diff --git a/Manuscript/CI-F12.aux b/Manuscript/CI-F12.aux index a49876a..8277d6c 100644 --- a/Manuscript/CI-F12.aux +++ b/Manuscript/CI-F12.aux @@ -22,34 +22,36 @@ \citation{Tenno12a,Tenno12b,Hattig12,Kong12} \citation{Tew07b} \citation{Huron73,Evangelisti83} -\citation{Heully92} -\citation{Giner13,Scemama13a,Scemama13b,Scemama14,Giner15,Caffarel16} +\citation{Heully92,Garniron18} +\citation{Giner13,Scemama13a,Scemama13b,Scemama14,Giner15,Caffarel16,Loos18b,Loos19} \citation{Shiozaki11} \citation{Kato51,Kato57} \citation{Pack66,Morgan93} \citation{Tenno04a} -\citation{Pulay82} \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}} -\newlabel{eq:D}{{1}{1}{}{equation.0.1}{}} -\newlabel{eq:WF-F12-CIPSI}{{2}{1}{}{equation.0.2}{}} -\newlabel{eq:Ja}{{4}{1}{}{equation.0.4}{}} -\newlabel{eq:DrH}{{8}{1}{}{equation.0.8}{}} -\newlabel{eq:IHF}{{9}{1}{}{equation.0.9}{}} -\newlabel{eq:tI}{{10}{1}{}{equation.0.10}{}} +\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Ansatz}{1}{section*.4}} +\newlabel{eq:D}{{1}{1}{}{equation.2.1}{}} +\newlabel{eq:WF-F12-CIPSI}{{2}{1}{}{equation.2.2}{}} +\newlabel{eq:Ja}{{4}{1}{}{equation.2.4}{}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Dressing}{1}{section*.5}} +\newlabel{eq:DrH}{{8}{1}{}{equation.3.8}{}} +\newlabel{eq:IHF}{{9}{1}{}{equation.3.9}{}} +\citation{Pulay82} \citation{SzaboBook} \citation{Kutzelnigg91,Klopper92,Persson97,Klopper02,Manby03,Werner03,Klopper04,Tenno04a,Tenno04b,May05,Manby06,Tenno07,Komornicki11,Reine12,GG16} \citation{Kutzelnigg91,Klopper92} \citation{3ERI1,3ERI2,4eRR,IntF12} \citation{Kutzelnigg91,Klopper02,Valeev04,Werner07,Hattig12} \citation{Klopper02,Valeev04} -\citation{PT2} +\citation{Garniron17b} \citation{Kutzelnigg91} \citation{Tenno04a} \citation{Persson96,Persson97,May04,Tenno04b,Tew05,May05} \citation{Tew05} -\citation{QP} +\citation{Garniron19} \citation{Kong12} \citation{Nakashima07} \citation{Puchalski09} @@ -64,6 +66,12 @@ \citation{AlmoraDiaz14} \citation{Yousaf08,Yousaf09} \citation{Giner13,Giner15,Caffarel16} +\newlabel{eq:tI}{{10}{2}{}{equation.3.10}{}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Matrix elements}{2}{section*.6}} +\newlabel{eq:RI}{{11}{2}{}{equation.4.11}{}} +\newlabel{eq:IHF-RI}{{12}{2}{}{equation.4.12}{}} +\@writefile{toc}{\contentsline {section}{\numberline {V}Conclusion}{2}{section*.7}} +\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{2}{section*.8}} \bibdata{CI-F12Notes,CI-F12} \bibcite{Kutzelnigg85}{{1}{1985}{{Kutzelnigg}}{{}}} \bibcite{Kutzelnigg91}{{2}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}} @@ -85,57 +93,58 @@ \bibcite{Huron73}{{18}{1973}{{Huron, Malrieu,\ and\ Rancurel}}{{}}} \bibcite{Evangelisti83}{{19}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}} \bibcite{Heully92}{{20}{1992}{{Heuilly\ and\ Malrieu}}{{}}} -\bibcite{Giner13}{{21}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}} -\bibcite{Scemama13a}{{22}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} -\bibcite{Scemama13b}{{23}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} -\bibcite{Scemama14}{{24}{2014}{{Scemama\ \emph {et~al.}}}{{Scemama, Applencourt, Giner,\ and\ Caffarel}}} -\newlabel{eq:RI}{{11}{2}{}{equation.0.11}{}} -\newlabel{eq:IHF-RI}{{12}{2}{}{equation.0.12}{}} -\bibcite{Giner15}{{25}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}} -\bibcite{Caffarel16}{{26}{2016}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} -\bibcite{Shiozaki11}{{27}{2011}{{Shiozaki, Knizia,\ and\ Werner}}{{}}} -\bibcite{Kato51}{{28}{1951}{{Kato}}{{}}} -\bibcite{Kato57}{{29}{1957}{{Kato}}{{}}} -\bibcite{Pack66}{{30}{1966}{{Pack\ and\ {Byers Brown}}}{{}}} -\bibcite{Morgan93}{{31}{1993}{{{Morgan III}\ and\ Kutzelnigg}}{{}}} -\bibcite{Tenno04a}{{32}{2004{}}{{Ten-no}}{{}}} -\bibcite{Pulay82}{{33}{1982}{{Pulay}}{{}}} -\bibcite{SzaboBook}{{34}{1989}{{Szabo\ and\ Ostlund}}{{}}} -\bibcite{Klopper92}{{35}{1992}{{Klopper\ and\ Rohse}}{{}}} -\bibcite{Klopper02}{{36}{2002}{{Klopper\ and\ Samson}}{{}}} -\bibcite{Manby03}{{37}{2003}{{Manby}}{{}}} -\bibcite{Werner03}{{38}{2003}{{Werner, Manby,\ and\ Knowles}}{{}}} -\bibcite{Klopper04}{{39}{2004}{{Klopper}}{{}}} -\bibcite{Manby06}{{40}{2006}{{Manby\ \emph {et~al.}}}{{Manby, Werner, Adler,\ and\ May}}} -\bibcite{Tenno07}{{41}{2007}{{Ten-no}}{{}}} -\bibcite{Komornicki11}{{42}{2011}{{Komornicki\ and\ King}}{{}}} -\bibcite{Reine12}{{43}{2012}{{Reine, Helgaker,\ and\ Lind}}{{}}} -\bibcite{GG16}{{44}{2016}{{Barca\ and\ Gill}}{{}}} -\bibcite{3ERI1}{{45}{2016}{{Barca, Loos,\ and\ Gill}}{{}}} -\bibcite{3ERI2}{{46}{tion}{{Barca, Loos,\ and\ Gill}}{{}}} -\bibcite{4eRR}{{47}{ress}{{Barca\ and\ Loos}}{{}}} -\bibcite{IntF12}{{48}{2017}{{Barca\ and\ Loos}}{{}}} -\bibcite{Valeev04}{{49}{2004}{{Valeev}}{{}}} -\bibcite{Werner07}{{50}{2007}{{Werner, Adler,\ and\ Manby}}{{}}} -\bibcite{PT2}{{51}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Loos,\ and\ Caffarel.}}} -\bibcite{QP}{{52}{2015}{{Scemama\ \emph {et~al.}}}{{Scemama, Giner, Applencourt, David,\ and\ Caffarel}}} -\bibcite{Nakashima07}{{53}{2007}{{Nakashima\ and\ Nakatsuji}}{{}}} -\bibcite{Puchalski09}{{54}{2009}{{Puchalski, Kedziera,\ and\ Pachucki}}{{}}} -\bibcite{Sharkey11}{{55}{2011}{{Sharkey\ and\ Adamowicz}}{{}}} -\bibcite{Bubin11}{{56}{2011}{{Bubin\ and\ Adamowicz}}{{}}} -\bibcite{Sharkey10}{{57}{2010}{{Sharkey, Bubin,\ and\ Adamowicz}}{{}}} -\bibcite{Klopper10}{{58}{2010}{{Klopper\ \emph {et~al.}}}{{Klopper, Bachorz, Tew,\ and\ Hattig}}} -\bibcite{Pachucki10}{{59}{2010}{{Pachucki}}{{}}} -\bibcite{Cleland12}{{60}{2012}{{Cleland\ \emph {et~al.}}}{{Cleland, Booth, Overy,\ and\ Alavi}}} -\bibcite{AlmoraDiaz14}{{61}{2014}{{Almora-Diaz}}{{}}} -\bibcite{Yousaf08}{{62}{2008}{{Yousaf\ and\ Peterson}}{{}}} -\bibcite{Yousaf09}{{63}{2009}{{Yousaf\ and\ Peterson}}{{}}} +\bibcite{Garniron18}{{21}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}} +\bibcite{Giner13}{{22}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}} +\bibcite{Scemama13a}{{23}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} +\bibcite{Scemama13b}{{24}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} +\bibcite{Scemama14}{{25}{2014}{{Scemama\ \emph {et~al.}}}{{Scemama, Applencourt, Giner,\ and\ Caffarel}}} +\bibcite{Giner15}{{26}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}} +\bibcite{Caffarel16}{{27}{2016}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} +\bibcite{Loos18b}{{28}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}} +\bibcite{Loos19}{{29}{2019}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}} +\bibcite{Shiozaki11}{{30}{2011}{{Shiozaki, Knizia,\ and\ Werner}}{{}}} +\bibcite{Kato51}{{31}{1951}{{Kato}}{{}}} +\bibcite{Kato57}{{32}{1957}{{Kato}}{{}}} +\bibcite{Pack66}{{33}{1966}{{Pack\ and\ {Byers Brown}}}{{}}} +\bibcite{Morgan93}{{34}{1993}{{{Morgan III}\ and\ Kutzelnigg}}{{}}} +\bibcite{Tenno04a}{{35}{2004{}}{{Ten-no}}{{}}} +\bibcite{Pulay82}{{36}{1982}{{Pulay}}{{}}} +\bibcite{SzaboBook}{{37}{1989}{{Szabo\ and\ Ostlund}}{{}}} +\bibcite{Klopper92}{{38}{1992}{{Klopper\ and\ Rohse}}{{}}} +\bibcite{Klopper02}{{39}{2002}{{Klopper\ and\ Samson}}{{}}} +\bibcite{Manby03}{{40}{2003}{{Manby}}{{}}} +\bibcite{Werner03}{{41}{2003}{{Werner, Manby,\ and\ Knowles}}{{}}} +\bibcite{Klopper04}{{42}{2004}{{Klopper}}{{}}} +\bibcite{Manby06}{{43}{2006}{{Manby\ \emph {et~al.}}}{{Manby, Werner, Adler,\ and\ May}}} +\bibcite{Tenno07}{{44}{2007}{{Ten-no}}{{}}} +\bibcite{Komornicki11}{{45}{2011}{{Komornicki\ and\ King}}{{}}} +\bibcite{Reine12}{{46}{2012}{{Reine, Helgaker,\ and\ Lind}}{{}}} +\bibcite{GG16}{{47}{2016}{{Barca\ and\ Gill}}{{}}} +\bibcite{3ERI1}{{48}{2016}{{Barca, Loos,\ and\ Gill}}{{}}} +\bibcite{3ERI2}{{49}{tion}{{Barca, Loos,\ and\ Gill}}{{}}} +\bibcite{4eRR}{{50}{ress}{{Barca\ and\ Loos}}{{}}} +\bibcite{IntF12}{{51}{2017}{{Barca\ and\ Loos}}{{}}} +\bibcite{Valeev04}{{52}{2004}{{Valeev}}{{}}} +\@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}{}} +\bibcite{Werner07}{{53}{2007}{{Werner, Adler,\ and\ Manby}}{{}}} +\bibcite{Garniron17b}{{54}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Loos,\ and\ Caffarel}}} +\bibcite{Garniron19}{{55}{2019}{{Garniron\ \emph {et~al.}}}{{Garniron, Gasperich, Applencourt, Benali, Fert{\'e}, Paquier, Pradines, Assaraf, Reinhardt, Toulouse, Barbaresco, Renon, David, Malrieu, V{\'e}ril, Caffarel, Loos, Giner,\ and\ Scemama}}} +\bibcite{Nakashima07}{{56}{2007}{{Nakashima\ and\ Nakatsuji}}{{}}} +\bibcite{Puchalski09}{{57}{2009}{{Puchalski, Kedziera,\ and\ Pachucki}}{{}}} +\bibcite{Sharkey11}{{58}{2011}{{Sharkey\ and\ Adamowicz}}{{}}} +\bibcite{Bubin11}{{59}{2011}{{Bubin\ and\ Adamowicz}}{{}}} +\bibcite{Sharkey10}{{60}{2010}{{Sharkey, Bubin,\ and\ Adamowicz}}{{}}} +\bibcite{Klopper10}{{61}{2010}{{Klopper\ \emph {et~al.}}}{{Klopper, Bachorz, Tew,\ and\ Hattig}}} +\bibcite{Pachucki10}{{62}{2010}{{Pachucki}}{{}}} +\bibcite{Cleland12}{{63}{2012}{{Cleland\ \emph {et~al.}}}{{Cleland, Booth, Overy,\ and\ Alavi}}} +\bibcite{AlmoraDiaz14}{{64}{2014}{{Almora-Diaz}}{{}}} +\bibcite{Yousaf08}{{65}{2008}{{Yousaf\ and\ Peterson}}{{}}} +\bibcite{Yousaf09}{{66}{2009}{{Yousaf\ and\ Peterson}}{{}}} \bibstyle{aipnum4-1} \citation{REVTEX41Control} \citation{aip41Control} -\@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}{}} -\newlabel{LastBibItem}{{63}{3}{}{table.2}{}} \@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}{}} +\newlabel{LastBibItem}{{66}{4}{}{section*.8}{}} \newlabel{LastPage}{{}{4}{}{}{}} diff --git a/Manuscript/CI-F12.bbl b/Manuscript/CI-F12.bbl index ed684e6..414a8f8 100644 --- a/Manuscript/CI-F12.bbl +++ b/Manuscript/CI-F12.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{63}% +\begin{thebibliography}{66}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -215,6 +215,18 @@ {Malrieu}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {199}},\ \bibinfo {pages} {545} (\bibinfo {year} {1992})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Garniron}\ \emph {et~al.}(2018)\citenamefont + {Garniron}, \citenamefont {Scemama}, \citenamefont {Giner}, \citenamefont + {Caffarel},\ and\ \citenamefont {Loos}}]{Garniron18}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {Garniron}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, + \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, \bibinfo + {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}}, \ and\ \bibinfo + {author} {\bibfnamefont {P.~F.}\ \bibnamefont {Loos}},\ }\href {\doibase + 10.1063/1.5044503} {\bibfield {journal} {\bibinfo {journal} {J. 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Chem. Theory Comput.}\ }\textbf {\bibinfo + {volume} {in press}} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Nakashima}\ and\ \citenamefont {Nakatsuji}(2007)}]{Nakashima07}% \BibitemOpen diff --git a/Manuscript/CI-F12.bib b/Manuscript/CI-F12.bib index c6359ce..429c95d 100644 --- a/Manuscript/CI-F12.bib +++ b/Manuscript/CI-F12.bib @@ -1,13 +1,133 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2017-07-21 21:57:53 +0200 +%% Created for Pierre-Francois Loos at 2019-03-10 21:06:16 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Garniron19, + Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama}, + Date-Added = {2019-03-10 21:06:09 +0100}, + Date-Modified = {2019-03-10 21:06:15 +0100}, + Journal = {J. Chem. Theory Comput.}, + Title = {Quantum Package 2.0: a open-source determinant-driven suite of programs}, + Volume = {in press}, + Year = {2019}} + +@article{Scemama18a, + Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos}, + Date-Added = {2019-03-10 21:04:59 +0100}, + Date-Modified = {2019-03-10 21:05:07 +0100}, + Doi = {10.1021/acs.jctc.7b01250}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1395}, + Title = {Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS}, + Volume = {14}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}} + +@article{Scemama18b, + Author = {Anthony Scemama and Anouar Benali and Denis Jacquemin and Michel Caffarel and Pierre-Fran{\c{c}}ois Loos}, + Date-Added = {2019-03-10 21:04:59 +0100}, + Date-Modified = {2019-03-10 21:05:12 +0100}, + Doi = {10.1063/1.5041327}, + Journal = {J. Chem. Phys.}, + Month = {jul}, + Number = {3}, + Pages = {034108}, + Publisher = {{AIP} Publishing}, + Title = {Excitation energies from diffusion Monte Carlo using selected configuration interaction nodes}, + Url = {https://doi.org/10.1063%2F1.5041327}, + Volume = {149}, + Year = 2018, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.5041327}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.5041327}} + +@article{Veril_2018, + Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos}, + Date-Added = {2019-03-10 21:04:59 +0100}, + Date-Modified = {2019-03-10 21:04:59 +0100}, + Doi = {10.1021/acs.jctc.8b00745}, + Journal = {J. Chem. Theory Comput.}, + Pages = {5220}, + Title = {Unphysical Discontinuities in {{GW}} Methods}, + Volume = {14}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00745}} + +@article{Garniron18, + Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos}, + Date-Added = {2019-03-10 21:04:56 +0100}, + Date-Modified = {2019-03-10 21:05:47 +0100}, + Doi = {10.1063/1.5044503}, + Journal = {J. Chem. Phys.}, + Pages = {064103}, + Title = {Selected Configuration Interaction Dressed by Perturbation}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}} + +@article{Loos18a, + Author = {P. F. Loos and P. Romaniello and J. A. Berger}, + Date-Added = {2019-03-10 21:04:56 +0100}, + Date-Modified = {2019-03-10 21:05:22 +0100}, + Doi = {10.1021/acs.jctc.8b00260}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3071}, + Title = {Green Functions and Self-Consistency: Insights From the Spherium Model}, + Volume = {14}, + Year = {2018}, + Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.8b00260}} + +@article{Loos18b, + Author = {Pierre-Fran{\c{c}}ois Loos and Anthony Scemama and Aymeric Blondel and Yann Garniron and Michel Caffarel and Denis Jacquemin}, + Date-Added = {2019-03-10 21:04:56 +0100}, + Date-Modified = {2019-03-10 21:05:31 +0100}, + Doi = {10.1021/acs.jctc.8b00406}, + Journal = {J. Chem. Theory Comput.}, + Month = {jul}, + Number = {8}, + Pages = {4360--4379}, + Publisher = {American Chemical Society ({ACS})}, + Title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks}, + Url = {https://doi.org/10.1021%2Facs.jctc.8b00406}, + Volume = {14}, + Year = 2018, + Bdsk-Url-1 = {https://doi.org/10.1021%2Facs.jctc.8b00406}, + Bdsk-Url-2 = {https://doi.org/10.1021/acs.jctc.8b00406}} + +@article{Loos19, + Author = {P. F. Loos and M. Boggio-Pasqua and A. Scemama and M. Caffarel and D. Jacquemin}, + Date-Added = {2019-03-10 21:04:56 +0100}, + Date-Modified = {2019-03-10 21:05:26 +0100}, + Doi = {10.1021/acs.jctc.8b01205}, + Journal = {J. Chem. Theory Comput.}, + Pages = {in press}, + Title = {Reference energies for double excitations}, + Volume = {15}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}} + +@article{Garniron17b, + Author = {Yann Garniron and Anthony Scemama and Pierre-Fran{\c{c}}ois Loos and Michel Caffarel}, + Date-Added = {2019-03-10 21:04:41 +0100}, + Date-Modified = {2019-03-10 21:05:44 +0100}, + Doi = {10.1063/1.4992127}, + Journal = {J. Chem. Phys.}, + Month = {jul}, + Number = {3}, + Pages = {034101}, + Publisher = {{AIP} Publishing}, + Title = {Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory}, + Url = {https://doi.org/10.1063%2F1.4992127}, + Volume = {147}, + Year = 2017, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4992127}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.4992127}} + @article{PT2, Author = {Y. Garniron and A. Scemama and P. F. Loos and M. 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LaTeX Info: Redefining \ldots on input line 472. LaTeX Info: Redefining \dots on input line 475. LaTeX Info: Redefining \cdots on input line 596. -\Mathstrutbox@=\box41 -\strutbox@=\box42 +\Mathstrutbox@=\box43 +\strutbox@=\box44 \big@size=\dimen117 LaTeX Font Info: Redeclaring font encoding OML on input line 712. LaTeX Font Info: Redeclaring font encoding OMS on input line 713. @@ -375,10 +369,10 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 713. LaTeX Info: Redefining \[ on input line 2817. LaTeX Info: Redefining \] on input line 2818. ) -(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +(/usr/local/texlive/2018/texmf-dist/tex/latex/amsfonts/amssymb.sty Package: amssymb 2013/01/14 v3.01 AMS font symbols -(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +(/usr/local/texlive/2018/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support \symAMSa=\mathgroup7 \symAMSb=\mathgroup8 @@ -387,42 +381,27 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' LaTeX Info: Redefining \frak on input line 111. 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-\l__iow_current_indentation_int=\count131 -L3 Module: l3skip 2016/08/19 v6692 L3 Dimensions and skips +\g__intarray_font_int=\count126 +\g__prg_map_int=\count127 +\c_log_iow=\count128 +\l_iow_line_count_int=\count129 +\l__iow_line_target_int=\count130 +\l__iow_one_indent_int=\count131 +\l__iow_indent_int=\count132 \c_zero_dim=\dimen124 \c_max_dim=\dimen125 \l_tmpa_dim=\dimen126 @@ -441,56 +420,110 @@ L3 Module: l3skip 2016/08/19 v6692 L3 Dimensions and skips \l_tmpb_muskip=\muskip15 \g_tmpa_muskip=\muskip16 \g_tmpb_muskip=\muskip17 -L3 Module: l3keys 2016/09/21 v6711 L3 Key-value interfaces -\l_keys_choice_int=\count132 -L3 Module: l3fp 2016/11/21 v6761 L3 Floating points -\c__fp_leading_shift_int=\count133 -\c__fp_middle_shift_int=\count134 -\c__fp_trailing_shift_int=\count135 -\c__fp_big_leading_shift_int=\count136 -\c__fp_big_middle_shift_int=\count137 -\c__fp_big_trailing_shift_int=\count138 -\c__fp_Bigg_leading_shift_int=\count139 -\c__fp_Bigg_middle_shift_int=\count140 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\l__coffin_left_corner_dim=\dimen149 \l__coffin_right_corner_dim=\dimen150 @@ -498,19 +531,17 @@ L3 Module: l3candidates 2016/08/19 v6688 L3 Experimental additions to l3kernel \l__coffin_top_corner_dim=\dimen152 \l__coffin_scaled_total_height_dim=\dimen153 \l__coffin_scaled_width_dim=\dimen154 -L3 Module: l3luatex 2016/08/18 v6679 L3 Experimental LuaTeX-specific functions ) -(/usr/share/texlive/texmf-dist/tex/latex/l3kernel/l3pdfmode.def -File: l3pdfmode.def 2016/08/18 v6679 L3 Experimental driver: PDF mode -\l__driver_color_stack_int=\count142 -\l__driver_tmp_box=\box58 +(/usr/local/texlive/2018/texmf-dist/tex/latex/l3kernel/l3pdfmode.def +File: l3pdfmode.def 2017/03/18 v L3 Experimental driver: PDF mode +\l__driver_color_stack_int=\count270 )) -Package: xparse 2016/11/21 v6760 L3 Experimental document command parser -\l__xparse_current_arg_int=\count143 -\l__xparse_m_args_int=\count144 -\l__xparse_mandatory_args_int=\count145 -\l__xparse_processor_int=\count146 -\l__xparse_v_nesting_int=\count147 +Package: xparse 2018/02/21 L3 Experimental document command parser +\l__xparse_current_arg_int=\count271 +\g__xparse_grabber_int=\count272 +\l__xparse_m_args_int=\count273 +\l__xparse_mandatory_args_int=\count274 +\l__xparse_v_nesting_int=\count275 ) ................................................. . LaTeX info: "xparse/define-command" @@ -548,6 +579,31 @@ Package: xparse 2016/11/21 v6760 L3 Experimental document command parser . Defining command \@quantity with sig. ' t\big t\Big t\bigg t\Bigg g o d() . d|| ' on line 62. ................................................. + +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\@quantity' should be a single +* character. +************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -680,12 +736,60 @@ Package: xparse 2016/11/21 v6760 L3 Experimental document command parser . Defining command \@braces with sig. ' s t\big t\Big t\bigg t\Bigg m m m ' on . line 156. ................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\@braces' should be a single +* character. +************************************************* ................................................. . LaTeX info: "xparse/define-command" . . Defining command \fbraces with sig. ' s t\big t\Big t\bigg t\Bigg m m m m ' . on line 174. ................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\fbraces' should be a single +* character. +************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -1217,6 +1321,12 @@ Package: xparse 2016/11/21 v6760 L3 Experimental document command parser . . Defining command \bra with sig. ' s m t\ket s g ' on line 526. ................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\ket ' for the command '\bra' should be a single +* character. +************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -1389,91 +1499,29 @@ Package: xparse 2016/11/21 v6760 L3 Experimental document command parser . . Defining command \homework with sig. 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LaTeX Info: Redefining \addtolength on input line 81. -\calc@Ccount=\count195 +\calc@Ccount=\count278 \calc@Cskip=\skip63 ) -(/usr/share/texlive/texmf-dist/tex/latex/chemgreek/chemgreek.sty +(/usr/local/texlive/2018/texmf-dist/tex/latex/chemgreek/chemgreek.sty Package: chemgreek 2016/12/20 v1.1 interfaceforuprightgreeklettersforuseinchemi stry (CN) -\l__chemgreek_tmpa_int=\count196 -\g__chemgreek_tmpa_int=\count197 +\l__chemgreek_tmpa_int=\count279 +\g__chemgreek_tmpa_int=\count280 ................................................. . LaTeX info: "xparse/define-command" . @@ -1525,7 +1573,7 @@ stry (CN) . Defining command \selectchemgreekmapping with sig. 'm' on line 491. ................................................. ) -\l__mhchem_ce_distanceFromLastComma_int=\count198 +\l__mhchem_ce_distanceFromLastComma_int=\count281 \__mhchem_arrow_extPgfLength_skip=\skip64 \mhchem@minispace@tmp=\skip65 \mhchem@minibackspace@tmp=\skip66 @@ -1536,7 +1584,7 @@ stry (CN) \mhchem@smallbondwidth=\skip71 \mhchem@bondtmp@minussidebearingleft=\skip72 \mhchem@bondtmp@minussidebearingright=\skip73 -\l__mhchem_option_version_int=\count199 +\l__mhchem_option_version_int=\count282 \mhchem@option@minusmathsidebearingleft=\dimen157 \mhchem@option@minusmathsidebearingright=\dimen158 \mhchem@option@minustextsidebearingleft=\dimen159 @@ -1550,27 +1598,27 @@ Package mhchem Warning: You did not specify a 'version' option for the mhchem (mhchem) different output with the same input in future versions . -) (/usr/share/texlive/texmf-dist/tex/latex/psnfss/mathpazo.sty +) (/usr/local/texlive/2018/texmf-dist/tex/latex/psnfss/mathpazo.sty Package: mathpazo 2005/04/12 PSNFSS-v9.2a Palatino w/ Pazo Math (D.Puga, WaS) \symupright=\mathgroup9 ) -(/usr/share/texlive/texmf-dist/tex/latex/libertine/libertine.sty -Package: libertine 2014/11/25 (Bob Tennent) Supports Libertine and Biolinum fon +(/usr/local/texlive/2018/texmf-dist/tex/latex/libertine/libertine.sty +Package: libertine 2018/01/03 (Bob Tennent) Supports Libertine and Biolinum fon ts for all LaTeX engines. -(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +(/usr/local/texlive/2018/texmf-dist/tex/latex/xkeyval/xkeyval.sty Package: xkeyval 2014/12/03 v2.7a package option processing (HA) -(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex -(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex +(/usr/local/texlive/2018/texmf-dist/tex/generic/xkeyval/xkeyval.tex +(/usr/local/texlive/2018/texmf-dist/tex/generic/xkeyval/xkvutils.tex \XKV@toks=\toks25 \XKV@tempa@toks=\toks26 ) -\XKV@depth=\count266 +\XKV@depth=\count283 File: xkeyval.tex 2014/12/03 v2.7a key=value parser (HA) )) -(/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty -Package: textcomp 2017/02/22 v2.0g Standard LaTeX package +(/usr/local/texlive/2018/texmf-dist/tex/latex/base/textcomp.sty +Package: textcomp 2017/04/05 v2.0i Standard LaTeX package Package textcomp Info: Sub-encoding information: (textcomp) 5 = only ISO-Adobe without \textcurrency (textcomp) 4 = 5 + \texteuro @@ -1584,9 +1632,84 @@ Package textcomp Info: Sub-encoding information: (textcomp) See the documentation for details. 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PDF statistics: - 327 PDF objects out of 1000 (max. 8388607) - 301 compressed objects within 4 object streams - 98 named destinations out of 1000 (max. 500000) - 45585 words of extra memory for PDF output out of 51595 (max. 10000000) + 360 PDF objects out of 1000 (max. 8388607) + 334 compressed objects within 4 object streams + 107 named destinations out of 1000 (max. 500000) + 48193 words of extra memory for PDF output out of 51595 (max. 10000000) diff --git a/Manuscript/CI-F12.out b/Manuscript/CI-F12.out index cc385f0..39c0e68 100644 --- a/Manuscript/CI-F12.out +++ b/Manuscript/CI-F12.out @@ -1,2 +1,8 @@ \BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2 \BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 +\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3 +\BOOKMARK [1][-]{section*.4}{Ansatz}{section*.2}% 4 +\BOOKMARK [1][-]{section*.5}{Dressing}{section*.2}% 5 +\BOOKMARK [1][-]{section*.6}{Matrix elements}{section*.2}% 6 +\BOOKMARK [1][-]{section*.7}{Conclusion}{section*.2}% 7 +\BOOKMARK [1][-]{section*.8}{Acknowledgments}{section*.2}% 8 diff --git a/Manuscript/CI-F12.pdf b/Manuscript/CI-F12.pdf index b963f7b..b4eddcd 100644 Binary files a/Manuscript/CI-F12.pdf and b/Manuscript/CI-F12.pdf differ diff --git a/Manuscript/CI-F12.tex b/Manuscript/CI-F12.tex index f4b0dc9..2977948 100644 --- a/Manuscript/CI-F12.tex +++ b/Manuscript/CI-F12.tex @@ -74,28 +74,28 @@ The performance of the newly-designed explicitly-correlated dressing CI method i \maketitle %---------------------------------------------------------------- -\textit{Introduction.---} +\section{Introduction} %---------------------------------------------------------------- One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set. -This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function \cite{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}. +This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94} This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis). -This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$ \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}. -The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets \cite{Tenno12a, Tenno12b, Hattig12, Kong12}. -For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis \cite{Tew07b}. +This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} +The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno12a, Tenno12b, Hattig12, Kong12} +For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew07b} -In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix \cite{Huron73, Evangelisti83}. -This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios \cite{Heully92}. +In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix. \cite{Huron73, Evangelisti83} +This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios. \cite{Heully92, Garniron18} Compared to other explicitly-correlated methods, this dressing strategy has the advantage of introducing the explicit correlation at a low computational cost. -The present explicitly-correlated dressing CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16}. +The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods. \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16, Loos18b, Loos19} However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix. %Here, we focus on systems well described by a single (reference) determinant $\kO$ assumed to be a Hartree-Fock (HF) determinant. %The multireference version of the present method will be reported in a separate study. Atomic units are used throughout. %---------------------------------------------------------------- -\textit{Ansatz.---} +\section{Ansatz} %---------------------------------------------------------------- -Inspired by a number of previous research \cite{Shiozaki11}, our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part +Inspired by a number of previous research, \cite{Shiozaki11} our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part \begin{equation} \label{eq:D} \kD = \sum_{I} c_I \kI @@ -131,7 +131,7 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various \end{equation} %---------------------------------------------------------------- -\textit{Dressing.---} +\section{Dressing} %---------------------------------------------------------------- Our primary goal is to introduce the explicit correlation between electrons at low computational cost. Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$, @@ -159,7 +159,7 @@ It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = At this stage, two key comments are in order. First, as one may have realized, the coefficients $t_I$ are unknown. -\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz) \cite{Tenno04a}.} +\alert{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} @@ -171,7 +171,7 @@ which can be easily solved using standard linear algebra packages.} Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $c_I$, 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 $t_I$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}]. In practice, we initially start with a CI vector obtained by the diagonalization of the undressed Hamiltonian, and convergence is usually reached within few cycles. -For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}. +For pathological cases, a DIIS-like procedure may be employed. \cite{Pulay82} %%% FIG 1 %%% %\begin{figure} @@ -184,21 +184,21 @@ For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}. %%% %%% %---------------------------------------------------------------- -\textit{Matrix elements.---} +\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 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 $f_{12}$. -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}. +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}. -We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals \cite{3ERI1, 3ERI2, 4eRR, IntF12}. +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} +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 explore here 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). -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}). +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). +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 \begin{equation} @@ -217,7 +217,7 @@ Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eq \end{split} \end{equation} 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{PT2}. +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} %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 $\kJ$. @@ -228,7 +228,7 @@ Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqre %iii) the pure singles $\ket*{_{i}^{\alpha}}$. 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}. +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} @@ -269,12 +269,12 @@ In all the calculations presented below, we consider the following Slater-type c \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 fo 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 ). \end{equation} The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study. -All the calculations have been performed with Quantum Package \cite{QP}. +All the calculations have been performed with Quantum Package. \cite{Garniron19} %%% TABLE 1 %%% \begin{table} @@ -283,68 +283,68 @@ All the calculations have been performed with Quantum Package \cite{QP}. 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.} \begin{ruledtabular} -\begin{tabular}{lcccd} - Atom & $N$ & FCI-F12 & CIPSI & \text{FCI} \\ +\begin{tabular}{lcdd} + Atom & $N$ & \mcc{FCI-F12} & \mcc{FCI} \\ \hline - \ce{He} & D & & & -2.887\,595 \footnotemark[1] \\ - (cc-pV$N$Z) & T & & & -2.900\,232 \footnotemark[1] \\ - & Q & & & -2.902\,411 \footnotemark[1] \\ - & 5 & & & -2.903\,152 \footnotemark[1] \\ - & 6 & & & -2.903\,432 \footnotemark[1] \\ - & $\infty$ & & & -2.903\,724 \footnotemark[2] \\ + \ce{He} & D & & -2.887\,595 \footnotemark[1] \\ + (cc-pV$N$Z) & T & & -2.900\,232 \footnotemark[1] \\ + & Q & & -2.902\,411 \footnotemark[1] \\ + & 5 & & -2.903\,152 \footnotemark[1] \\ + & 6 & & -2.903\,432 \footnotemark[1] \\ + & $\infty$ & & -2.903\,724 \footnotemark[2] \\ \hline - \ce{Li} & D & & & -7.466\,025 (FCI) \\ - (cc-pCV$N$Z) & T & & & -7.474\,251 (FCI) \\ - & Q & & & -7.476\,373 (FCI) \\ - & $\infty$ & & & -7.478\,060 \footnotemark[3] \\ + \ce{Li} & D & & -7.466\,025 (FCI) \\ + (cc-pCV$N$Z) & T & & -7.474\,251 (FCI) \\ + & Q & & -7.476\,373 (FCI) \\ + & $\infty$ & & -7.478\,060 \footnotemark[3] \\ \hline - \ce{Be} & D & & & -14.651\,833 (FCI) \\ - (cc-pCV$N$Z) & T & & & -14.662\,368 (FCI) \\ - & Q & & & -14.665\,566 (CIPSI) \\ - & $\infty$ & & & -14.667\,356 \footnotemark[4] \\ - & $\infty$ & & & -14.667\,39 (TOTO) \\ + \ce{Be} & D & & -14.651\,833 (FCI) \\ + (cc-pCV$N$Z) & T & & -14.662\,368 (FCI) \\ + & Q & & -14.665\,566 (CIPSI) \\ + & $\infty$ & & -14.667\,356 \footnotemark[4] \\ + & $\infty$ & & -14.667\,39 (TOTO) \\ \hline - \ce{B} & D & & & -24.619\,101 (FCI) \\ - (cc-pwCV$N$Z) & T & & & -24.643\,222 (CIPSI) \\ - & Q & & & -24.650\,331 (CIPSI) \\ - & 5 & & & -24.652\,309 (CIPSI) \\ - & $\infty$ & & & -24.653\,866 \footnotemark[5] \\ - & $\infty$ & & & -24.653\,90 (TOTO) \\ + \ce{B} & D & & -24.619\,101 (FCI) \\ + (cc-pwCV$N$Z) & T & & -24.643\,222 (CIPSI) \\ + & Q & & -24.650\,331 (CIPSI) \\ + & 5 & & -24.652\,309 (CIPSI) \\ + & $\infty$ & & -24.653\,866 \footnotemark[5] \\ + & $\infty$ & & -24.653\,90 (TOTO) \\ \hline - \ce{C} & D & & & -37.792\,469 (FCI) \\ - (cc-pwCV$N$Z) & T & & & -37.829\,847 (CIPSI) \\ - & Q & & & -37.839\,816 (CIPSI) \\ - & 5 & & & -37.842\,731 (CIPSI) \\ - & $\infty$ & & & -37.840\,129 6 \\ - & $\infty$ & & & -37.845\,0 (TOTO) \\ + \ce{C} & D & & -37.792\,469 (FCI) \\ + (cc-pwCV$N$Z) & T & & -37.829\,847 (CIPSI) \\ + & Q & & -37.839\,816 (CIPSI) \\ + & 5 & & -37.842\,731 (CIPSI) \\ + & $\infty$ & & -37.840\,129 6 \\ + & $\infty$ & & -37.845\,0 (TOTO) \\ \hline - \ce{N} & D & & & -54.517\,650 (FCI) \\ - (cc-pwCV$N$Z) & T & & & \\ - & Q & & & \\ - & 5 & & & -54.585\,926 (CIPSI) \\ - & $\infty$ & & & -54.588\,917 \footnotemark[7] \\ - & $\infty$ & & & -54.589\,3 (TOTO) \\ + \ce{N} & D & & -54.517\,650 (FCI) \\ + (cc-pwCV$N$Z) & T & & \\ + & Q & & \\ + & 5 & & -54.585\,926 (CIPSI) \\ + & $\infty$ & & -54.588\,917 \footnotemark[7] \\ + & $\infty$ & & -54.589\,3 (TOTO) \\ \hline - \ce{O} & D & & & \\ - (cc-pwCV$N$Z) & T & & & \\ - & Q & & & -75.054\,737 (CIPSI) \\ - & 5 & & & -75.062\,002 (CIPSI) \\ - & $\infty$ & & & -75.066\,892 \footnotemark[7] \\ - & $\infty$ & & & -75.067\,4 (TOTO) \\ + \ce{O} & D & & \\ + (cc-pwCV$N$Z) & T & & \\ + & Q & & -75.054\,737 (CIPSI) \\ + & 5 & & -75.062\,002 (CIPSI) \\ + & $\infty$ & & -75.066\,892 \footnotemark[7] \\ + & $\infty$ & & -75.067\,4 (TOTO) \\ \hline - \ce{F} & D & & & -99.566\,902 (CIPSI) \\ - (cc-pwCV$N$Z) & T & & & -99.682\,616 (CIPSI) \\ - & Q & & & -99.715\,563 (CIPSI) \\ - & 5 & & & -99.726\,249 (CIPSI) \\ - & $\infty$ & & & -99.733\,424 \footnotemark[7] \\ - & $\infty$ & & & -99.734\,1 (TOTO) \\ + \ce{F} & D & & -99.566\,902 (CIPSI) \\ + (cc-pwCV$N$Z) & T & & -99.682\,616 (CIPSI) \\ + & Q & & -99.715\,563 (CIPSI) \\ + & 5 & & -99.726\,249 (CIPSI) \\ + & $\infty$ & & -99.733\,424 \footnotemark[7] \\ + & $\infty$ & & -99.734\,1 (TOTO) \\ \hline - \ce{Ne} & D & & & \\ - (cc-pwCV$N$Z) & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & -128.937\,274 \footnotemark[7] \\ - & $\infty$ & & & -128.938\,3 (TOTO) \\ + \ce{Ne} & D & & \\ + (cc-pwCV$N$Z) & T & & \\ + & Q & & \\ + & 5 & & \\ + & $\infty$ & & -128.937\,274 \footnotemark[7] \\ + & $\infty$ & & -128.938\,3 (TOTO) \\ \end{tabular} \end{ruledtabular} \footnotetext[1]{Reference \onlinecite{Kong12}} @@ -391,12 +391,12 @@ Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & %%% In Table \ref{tab:atoms}, we report the total atomic energy of the neutral atoms from $Z = 2$ to $10$ for various Dunning's basis sets. -In all calculations, the associated OPTRI basis is used as CABS \cite{Yousaf08, Yousaf09}. +In all calculations, the associated OPTRI basis is used as CABS. \cite{Yousaf08, Yousaf09} -In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2} and \ce{H2O} molecules at experimental geometry \cite{Giner13, Giner15, Caffarel16}. +In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2} and \ce{H2O} molecules at experimental geometry. \cite{Giner13, Giner15, Caffarel16} %---------------------------------------------------------------- -\textit{Conclusion.---} +\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. @@ -404,8 +404,10 @@ In particular, one eschew the computation of four-electron integrals as well as We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated full CI methods. %---------------------------------------------------------------- -\textit{Acknowledgments.---} -This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015). +\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. +\end{acknowledgments} %---------------------------------------------------------------- \bibliography{CI-F12}