working on the theory section

Manuscript/srDFT_SC.aux

@@ -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}{}}

Manuscript/srDFT_SC.out

@@ -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

Manuscript/srDFT_SC.tex

@@ -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}
 %%%%%%%%%%%%%%%%%%%%%%%%