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minor corrections following discussion with David

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Pierre-Francois Loos 2019-05-11 23:11:39 +02:00
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41
Manuscript/CI-F12.aux
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@ -29,15 +29,16 @@
\citation{Kato51,Kato57}
\citation{Pack66,Morgan93,Tew08,ExSpherium10,eee15}
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\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}{}}
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\@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}
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\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}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {VI}Results}{3}{section*.8}}
\@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}}
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\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}}}

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Manuscript/CI-F12.tex
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@ -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
\begin{equation}
\gamma_{ij} =
\begin{cases}
1/2, & \text{for opposite-spin electrons},
\\
1/4, & \text{for same-spin electrons}.
\end{cases}
\end{equation}
\alert{The correlation factor \eqref{eq:Ja} is not size-consistent.}
is a (linear) correlation factor.% with
%\begin{equation}
% \gamma_{ij} =
% \begin{cases}
% 1/2, & \text{for opposite-spin electrons},
% \\
% 1/4, & \text{for same-spin electrons}.
% \end{cases}
%\end{equation}
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
\begin{equation}
\label{eq:tI}
\sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
\end{equation}
which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
\alert{T2: Here include the rules to determine the coefficients $\cF{I}$.}
%\alert{This yields the following linear system of equations
%\begin{equation}
%\label{eq:tI}
% \sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
%\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}.
In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ shuffling around the dressing term.
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 $\cD{I}^{-1}$ 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.