{figure.2}}
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\newlabel{fig:N2_avtz}{{2}{3}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{4}{figure.3}}
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\newlabel{fig:F2_avdz}{{3}{4}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.4}}
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\newlabel{fig:F2_avtz}{{4}{5}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.5}}
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\newlabel{fig:H10_vdz}{{5}{6}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.6}}
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\newlabel{fig:H10_vtz}{{6}{6}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
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\newlabel{LastPage}{{}{6}{}{}{}}
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\citation{GinPraFerAssSavTou-JCP-18}
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\citation{GinPraFerAssSavTou-JCP-18}
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\citation{GinPraFerAssSavTou-JCP-18}
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\citation{G2,excited}
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\citation{kato}
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\citation{GinPraFerAssSavTou-JCP-18}
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\bibdata{srDFT_SCNotes,srDFT_SC}
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\bibstyle{aipnum4-1}
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\citation{REVTEX41Control}
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\citation{aip41Control}
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\@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{2}{section*.5}}
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\newlabel{eq:levy}{{1}{2}{}{equation.2.1}{}}
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\newlabel{eq:levy_func}{{2}{2}{}{equation.2.2}{}}
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\newlabel{eq:e0approx}{{5}{2}{}{equation.2.5}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {B}Definition of an effective interaction within $\mathcal {B}$}{2}{section*.6}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{2}{section*.7}}
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\newlabel{LastBibItem}{{0}{2}{}{figure.7}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{3}{figure.1}}
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\newlabel{fig:N2_avdz}{{1}{3}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{4}{figure.2}}
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\newlabel{fig:N2_avtz}{{2}{4}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.3}}
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\newlabel{fig:F2_avdz}{{3}{5}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.4}}
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\newlabel{fig:F2_avtz}{{4}{6}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.5}}
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\newlabel{fig:H10_vdz}{{5}{7}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.6}}
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\newlabel{fig:H10_vtz}{{6}{7}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.7}}
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\newlabel{fig:H10_vqz}{{7}{8}{H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.7}{}}
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\newlabel{LastPage}{{}{8}{}{}{}}
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@ -2,4 +2,6 @@
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\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
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\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3
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\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4
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\BOOKMARK [1][-]{section*.5}{Results}{section*.2}% 5
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\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5
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\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6
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\BOOKMARK [1][-]{section*.7}{Results}{section*.2}% 7
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@ -62,6 +62,7 @@
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\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
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\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
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\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}}
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\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]}
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\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
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\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
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\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
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@ -85,8 +86,7 @@
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% numbers
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\newcommand{\rnum}[0]{{\rm I\!R}}
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\newcommand{\bfr}[1]{{\bf x}_{#1}}
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\newcommand{\bfrb}[1]{{\bf r}_{#1}}
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\newcommand{\bfr}[1]{{\bf r}_{#1}}
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\newcommand{\dr}[1]{\text{d}\bfr{#1}}
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\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}}
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\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} }
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@ -109,9 +109,9 @@
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\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\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)}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\twodmrpsi}[0]{ n^{2}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{2}_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n^{2}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
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\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
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\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
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@ -133,6 +133,7 @@
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\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
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\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
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\newcommand{\denfci}[0]{\den_{\psifci}}
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\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
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\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
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\newcommand{\denrfci}[0]{\denr_{\psifci}}
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\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
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@ -188,6 +189,7 @@
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\newcommand{\modY}{\text{Y}}
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% basis sets
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\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}}
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\Basval}{\mathcal{B}_\text{val}}
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\newcommand{\Val}{\mathcal{V}}
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@ -200,10 +202,9 @@
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
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\newcommand{\bx}[1]{\mathbf{x}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\dbx}[1]{d\bx{#1}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
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@ -273,7 +274,60 @@ When the molecular system
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%%%%%%%%%%%%%%%%%%%%%%%%
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The theoretical framework of the basis set correction have been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study.
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\subsection{Basic formal equations}
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The exact ground state energy $E_0$ of a $N-$electron system can be obtained by the Levy-Lieb constrained search formalism which is an elegant mathematical framework connecting WFT and DFT
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\begin{equation}
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\label{eq:levy}
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E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
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\end{equation}
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where $(v_{ne}(\br)|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
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\begin{equation}
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\label{eq:levy_func}
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F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
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\end{equation}
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The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
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As in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
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Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ as
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\begin{equation}
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F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}
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\end{equation}
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where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as
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\begin{equation}
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\begin{aligned}
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\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\
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&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}},
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\end{aligned}
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\end{equation}
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and $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$.
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The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$.
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Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18}), one obtains the following approximation for the exact ground state density
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\begin{equation}
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\label{eq:e0approx}
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E_0 = \efci + \efuncbasisFCI
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\end{equation}
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where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in \cite{GinPraFerAssSavTou-JCP-18} and further emphasized in \cite{G2,excited}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set.
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The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity.
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\subsection{Definition of an effective interaction within $\Bas$}
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As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, the lack of cusp in any wave function $\wf{}{\Bas}$ could also originate from an effective non-divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence at the electron coalescence point.
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As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction.
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More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as
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\begin{equation}
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\wbasis = \fbasis/\twodmrdiagpsi
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\end{equation}
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where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$
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\begin{equation}
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\twodmrdiagpsi = \sum_{pqrs} \phi_{p}(\br) \phi_{q}(\br) \Gam{pq}{rs} \phi_{r}(\br) \phi_{s}(\br),
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\end{equation}
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$\Gam{pq}{rs}$
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\begin{equation}
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\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \elemm{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
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\end{equation}
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where $\twodmrdiagpsi$ is the two-body density of
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -338,6 +392,16 @@ The theoretical framework of the basis set correction have been derived in detai
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\end{figure}
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\begin{figure}
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\includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}\\
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\includegraphics[width=\linewidth]{data/H10/DFT_vqzE_error.eps}\\
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% \includegraphics[width=\linewidth]{fig2c}
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\caption{
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H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
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\label{fig:H10_vqz}}
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\end{figure}
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