From feeeeb719055e9579eda8831553d4d73cf8a8c85 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 11 May 2019 23:11:39 +0200 Subject: [PATCH] minor corrections following discussion with David --- Manuscript/CI-F12.aux | 41 ++++++++++++++++---------------- Manuscript/CI-F12.tex | 55 ++++++++++++++++++++++--------------------- 2 files changed, 48 insertions(+), 48 deletions(-) diff --git a/Manuscript/CI-F12.aux b/Manuscript/CI-F12.aux index 02ab06c..4959a2d 100644 --- a/Manuscript/CI-F12.aux +++ b/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 {abstract}{Abstract}{1}{section*.1}} -\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}} -\@writefile{toc}{\contentsline {section}{\numberline {II}Ans{\"a}tz}{1}{section*.4}} +\@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}\protected@file@percent } +\@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}\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}} -\@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}} +\@writefile{toc}{\contentsline {section}{\numberline {V}Computational details}{3}{section*.7}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {VI}Results}{3}{section*.8}\protected@file@percent } +\@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}\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}}} diff --git a/Manuscript/CI-F12.tex b/Manuscript/CI-F12.tex index 3cb69cb..c54dbe3 100644 --- a/Manuscript/CI-F12.tex +++ b/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 -\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.