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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}}
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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}{}}
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\@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}{}}
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\bibdata{CI-F12Notes,CI-F12,CI-F12-control}
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\bibcite{Kutzelnigg85}{{1}{1985}{{Kutzelnigg}}{{}}}
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\bibcite{Kutzelnigg91}{{2}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
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\bibcite{Persson97}{{8}{1997}{{Persson\ and\ Taylor}}{{}}}
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\bibcite{May04}{{9}{2004}{{May\ and\ Manby}}{{}}}
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\bibcite{Tew05}{{11}{2005}{{Tew\ and\ Klopper}}{{}}}
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\bibcite{May05}{{12}{2005}{{May\ \emph {et~al.}}}{{May, Valeev, Polly,\ and\ Manby}}}
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\bibcite{Tenno12a}{{13}{2012}{{Ten-no\ and\ Noga}}{{}}}
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\bibcite{Tenno12b}{{14}{2012}{{Ten-no}}{{}}}
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\bibcite{Hattig12}{{15}{2012}{{Hattig\ \emph {et~al.}}}{{Hattig, Klopper, Kohn,\ and\ Tew}}}
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\bibcite{Kong12}{{16}{2012}{{Kong, Bischo,\ and\ Valeev}}{{}}}
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\bibcite{Tew07b}{{17}{2007}{{Tew\ \emph {et~al.}}}{{Tew, Klopper, Neiss,\ and\ Hattig}}}
|
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\bibcite{Malrieu85}{{18}{1985}{{Malrieu, Durand,\ and\ Daudey}}{{}}}
|
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\bibcite{Heully92}{{19}{1992}{{Heuilly\ and\ Malrieu}}{{}}}
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\bibcite{Garniron18}{{20}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
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\bibcite{Knowles84}{{21}{1984}{{Knowles\ and\ Handy}}{{}}}
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\bibcite{Knowles89}{{22}{1989}{{Knowles\ and\ Handy}}{{}}}
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\bibcite{Huron73}{{23}{1973}{{Huron, Malrieu,\ and\ Rancurel}}{{}}}
|
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\bibcite{Giner13}{{24}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
|
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\bibcite{Giner15}{{25}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
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\bibcite{Caffarel16}{{26}{2016}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}}
|
||||
\bibcite{Scemama18a}{{27}{2018}{{Scemama\ \emph {et~al.}}}{{Scemama, Garniron, Caffarel,\ and\ Loos}}}
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\bibcite{Scemama18b}{{28}{ress}{{Scemama\ \emph {et~al.}}}{{Scemama, Benali, Jacquemin, Caffarel,\ and\ Loos}}}
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\bibcite{Loos18b}{{29}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
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\bibcite{Loos19}{{30}{2019}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
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\bibcite{Shiozaki11}{{31}{2011}{{Shiozaki, Knizia,\ and\ Werner}}{{}}}
|
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\bibcite{Kato51}{{32}{1951}{{Kato}}{{}}}
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\bibcite{Kato57}{{33}{1957}{{Kato}}{{}}}
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\bibcite{Pack66}{{34}{1966}{{Pack\ and\ {Byers Brown}}}{{}}}
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\bibcite{Morgan93}{{35}{1993}{{{Morgan III}\ and\ Kutzelnigg}}{{}}}
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\bibcite{Tew08}{{36}{2008}{{Tew}}{{}}}
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\bibcite{ExSpherium10}{{37}{2010}{{Loos\ and\ Gill}}{{}}}
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\bibcite{eee15}{{38}{2015}{{Loos, Bloomfield,\ and\ Gill}}{{}}}
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||||
\bibcite{Tenno04a}{{39}{2004{}}{{Ten-no}}{{}}}
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||||
\bibcite{SzaboBook}{{40}{1989}{{Szabo\ and\ Ostlund}}{{}}}
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\bibcite{Klopper92}{{41}{1992}{{Klopper\ and\ Rohse}}{{}}}
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\bibcite{Klopper02}{{42}{2002}{{Klopper\ and\ Samson}}{{}}}
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\bibcite{Manby03}{{43}{2003}{{Manby}}{{}}}
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\bibcite{Werner03}{{44}{2003}{{Werner, Manby,\ and\ Knowles}}{{}}}
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\bibcite{Klopper04}{{45}{2004}{{Klopper}}{{}}}
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\bibcite{Manby06}{{46}{2006}{{Manby\ \emph {et~al.}}}{{Manby, Werner, Adler,\ and\ May}}}
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\bibcite{Tenno07}{{47}{2007}{{Ten-no}}{{}}}
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\@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}}
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\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}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{4}{section*.9}}
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\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{4}{section*.10}}
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\bibcite{Komornicki11}{{48}{2011}{{Komornicki\ and\ King}}{{}}}
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\bibcite{Reine12}{{49}{2012}{{Reine, Helgaker,\ and\ Lind}}{{}}}
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\bibcite{GG16}{{50}{2016}{{Barca\ and\ Gill}}{{}}}
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\bibcite{3ERI1}{{51}{2016}{{Barca, Loos,\ and\ Gill}}{{}}}
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\bibcite{3ERI2}{{52}{tion}{{Barca, Loos,\ and\ Gill}}{{}}}
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\bibcite{4eRR}{{53}{ress}{{Barca\ and\ Loos}}{{}}}
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\bibcite{IntF12}{{54}{2017}{{Barca\ and\ Loos}}{{}}}
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\bibcite{Valeev04}{{55}{2004}{{Valeev}}{{}}}
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\bibcite{Werner07}{{56}{2007}{{Werner, Adler,\ and\ Manby}}{{}}}
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||||
\bibcite{Garniron17b}{{57}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Loos,\ and\ Caffarel}}}
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\bibcite{Nakashima07}{{58}{2007}{{Nakashima\ and\ Nakatsuji}}{{}}}
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\bibcite{Puchalski09}{{59}{2009}{{Puchalski, Kedziera,\ and\ Pachucki}}{{}}}
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\bibcite{Sharkey11}{{60}{2011}{{Sharkey\ and\ Adamowicz}}{{}}}
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\bibcite{Bubin11}{{61}{2011}{{Bubin\ and\ Adamowicz}}{{}}}
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\bibcite{Sharkey10}{{62}{2010}{{Sharkey, Bubin,\ and\ Adamowicz}}{{}}}
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||||
\bibcite{Klopper10}{{63}{2010}{{Klopper\ \emph {et~al.}}}{{Klopper, Bachorz, Tew,\ and\ Hattig}}}
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\bibcite{Pachucki10}{{64}{2010}{{Pachucki}}{{}}}
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\bibcite{Cleland12}{{65}{2012}{{Cleland\ \emph {et~al.}}}{{Cleland, Booth, Overy,\ and\ Alavi}}}
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||||
\bibcite{AlmoraDiaz14}{{66}{2014}{{Almora-Diaz}}{{}}}
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\bibcite{Yousaf08}{{67}{2008}{{Yousaf\ and\ Peterson}}{{}}}
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\bibcite{Yousaf09}{{68}{2009}{{Yousaf\ and\ Peterson}}{{}}}
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@ -1,6 +1,5 @@
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\documentclass[aip,jcp,reprint,showkeys]{revtex4-1}
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\usepackage{mathpazo,libertine}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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@ -56,7 +55,7 @@
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\MPI}{Max-Planck-Institut f\"ur Festk\"orperforschung, Heisenbergstra{\ss}e 1, 70569 Stuttgart, Germany}
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\begin{document}
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\title{Dressing the configuration interaction matrix with explicit correlation}
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@ -65,6 +64,8 @@
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\affiliation{\LCPQ}
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\author{Michel Caffarel}
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\affiliation{\LCPQ}
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\author{David P. Tew}
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\affiliation{\MPI}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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@ -133,19 +134,18 @@ The projector
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ensures the orthogonality between $\kD$ and $\kF$ (where $\hI$ is the identity operator), and
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\begin{equation}
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\label{eq:Ja}
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f = \sum_{i < j} \gamma_{ij} f_{ij}
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f = \sum_{i < j} f_{ij}
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% f = \sum_{i < j} \gamma_{ij} f_{ij}
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\end{equation}
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is a correlation factor with
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\begin{equation}
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\gamma_{ij} =
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\begin{cases}
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1/2, & \text{for opposite-spin electrons},
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\\
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1/4, & \text{for same-spin electrons}.
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\end{cases}
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\end{equation}
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\alert{The correlation factor \eqref{eq:Ja} is not size-consistent.}
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is a (linear) correlation factor.% with
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%\begin{equation}
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% \gamma_{ij} =
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% \begin{cases}
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% 1/2, & \text{for opposite-spin electrons},
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% \\
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% 1/4, & \text{for same-spin electrons}.
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% \end{cases}
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%\end{equation}
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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
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\begin{equation}
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f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
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@ -157,7 +157,7 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
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Our primary goal is to introduce the explicit correlation between electrons at relatively low computational cost.
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Therefore, assuming that $\hH \ket{\Psi} = E\,\Psi$, one can write, by projection over $\bra{I}$,
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\begin{equation}
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\cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0.
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\cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0,
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\end{equation}
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where $H_{IJ} = \mel{I}{\hH}{J}$.
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Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
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@ -176,19 +176,21 @@ with
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\mel{I}{\hH}{F} = \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
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\end{equation}
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and $f_{IJ} = \mel{I}{f}{J}$.
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We refer to this strategy as diagonal dressing as only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}.
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Because only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}, we refer to this strategy as diagonal dressing.
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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.
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Moreover, because the CI-F12 energy is obtained via projection, the present method is not variational.
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At this stage, two key comments are in order.
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First, as one may have realized, the coefficients $\cF{I}$ are unknown.
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\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}}
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However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}
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\alert{This yields the following linear system of equations
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\begin{equation}
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\label{eq:tI}
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\sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
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\end{equation}
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which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
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\alert{T2: Here include the rules to determine the coefficients $\cF{I}$.}
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%\alert{This yields the following linear system of equations
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%\begin{equation}
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%\label{eq:tI}
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||||
% \sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
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%\end{equation}
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%which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
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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 $\oH$.
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At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
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@ -196,8 +198,8 @@ In practice, we initially start with a CI vector obtained by the diagonalization
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We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme.
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Note that the present formalism is state-specific and only focus on the ground state.
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Multi-state can potentially developed following our work in Ref.~\onlinecite{Garniron18}.
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In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ shuffling around the dressing term.
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A multi-state strategy can be applied following our work in Ref.~\onlinecite{Garniron18}.
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In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ by shuffling around the dressing term.
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Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\cD{I}$, we have
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\begin{equation}
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||||
\label{eq:DrH}
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@ -212,7 +214,6 @@ Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\c
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H_{IJ}, & \text{otherwise}.
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||||
\end{cases}
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||||
\end{equation}
|
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It is important to mention that, because the CI-F12 energy is obtained via projection, the present method is not variational.
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|
||||
%%% FIG 1 %%%
|
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\begin{figure}
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@ -228,7 +229,7 @@ It is important to mention that, because the CI-F12 energy is obtained via proje
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||||
\section{Matrix elements}
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||||
%----------------------------------------------------------------
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||||
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}
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||||
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 $f_{12}$.
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||||
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.
|
||||
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