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working on the theory section

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Manuscript/srDFT_SC.aux
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@ -32,22 +32,29 @@
\citation{GinPraFerAssSavTou-JCP-18}
\citation{GinPraFerAssSavTou-JCP-18}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{2}{section*.5}}
\newlabel{sec:basic}{{II\tmspace +\thinmuskip {.1667em}A}{2}{}{section*.5}{}}
\newlabel{eq:levy}{{1}{2}{}{equation.2.1}{}}
\newlabel{eq:levy_func}{{2}{2}{}{equation.2.2}{}}
\newlabel{eq:e0approx}{{5}{2}{}{equation.2.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B}Definition of an effective interaction within $\mathcal {B}$}{2}{section*.6}}
\newlabel{sec:wee}{{II\tmspace +\thinmuskip {.1667em}B}{2}{}{section*.6}{}}
\newlabel{eq:wbasis}{{6}{2}{}{equation.2.6}{}}
\newlabel{eq:fbasis}{{8}{2}{}{equation.2.8}{}}
\newlabel{eq:cbs_wbasis}{{10}{2}{}{equation.2.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{2}{section*.7}}
\citation{GinPraFerAssSavTou-JCP-18}
\bibdata{srDFT_SCNotes,srDFT_SC}
\bibstyle{aipnum4-1}
\citation{REVTEX41Control}
\citation{aip41Control}
\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{3}{section*.7}}
\newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{3}{}{section*.7}{}}
\newlabel{eq:weelr}{{11}{3}{}{equation.2.11}{}}
\newlabel{eq:cbs_mu}{{14}{3}{}{equation.2.14}{}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{3}{section*.8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{3}{section*.8}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form of the approximated functionals}{3}{section*.9}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{3}{section*.10}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{3}{section*.11}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{3}{section*.12}}
\newlabel{LastBibItem}{{0}{3}{}{figure.7}{}}
\@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. }}{4}{figure.1}}
\newlabel{fig:N2_avdz}{{1}{4}{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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Manuscript/srDFT_SC.out
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@ -5,4 +5,8 @@
\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5
\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6
\BOOKMARK [2][-]{section*.7}{Definition of an range-separation parameter varying in real space}{section*.4}% 7
\BOOKMARK [1][-]{section*.8}{Results}{section*.2}% 8
\BOOKMARK [2][-]{section*.8}{Approximation for B[n\(r\)] : link with RSDFT}{section*.4}% 8
\BOOKMARK [3][-]{section*.9}{Generic form of the approximated functionals}{section*.8}% 9
\BOOKMARK [3][-]{section*.10}{Introduction of the effective spin-density}{section*.8}% 10
\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12

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Manuscript/srDFT_SC.tex
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@ -276,8 +276,9 @@ When the molecular system
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
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.
First in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}. Thanks to the range separation parameter, we make a mapping with a specific class of RSDFT functionals and propose practical approximations for the unknown density functional complementary to a basis set $\Bas$, for which new approximations for the strong correlation regime are given in section \ref{sec:functional}.
\subsection{Basic formal equations}
\label{sec:basic}
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
\begin{equation}
\label{eq:levy}
@ -314,6 +315,7 @@ where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally
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.
\subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee}
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.
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.
@ -350,6 +352,7 @@ As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $
\end{equation}
The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
\subsection{Definition of an range-separation parameter varying in real space}
\label{sec:mur}
As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$
\begin{equation}
\label{eq:weelr}
@ -369,6 +372,11 @@ Because of the very definition of $\wbasis$, one has the following properties at
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty,
\end{equation}
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
\subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT}
\subsubsection{Generic form of the approximated functionals}
\subsubsection{Introduction of the effective spin-density}
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%