minor corrections following discussion with David

Manuscript/CI-F12.aux

@@ -29,15 +29,16 @@
 \citation{Kato51,Kato57}
 \citation{Pack66,Morgan93,Tew08,ExSpherium10,eee15}
 \newlabel{FirstPage}{{}{1}{}{section*.1}{}}
-\@writefile{toc}{\contentsline {title}{Dressing the configuration interaction matrix with explicit correlation}{1}{section*.2}}
+\@writefile{toc}{\contentsline {title}{Dressing the configuration interaction matrix with explicit correlation}{1}{section*.2}\protected@file@percent }
-\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}}
+\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\numberline {II}Ans{\"a}tz}{1}{section*.4}}
+\@writefile{toc}{\contentsline {section}{\numberline {II}Ans{\"a}tz}{1}{section*.4}\protected@file@percent }
 \newlabel{eq:ansatz}{{1}{1}{}{equation.2.1}{}}
 \newlabel{eq:D}{{2}{1}{}{equation.2.2}{}}
 \newlabel{eq:WF-F12-CIPSI}{{3}{1}{}{equation.2.3}{}}
 \newlabel{eq:Ja}{{5}{1}{}{equation.2.5}{}}
-\@writefile{toc}{\contentsline {section}{\numberline {III}Effective Hamiltonian}{1}{section*.5}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}Effective Hamiltonian}{1}{section*.5}\protected@file@percent }
+\newlabel{eq:DrH}{{8}{1}{}{equation.3.8}{}}
 \citation{Tenno04a}
 \citation{Garniron18}
 \citation{Garniron18}
@@ -47,15 +48,14 @@
 \citation{3ERI1,3ERI2,4eRR,IntF12}
 \citation{Kutzelnigg91,Klopper02,Valeev04,Werner07,Hattig12}
 \citation{Klopper02,Valeev04}
-\newlabel{eq:DrH}{{9}{2}{}{equation.3.9}{}}
-\newlabel{eq:IHF}{{10}{2}{}{equation.3.10}{}}
-\newlabel{eq:tI}{{11}{2}{}{equation.3.11}{}}
-\newlabel{eq:DrH}{{12}{2}{}{equation.3.12}{}}
-\@writefile{toc}{\contentsline {section}{\numberline {IV}Matrix elements}{2}{section*.6}}
-\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces   Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange).}}{2}{figure.1}}
-\newlabel{fig:CBS}{{1}{2}{Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange)}{figure.1}{}}
-\newlabel{eq:RI}{{13}{2}{}{equation.4.13}{}}
 \citation{Garniron17b}
+\newlabel{eq:IHF}{{9}{2}{}{equation.3.9}{}}
+\newlabel{eq:DrH}{{10}{2}{}{equation.3.10}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {IV}Matrix elements}{2}{section*.6}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces   Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange).}}{2}{figure.1}\protected@file@percent }
+\newlabel{fig:CBS}{{1}{2}{Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange)}{figure.1}{}}
+\newlabel{eq:RI}{{11}{2}{}{equation.4.11}{}}
+\newlabel{eq:IHF-RI}{{12}{2}{}{equation.4.12}{}}
 \citation{Kutzelnigg91}
 \citation{Tenno04a}
 \citation{Persson96,Persson97,May04,Tenno04b,Tew05,May05}
@@ -74,10 +74,10 @@
 \citation{AlmoraDiaz14}
 \citation{Yousaf08,Yousaf09}
 \citation{Giner13,Giner15,Caffarel16}
-\newlabel{eq:IHF-RI}{{14}{3}{}{equation.4.14}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {V}Computational details}{3}{section*.7}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\numberline {V}Computational details}{3}{section*.7}}
+\@writefile{toc}{\contentsline {section}{\numberline {VI}Results}{3}{section*.8}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\numberline {VI}Results}{3}{section*.8}}
+\@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{3}{section*.9}\protected@file@percent }
-\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces   FCI-F12 and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pCVXZ and cc-pVXZ basis sets. \leavevmode {\color  {red}The corresponding cc-pVXZ\_OPTRI or cc-pCVXZ\_OPTRI auxiliary basis is used as CABS.}}}{3}{table.1}}
+\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces   FCI-F12 and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pCVXZ and cc-pVXZ basis sets. \leavevmode {\color  {red}The corresponding cc-pVXZ\_OPTRI or cc-pCVXZ\_OPTRI auxiliary basis is used as CABS.}}}{3}{table.1}\protected@file@percent }
 \newlabel{tab:atoms}{{I}{3}{FCI-F12 and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pCVXZ and cc-pVXZ basis sets. \alert {The corresponding cc-pVXZ\_OPTRI or cc-pCVXZ\_OPTRI auxiliary basis is used as CABS.}}{table.1}{}}
 \bibdata{CI-F12Notes,CI-F12,CI-F12-control}
 \bibcite{Kutzelnigg85}{{1}{1985}{{Kutzelnigg}}{{}}}
@@ -127,10 +127,6 @@
 \bibcite{Klopper04}{{45}{2004}{{Klopper}}{{}}}
 \bibcite{Manby06}{{46}{2006}{{Manby\ \emph  {et~al.}}}{{Manby, Werner, Adler,\ and\ May}}}
 \bibcite{Tenno07}{{47}{2007}{{Ten-no}}{{}}}
-\@writefile{lot}{\contentsline {table}{\numberline {II}{\ignorespaces   CIPSI, FCI-F12, i-FCIQMC and FCI total ground-state energy of the \ce {H2}, \ce {F2} and \ce {H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}}{4}{table.2}}
-\newlabel{tab:molecules}{{II}{4}{CIPSI, FCI-F12, i-FCIQMC and FCI total ground-state energy of the \ce {H2}, \ce {F2} and \ce {H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS}{table.2}{}}
-\@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{4}{section*.9}}
-\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{4}{section*.10}}
 \bibcite{Komornicki11}{{48}{2011}{{Komornicki\ and\ King}}{{}}}
 \bibcite{Reine12}{{49}{2012}{{Reine, Helgaker,\ and\ Lind}}{{}}}
 \bibcite{GG16}{{50}{2016}{{Barca\ and\ Gill}}{{}}}
@@ -146,6 +142,9 @@
 \bibcite{Sharkey11}{{60}{2011}{{Sharkey\ and\ Adamowicz}}{{}}}
 \bibcite{Bubin11}{{61}{2011}{{Bubin\ and\ Adamowicz}}{{}}}
 \bibcite{Sharkey10}{{62}{2010}{{Sharkey, Bubin,\ and\ Adamowicz}}{{}}}
+\@writefile{lot}{\contentsline {table}{\numberline {II}{\ignorespaces   CIPSI, FCI-F12, i-FCIQMC and FCI total ground-state energy of the \ce {H2}, \ce {F2} and \ce {H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}}{4}{table.2}\protected@file@percent }
+\newlabel{tab:molecules}{{II}{4}{CIPSI, FCI-F12, i-FCIQMC and FCI total ground-state energy of the \ce {H2}, \ce {F2} and \ce {H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS}{table.2}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{4}{section*.10}\protected@file@percent }
 \bibcite{Klopper10}{{63}{2010}{{Klopper\ \emph  {et~al.}}}{{Klopper, Bachorz, Tew,\ and\ Hattig}}}
 \bibcite{Pachucki10}{{64}{2010}{{Pachucki}}{{}}}
 \bibcite{Cleland12}{{65}{2012}{{Cleland\ \emph  {et~al.}}}{{Cleland, Booth, Overy,\ and\ Alavi}}}

Manuscript/CI-F12.tex

@@ -1,6 +1,5 @@
 \documentclass[aip,jcp,reprint,showkeys]{revtex4-1}
 \usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem}
-
 \usepackage{mathpazo,libertine}
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
 \newcommand{\cdash}{\multicolumn{1}{c}{---}}
@@ -56,7 +55,7 @@
    
 % addresses
 \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
-   
+\newcommand{\MPI}{Max-Planck-Institut f\"ur Festk\"orperforschung, Heisenbergstra{\ss}e 1, 70569 Stuttgart, Germany}
 \begin{document}    
 
 \title{Dressing the configuration interaction matrix with explicit correlation}    
@@ -65,6 +64,8 @@
 \affiliation{\LCPQ}
 \author{Michel Caffarel}
 \affiliation{\LCPQ}
+\author{David P. Tew}
+\affiliation{\MPI}
 \author{Pierre-Fran\c{c}ois Loos}
 \email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
 \affiliation{\LCPQ}
@@ -132,19 +133,18 @@ The projector
 ensures the orthogonality between $\kD$ and $\kF$ (where $\hI$ is the identity operator), and
 \begin{equation}
 \label{eq:Ja}
-    f = \sum_{i < j} \gamma_{ij} f_{ij}
+    f = \sum_{i < j} f_{ij}
+%    f = \sum_{i < j} \gamma_{ij} f_{ij}
 \end{equation}
-is a correlation factor with
+is a (linear) correlation factor.% with
-\begin{equation}
+%\begin{equation}
-    \gamma_{ij} = 
+%    \gamma_{ij} = 
-    \begin{cases}
+%    \begin{cases}
-        1/2,    &    \text{for opposite-spin electrons},
+%        1/2,    &    \text{for opposite-spin electrons},
-        \\
+%        \\
-        1/4,    &    \text{for same-spin electrons}.
+%        1/4,    &    \text{for same-spin electrons}.
-    \end{cases}
+%    \end{cases}
-\end{equation}
+%\end{equation}
-\alert{The correlation factor \eqref{eq:Ja} is not size-consistent.}
-
 As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93, Tew08, ExSpherium10, eee15}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
 \begin{equation}
     f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
@@ -156,7 +156,7 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
 Our primary goal is to introduce the explicit correlation between electrons at relatively low computational cost.
 Therefore, assuming that $\hH \ket{\Psi} = E\,\Psi$, one can write, by projection over $\bra{I}$,
 \begin{equation}
-    \cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0.
+    \cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0,
 \end{equation} 
 where $H_{IJ} = \mel{I}{\hH}{J}$.
 Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
@@ -175,19 +175,21 @@ with
     \mel{I}{\hH}{F} =  \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K}  H_{IK} f_{KJ} ],
 \end{equation}
 and $f_{IJ} = \mel{I}{f}{J}$.
-We refer to this strategy as diagonal dressing as only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}.
+Because only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}, we refer to this strategy as diagonal dressing.
 It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that the dressed term vanishes in the limit of a complete one-electron basis, as one would expect.
+Moreover, because the CI-F12 energy is obtained via projection, the present method is not variational.
 
 At this stage, two key comments are in order.
 First, as one may have realized, the coefficients $\cF{I}$ are unknown. 
-\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}}
+However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}
 
-\alert{This yields the following linear system of equations
+\alert{T2: Here include the rules to determine the coefficients $\cF{I}$.}
-\begin{equation}
+%\alert{This yields the following linear system of equations
-\label{eq:tI}
+%\begin{equation}
-    \sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
+%\label{eq:tI}
-\end{equation}
+%    \sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
-which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
+%\end{equation}
+%which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
 
 Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $\cD{I}$, one must iterate the diagonalization process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$.
 At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
@@ -195,8 +197,8 @@ In practice, we initially start with a CI vector obtained by the diagonalization
 We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme.
 
 Note that the present formalism is state-specific and only focus on the ground state.
-Multi-state can potentially developed following our work in Ref.~\onlinecite{Garniron18}.
+A multi-state strategy can be applied following our work in Ref.~\onlinecite{Garniron18}.
-In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ shuffling around the dressing term.
+In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ by shuffling around the dressing term.
 Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\cD{I}$, we have
 \begin{equation}
 \label{eq:DrH}
@@ -211,7 +213,6 @@ Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\c
             H_{IJ},                                   &    \text{otherwise}.
         \end{cases}
 \end{equation}
-It is important to mention that, because the CI-F12 energy is obtained via projection, the present method is not variational.
 
 %%% FIG 1 %%%
 \begin{figure}
@@ -227,7 +228,7 @@ It is important to mention that, because the CI-F12 energy is obtained via proje
 \section{Matrix elements}
 %----------------------------------------------------------------
 Compared to a conventional CI calculation, new matrix elements are required.
-The simplest of them $f_{IJ}$ --- required in Eqs.~\eqref{eq:IHF} and \eqref{eq:tI} --- can be easily computed by applying Condon-Slater rules. \cite{SzaboBook}
+The simplest of them $f_{IJ}$ --- required in Eq.~\eqref{eq:IHF} --- can be easily computed by applying Condon-Slater rules. \cite{SzaboBook}
 They involve two-electron integrals over the correlation factor $f_{12}$.  
 Their computation has been thoroughly studied in the literature in the last thirty years. \cite{Kutzelnigg91, Klopper92, Persson97, Klopper02, Manby03, Werner03, Klopper04, Tenno04a, Tenno04b, May05, Manby06, Tenno07, Komornicki11, Reine12, GG16}
 These can be more or less expensive to compute depending on the choice of the correlation factor.