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Manuscript/CI-F12.aux
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@ -37,7 +37,7 @@
\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}Dressing}{1}{section*.5}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Effective Hamiltonian}{1}{section*.5}}
\citation{Tenno04a}
\citation{Garniron18}
\citation{Garniron18}
@ -47,16 +47,15 @@
\citation{3ERI1,3ERI2,4eRR,IntF12}
\citation{Kutzelnigg91,Klopper02,Valeev04,Werner07,Hattig12}
\citation{Klopper02,Valeev04}
\citation{Garniron17b}
\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: 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: 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}{}}
\@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}{}}
\newlabel{eq:IHF-RI}{{14}{2}{}{equation.4.14}{}}
\citation{Garniron17b}
\citation{Kutzelnigg91}
\citation{Tenno04a}
\citation{Persson96,Persson97,May04,Tenno04b,Tew05,May05}
@ -75,11 +74,11 @@
\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}}
\@writefile{toc}{\contentsline {section}{\numberline {VI}Results}{3}{section*.8}}
\@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{3}{section*.9}}
\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}}{3}{table.1}}
\newlabel{tab:atoms}{{I}{3}{FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS}{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}}
\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}}{{}}}
\bibcite{Kutzelnigg91}{{2}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
@ -128,14 +127,15 @@
\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}}{{}}}
\bibcite{3ERI1}{{51}{2016}{{Barca, Loos,\ and\ Gill}}{{}}}
\bibcite{3ERI2}{{52}{tion}{{Barca, Loos,\ and\ Gill}}{{}}}
\@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 {}Acknowledgments}{4}{section*.10}}
\bibcite{4eRR}{{53}{ress}{{Barca\ and\ Loos}}{{}}}
\bibcite{IntF12}{{54}{2017}{{Barca\ and\ Loos}}{{}}}
\bibcite{Valeev04}{{55}{2004}{{Valeev}}{{}}}

2
Manuscript/CI-F12.out
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@ -2,7 +2,7 @@
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3
\BOOKMARK [1][-]{section*.4}{Ans\344tz}{section*.2}% 4
\BOOKMARK [1][-]{section*.5}{Dressing}{section*.2}% 5
\BOOKMARK [1][-]{section*.5}{Effective Hamiltonian}{section*.2}% 5
\BOOKMARK [1][-]{section*.6}{Matrix elements}{section*.2}% 6
\BOOKMARK [1][-]{section*.7}{Computational details}{section*.2}% 7
\BOOKMARK [1][-]{section*.8}{Results}{section*.2}% 8

53
Manuscript/CI-F12.tex
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@ -151,7 +151,7 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
\end{equation}
%----------------------------------------------------------------
\section{Dressing}
\section{Effective Hamiltonian}
%----------------------------------------------------------------
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 $\bra{I}$,
@ -211,13 +211,15 @@ 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}
\includegraphics[width=\linewidth]{fig1}
\caption{
\label{fig:CBS}
Schematic representation of the various orbital spaces and their notation.
The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
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).}
\end{figure}
%%% %%%
@ -226,16 +228,16 @@ Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\c
%----------------------------------------------------------------
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}
They involve two-electron integrals over the geminal factor $f_{12}$.
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.
As shown in Eq.~\eqref{eq:IHF}, the present explicitly-correlated CI method also requires matrix elements of the form $\mel{I}{\hH f}{ J}$.
These are more problematic, as they involve the computation of numerous three-electron integrals over the operator $r_{12}^{-1}f_{13}$, as well as new two-electron integrals. \cite{Kutzelnigg91, Klopper92}
These are more problematic, as they involve the computation of numerous three-electron integrals over, for instance, the operator $r_{12}^{-1}f_{13}$, as well as new two-electron integrals. \cite{Kutzelnigg91, Klopper92}
We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals. \cite{3ERI1, 3ERI2, 4eRR, IntF12}
However, we will here explore a different route.
We propose to compute them using the resolution of the identity (RI) approximation, \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12} which requires a complete basis set (CBS).
We propose to compute them using the resolution of the identity (RI) approximation, \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12} which requires (at least formally) a complete basis set (CBS).
This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided as occupied $\qty{i}$ and virtual $\qty{a}$ subspaces) augmented by a complementary auxiliary basis set (CABS) $\qty{\alpha}$, such as $ \qty{p} \cap \qty{\alpha} = \varnothing$ and $\braket{p}{\alpha} = 0$. \cite{Klopper02, Valeev04} (see Fig.~\ref{fig:CBS}).
In the CBS, one can write
@ -257,16 +259,16 @@ Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eq
where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqref{eq:IHF-RI} is actually very similar to what is done within second-order multireference perturbation theory. \cite{Garniron17b}
\alert{The set $\mC$ is defined by two simple rules.
The set $\mC$ is defined by two simple rules.
First, because $f$ is a two-electron operator [and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}], we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
Three types of determinants have these two properties (see Fig.~\ref{fig:CBS}).:
i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$;
ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$;
iii) the pure singles $\ket*{_{i}^{\alpha}}$.}
i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$,
ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$, and
iii) the pure singles $\ket*{_{i}^{\alpha}}$.
\alert{Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}}
Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}
%\begin{gather}
% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
@ -303,15 +305,15 @@ Here, we will eschew the generalized Brillouin condition (GBC) which set these t
%----------------------------------------------------------------
\section{Computational details}
%----------------------------------------------------------------
In all the calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
In all the CI-F12 calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
\begin{equation}
f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
\end{equation}
which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience, \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} i.e.
which is fitted using $N_\text{GG}$ Gaussian geminals for computational convenience, \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} i.e.
\begin{equation}
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} d_\nu \exp( - \la_\nu r_{12}^2 ).
\end{equation}
The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
The contraction coefficients $d_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
Unless otherwise stated, all the calculations have been performed with \textsc{QCaml}, an electronic structure software written in \textsc{OCaml} specifically designed for the present study.
%----------------------------------------------------------------
@ -323,8 +325,8 @@ Unless otherwise stated, all the calculations have been performed with \textsc{Q
\begin{table}
\caption{
\label{tab:atoms}
FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set.
The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
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.}}
\begin{ruledtabular}
\begin{tabular}{lcdd}
Atom & X & \tabc{FCI-F12} & \tabc{FCI} \\
@ -362,14 +364,14 @@ The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
& $\infty$ & & -37.845\,0 (TOTO) \\
\hline
\ce{N} & D & & -54.517\,650 (FCI) \\
(cc-pwCV$N$Z) & T & & -54.567\,764 (CIPSI) \\
(cc-pwCVXZ) & T & & -54.567\,764 (CIPSI) \\
& Q & & -54.581\,885 (CIPSI) \\
& 5 & & -54.585\,926 (CIPSI) \\
& $\infty$ & & -54.588\,917 \fnm[7] \\
& $\infty$ & & -54.589\,3 (TOTO) \\
\hline
\ce{O} & D & & -74.946\,393 (CIPSI) \\
(cc-pwCV$N$Z) & T & & -75.031\,607 (CIPSI) \\
(cc-pwCVXZ) & T & & -75.031\,607 (CIPSI) \\
& Q & & -75.054\,737 (CIPSI) \\
& 5 & & -75.062\,002 (CIPSI) \\
& $\infty$ & & -75.066\,892 \fnm[7] \\
@ -383,10 +385,10 @@ The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
& $\infty$ & & -99.734\,1 (TOTO) \\
\hline
\ce{Ne} & D & & -128.721\,575 (CIPSI) \\
(cc-pwCV$N$Z) & T & & -128.869\,425 (CIPSI) \\
(cc-pwCVXZ) & T & & -128.869\,425 (CIPSI) \\
& Q & & -128.913\,064 (CIPSI) \\
& 5 & & -128.927\,705 (CIPSI) \\
& $\infty$ & & -128.937\,274 \footnotemark[7] \\
& $\infty$ & & -128.937\,274 \fnm[7] \\
& $\infty$ & & -128.938\,3 (TOTO) \\
\end{tabular}
\end{ruledtabular}
@ -404,7 +406,7 @@ The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
\begin{table*}
\caption{
\label{tab:molecules}
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.
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.}
\begin{ruledtabular}
\begin{tabular}{lcdddd}
@ -442,14 +444,15 @@ In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2}
\section{Conclusion}
%----------------------------------------------------------------
We have introduced a dressed version of the well-established CI method to incorporate explicitly the correlation between electrons.
We have shown that the new CI-F12 method allows to fix one of the main issue of conventional CI methods, i.e.~the slow convergence of the electronic energy with respect to the size of the one-electron basis set. Albeit not variational, our method is able to catch a large fraction of the basis set incompleteness error at a low computational cost compared to other variants.
We have shown that the new CI-F12 method allows to fix one of the main issue of conventional CI methods, i.e.~the slow convergence of the electronic energy with respect to the size of the one-electron basis set.
Albeit not variational, our method is able to catch a large fraction of the basis set incompleteness error at a low computational cost compared to other variants.
In particular, one eschew the computation of four-electron integrals as well as some types of three-electron integrals.
We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated full CI methods.
We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated FCI method.
%----------------------------------------------------------------
\begin{acknowledgments}
The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) for funding.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocations 2018-0510, 2018-18005 and 2019-18005.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocation 2019-18005.
\end{acknowledgments}
%----------------------------------------------------------------