commit 3c3bd8b4236ad03bc5876829a1a133151c3b18c3 Author: Pierre-Francois Loos Date: Wed Feb 27 11:32:52 2019 +0100 First commit diff --git a/Manuscript/CI-F12.aux b/Manuscript/CI-F12.aux new file mode 100644 index 0000000..a49876a --- /dev/null +++ b/Manuscript/CI-F12.aux @@ -0,0 +1,141 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\citation{Kutzelnigg85} 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Caffarel}, + Date-Added = {2017-03-19 20:02:16 +0000}, + Date-Modified = {2017-03-19 20:02:24 +0000}, + Doi = {10.5281/zenodo.30624}, + Month = sep, + Note = {doi:10.5281/zenodo.30624}, + Title = {Quantum Package v0.6}, + Url = {http://dx.doi.org/10.5281/zenodo.30624}, + Year = 2015, + Bdsk-Url-1 = {http://dx.doi.org/10.5281/zenodo.30624}} + +@article{Werner03, + Author = {H. J. Werner and F. R. Manby and P. J. Knowles}, + Date-Added = {2017-03-02 21:38:20 +0000}, + Date-Modified = {2017-03-02 21:38:20 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {8149}, + Volume = {118}, + Year = {2003}} + +@article{Manby06, + Author = {F. R. Manby and H. J. Werner and T. B. Adler and A. J. May}, + Date-Added = {2017-03-02 21:38:01 +0000}, + Date-Modified = {2017-03-02 21:38:01 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {094103}, + Volume = {124}, + Year = {2006}} + +@article{Klopper04, + Author = {W. 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Phys.}, + Month = {dec}, + Number = {24}, + Pages = {244110}, + Publisher = {{AIP} Publishing}, + Title = {{Accurate nonrelativistic ground-state energies of 3d transition metal atoms}}, + Url = {http://dx.doi.org/10.1063/1.4903985}, + Volume = {141}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4903985}} + +@article{Scemama13b, + Author = {A. Scemama and M. Caffarel and E. Oseret and W. Jalby}, + Date-Added = {2017-02-28 21:27:28 +0000}, + Date-Modified = {2017-02-28 21:35:21 +0000}, + Doi = {10.1002/jcc.23216}, + Journal = {J. Comput. Chem.}, + Month = {jan}, + Number = {11}, + Pages = {938--951}, + Publisher = {Wiley-Blackwell}, + Title = {{Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond}}, + Url = {http://dx.doi.org/10.1002/jcc.23216}, + Volume = {34}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.23216}} + +@incollection{Scemama13a, + Author = {A. Scemama and M. Caffarel and E. Oseret and W. Jalby}, + Booktitle = {High Performance Computing for Computational Science - {VECPAR} 2012}, + Date-Added = {2017-02-28 21:27:25 +0000}, + Date-Modified = {2017-02-28 21:35:26 +0000}, + Doi = {10.1007/978-3-642-38718-0_14}, + Pages = {118--127}, + Publisher = {Springer Science Business Media}, + Title = {{{QMC}=Chem: A Quantum Monte Carlo Program for Large-Scale Simulations in Chemistry at the Petascale Level and beyond}}, + Url = {http://dx.doi.org/10.1007/978-3-642-38718-0\_14}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1007/978-3-642-38718-0%5C_14}, + Bdsk-Url-2 = {http://dx.doi.org/10.1007/978-3-642-38718-0_14}} + +@misc{qmcchem, + Author = {A. Scemama and E. Giner and T. Applencourt and M. 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Rev.}, + Month = {Nov}, + Number = {3}, + Numpages = {7}, + Pages = {507--514}, + Publisher = {American Physical Society}, + Title = {Correlation Energy of an Electron Gas at High Density: Plasma Oscillations}, + Volume = {108}, + Year = {1957}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRev.108.507}} + +@article{Seidl04, + Author = {Seidl, Michael}, + Date-Added = {2015-08-25 04:11:41 +0000}, + Date-Modified = {2015-08-25 04:11:41 +0000}, + Doi = {10.1103/PhysRevB.70.073101}, + Journal = {Phys. Rev. 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Griebel}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:00 +0000}, + Journal = {Acta Numerica}, + Keywords = {quadrature, maths}, + Pages = {147--269}, + Volume = {13}, + Year = {2004}} + +@article{Griebel98, + Author = {T. Gerstner and M. Griebel}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:02 +0000}, + Journal = {Numer. Algorithms}, + Keywords = {quadrature, maths}, + Pages = {209--232}, + Volume = {18}, + Year = {1998}} + +@article{Hattig12, + Author = {C. Hattig and W. Klopper and A. Kohn and D. P. Tew}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:10 +0000}, + Journal = {Chem. Rev.}, + Pages = {4}, + Volume = {112}, + Year = {2012}} + +@article{Heo00, + Author = {S. Heo and Y. Xu}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:25 +0000}, + Journal = {Math. Comp.}, + Keywords = {quadrature, maths}, + Pages = {269--279}, + Volume = {70}, + Year = {2000}} + +@article{Hinrichs07, + Author = {A. Hinrichs and E. Novak}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:29 +0000}, + Journal = {Math. Comp.}, + Keywords = {quadrature}, + Pages = {1357--1372}, + Volume = {76}, + Year = {2007}} + +@article{Hirao09, + Author = {M. Hirao and M. Sawa}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-27 11:32:31 +0000}, + Journal = {SIAM J. Numer. Anal.}, + Keywords = {quadrature}, + Pages = {3195--3211}, + Volume = {47}, + Year = {2009}} + +@article{Hollett11, + Author = {J. W. Hollett and P. M. W. Gill}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-24 03:22:36 +0000}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {2972}, + Volume = {13}, + Year = {2011}} + +@book{HRSPbook, + Author = {W. J. Hehre and L. Radom and {P. v. R.} Schleyer and J. A. Pople}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-24 03:22:36 +0000}, + Keywords = {qmech}, + Publisher = {John Wiley \& Sons Inc}, + Title = {Ab initio molecular orbital theory}, + Year = {1986}} + +@article{Hylleraas28, + Author = {E. A. Hylleraas}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-24 03:22:36 +0000}, + Journal = {Z. Phys.}, + Pages = {469}, + Volume = {48}, + Year = {1928}} + +@article{Karton12, + Author = {A. Karton and J. M. L. Martin}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-24 03:22:36 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {124114}, + Volume = {136}, + Year = {2012}} + +@article{Klopper02, + Author = {W. Klopper and C. C. M. Samson}, + Date-Added = {2015-06-24 03:22:36 +0000}, + Date-Modified = {2015-06-24 03:22:36 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {6397}, + Volume = {116}, + Year = {2002}} + +@article{Kong12, + Author = {L. Kong and F. A. Bischo and E. F. 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Loos}, + Date-Added = {2015-06-23 22:20:24 +0000}, + Date-Modified = {2015-09-21 23:56:57 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {084114}, + Volume = {143}, + Year = {2015}} + +@article{Yanium11, + Author = {Y. Zhao and P. F. Loos and P. M. W. Gill}, + Date-Added = {2015-06-02 23:50:55 +0000}, + Date-Modified = {2015-06-27 11:28:14 +0000}, + Journal = {Phys. Rev. A}, + Keywords = {Gill, hookium, correlation}, + Pages = {032513/1--5}, + Volume = {84}, + Year = {2011}} + +@article{2DEG11, + Author = {P. F. Loos and P. M. W. Gill}, + Date-Added = {2015-06-02 23:50:44 +0000}, + Date-Modified = {2015-06-27 11:26:24 +0000}, + Journal = {Phys. Rev. B}, + Keywords = {Gill, jellium}, + Pages = {233102}, + Volume = {83}, + Year = {2011}} + +@article{3DEG11, + Author = {P. F. Loos and P. M. W. Gill}, + Date-Added = {2015-06-02 23:50:44 +0000}, + Date-Modified = {2015-06-27 11:26:30 +0000}, + Journal = {Phys. Rev. B}, + Keywords = {Gill, jellium}, + Pages = {033103}, + Volume = {84}, + Year = {2011}} + +@article{Ballium10, + Author = {P. F. Loos and P. M. W. Gill}, + Date-Added = {2015-06-02 23:50:44 +0000}, + Date-Modified = {2015-06-27 11:26:34 +0000}, + Journal = {J. Chem. Phys.}, + Keywords = {Gill, correlation}, + Pages = {234111}, + Volume = {132}, + Year = {2010}} + +@article{1DEG13, + Author = {P. F. Loos}, + Date-Added = {2015-06-02 23:50:40 +0000}, + Date-Modified = {2015-06-27 11:26:20 +0000}, + Journal = {J. Chem. Phys.}, + Pages = {064108}, + Volume = {138}, + Year = {2013}} + +@article{1DChem15, + Author = {P. F. Loos and C. J. Ball and P. M. W. Gill}, + Date-Added = {2015-06-02 23:28:34 +0000}, + Date-Modified = {2015-06-27 11:26:14 +0000}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {3196}, + Volume = {17}, + Year = {2015}} + +@article{Concentric10, + Author = {P. F. Loos and P. M. W. Gill}, + Date-Added = {2015-06-02 23:28:34 +0000}, + Date-Modified = {2015-06-27 11:26:37 +0000}, + Journal = {Phys. Rev. A}, + Keywords = {Gill, spherium, intracule}, + Pages = {052510/1--9}, + Volume = {81}, + Year = {2010}} + +@article{Exciton12, + Author = {P. F. Loos}, + Date-Added = {2015-06-02 23:28:34 +0000}, + Date-Modified = {2015-06-27 11:26:52 +0000}, + Journal = {Phys. Lett. A}, + Pages = {1997}, + Volume = {376}, + Year = {2012}} + +@article{gLDA14, + Author = {P. F. Loos and C. J. Ball and P. M. W. Gill}, + Date-Added = {2015-06-02 23:28:34 +0000}, + Date-Modified = {2015-06-27 11:27:01 +0000}, + Journal = {J. Chem. Phys.}, + Keywords = {Gill, jellium, correlation}, + Pages = {18A524}, + Volume = {140}, + Year = {2014}} + +@article{Nodes15, + Author = {P. F. Loos and D. Bressanini}, + Date-Added = {2015-06-02 23:28:34 +0000}, + Date-Modified = {2015-06-27 11:27:49 +0000}, + Journal = {J. Chem. 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Lett.}, + Pages = {190}, + Volume = {395}, + Year = {2004}} + +@article{Wind01, + Author = {Wind, P. and Helgaker, T. and Klopper, W.}, + Journal = {Theor. Chem. Acc.}, + Pages = {280}, + Volume = {106}, + Year = {2001}} + +@article{Limpanuparb13, + Author = {T. Limpanuparb and J. Milthorpe and A. P. Rendell and P. M. W. Gill}, + Date-Added = {2013-01-30 13:54:17 +1100}, + Date-Modified = {2015-06-27 11:33:28 +0000}, + Journal = {J. Chem. Theory Comput.}, + Keywords = {Gill, coulomb, exchange, hartreefock}, + Pages = {863--867}, + Volume = {9}, + Year = {2013}} + +@article{Gill09, + Author = {P. M. W. Gill and A. T. B. Gilbert}, + Date-Added = {2011-10-12 09:04:46 +1100}, + Date-Modified = {2015-06-27 11:31:33 +0000}, + Journal = {Chem. Phys.}, + Keywords = {Gill, coulomb, fitting}, + Pages = {86--90}, + Volume = {356}, + Year = {2009}} + +@article{Limpanuparb09, + Author = {T. Limpanuparb and P. M. W. Gill}, + Date-Added = {2011-10-12 09:05:17 +1100}, + Date-Modified = {2015-06-27 11:33:25 +0000}, + Journal = {Phys. Chem. Chem. Phys.}, + Keywords = {Gill, fitting}, + Pages = {9176--9181}, + Volume = {11}, + Year = {2009}} + +@article{Varganov08, + Author = {S. A. Varganov and A. T. B. Gilbert and E. Deplazes and P. M. W. Gill}, + Date-Added = {2011-10-12 09:03:50 +1100}, + Date-Modified = {2015-06-27 11:28:49 +0000}, + Journal = {J. Chem. Phys.}, + Keywords = {Gill, coulomb, fitting}, + Pages = {201104}, + Volume = {128}, + Year = {2008}} + +@article{qchem, + Author = {Shao, Yihan and Molnar, Laszlo Fusti and Jung, Yousung and Kussmann, Jorg and Ochsenfeld, Christian and Brown, Shawn T. and Gilbert, Andrew T.B. and Slipchenko, Lyudmila V. and Levchenko, Sergey V. and O{'}Neill, Darragh P. and DiStasio Jr, Robert A. and Lochan, Rohini C. and Wang, Tao and Beran, Gregory J.O. and Besley, Nicholas A. and Herbert, John M. and Yeh Lin, Ching and Van Voorhis, Troy and Hung Chien, Siu and Sodt, Alex and Steele, Ryan P. and Rassolov, Vitaly A. and Maslen, Paul E. and Korambath, Prakashan P. and Adamson, Ross D. and Austin, Brian and Baker, Jon and Byrd, Edward F. C. and Dachsel, Holger and Doerksen, Robert J. and Dreuw, Andreas and Dunietz, Barry D. and Dutoi, Anthony D. and Furlani, Thomas R. and Gwaltney, Steven R. and Heyden, Andreas and Hirata, So and Hsu, Chao-Ping and Kedziora, Gary and Khalliulin, Rustam Z. and Klunzinger, Phil and Lee, Aaron M. and Lee, Michael S. and Liang, WanZhen and Lotan, Itay and Nair, Nikhil and Peters, Baron and Proynov, Emil I. and Pieniazek, Piotr A. and Min Rhee, Young and Ritchie, Jim and Rosta, Edina and David Sherrill, C. and Simmonett, Andrew C. and Subotnik, Joseph E. and Lee Woodcock III, H. and Zhang, Weimin and Bell, Alexis T. and Chakraborty, Arup K. and Chipman, Daniel M. and Keil, Frerich J. and Warshel, Arieh and Hehre, Warren J. and Schaefer III, Henry F. and Kong, Jing and Krylov, Anna I. and Gill, Peter M. W. and Head-Gordon, Martin}, + Date-Modified = {2015-06-27 11:37:08 +0000}, + Issue = {27}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {3172-3191}, + Publisher = {The Royal Society of Chemistry}, + Volume = {8}, + Year = {2006}} + +@article{Limpanuparb12, + Author = {T. Limpanuparb and J. W. Hollett and P. M. W. Gill}, + Journal = {J. Chem. Phys.}, + Pages = {104102}, + Volume = {136}, + Year = {2012}} + +@article{Limpanuparb11b, + Author = {T. Limpanuparb and P. M. W. Gill}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2353}, + Volume = {7}, + Year = {2011}} + +@article{Limpanuparb11a, + Author = {T. Limpanuparb and A. T. B. Gilbert and P. M. W. Gill}, + Journal = {J. Chem. Theory Comput.}, + Pages = {830}, + Volume = {7}, + Year = {2011}} + +@article{Tenno04, + Author = {S. Ten-no}, + Journal = {Chem. Phys. Lett.}, + Pages = {56}, + Volume = {398}, + Year = {2004}} + +@article{Tenno12a, + Author = {S. Ten-no and J. Noga}, + Journal = {WIREs Comput. Mol. Sci.}, + Pages = {114}, + Volume = {2}, + Year = {2012}} + +@article{Tenno12b, + Author = {S. Ten-no}, + Date-Modified = {2016-12-23 01:45:55 +0000}, + Journal = {Theor. Chem. Acc.}, + Pages = {1070}, + Volume = {131}, + Year = {2012}} + +@book{NiederreiterBook, + Address = {Philadelphia}, + Author = {H. Niederreiter}, + Date-Added = {2013-01-11 15:35:24 +1100}, + Date-Modified = {2013-01-11 15:35:24 +1100}, + Keywords = {quadrature}, + Publisher = {SIAM}, + Title = {Random number generation and quasi-Monte Carlo methods}, + Year = {1992}} + +@article{Sloan05, + Author = {F. Y. Kuo and I. H. Sloan}, + Date-Added = {2013-01-11 12:59:07 +1100}, + Date-Modified = {2015-06-27 11:38:11 +0000}, + Journal = {Not. Am. Math. Soc.}, + Keywords = {quadrature}, + Pages = {1320--1328}, + Volume = {52}, + Year = {2005}} + +@article{Nakatsuji04, + Author = {H. Nakatsuji}, + Date-Added = {2013-01-03 15:28:59 +1100}, + Date-Modified = {2015-06-27 11:34:18 +0000}, + Journal = {Phys. Rev. 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Rabinowitz}, + Date-Added = {2012-12-19 11:05:54 +1100}, + Date-Modified = {2012-12-19 11:09:56 +1100}, + Edition = {2nd}, + Keywords = {quadrature, maths}, + Publisher = {Academic}, + Title = {Methods of numerical integration}, + Year = {1984}} + +@book{SecrestBook, + Address = {Englewood Cliffs, New Jersey}, + Author = {A. H. Stroud and D. Secrest}, + Date-Added = {2012-12-19 10:55:24 +1100}, + Date-Modified = {2012-12-19 10:55:24 +1100}, + Keywords = {quadrature, maths}, + Publisher = {Prentice-Hall}, + Title = {Gaussian Quadrature Formulas}, + Year = {1966}} + +@article{Sloan98, + Author = {I. H. Sloan and H. Wozniakowski}, + Date-Added = {2012-12-15 15:18:35 +1100}, + Date-Modified = {2015-06-27 11:38:17 +0000}, + Journal = {J. Complexity}, + Keywords = {quadrature, maths}, + Pages = {1--33}, + Volume = {14}, + Year = {1998}} + +@book{SloanBook, + Address = {Oxford}, + Author = {I. H. Sloan and S. Joe}, + Date-Added = {2012-12-15 14:43:25 +1100}, + Date-Modified = {2012-12-15 14:43:25 +1100}, + Keywords = {quadrature, maths}, + Publisher = {Clarendon Press}, + Title = {Lattice methods for multiple integration}, + Year = {1994}} + +@article{Trefethen08, + Author = {L. N. Trefethen}, + Date-Added = {2012-12-14 15:10:15 +1100}, + Date-Modified = {2015-06-27 11:28:32 +0000}, + Journal = {SIAM Review}, + Keywords = {quadrature, maths}, + Pages = {67--87}, + Volume = {50}, + Year = {2008}} + +@article{Xiao10, + Author = {H. Xiao and Z. Gimbutas}, + Date-Added = {2012-12-14 11:14:57 +1100}, + Date-Modified = {2015-06-27 11:29:10 +0000}, + Journal = {Comp. Math. Appl.}, + Keywords = {quadrature, maths}, + Pages = {663--676}, + Volume = {59}, + Year = {2010}} + +@book{StroudBook, + Address = {Englewood Cliffs, New Jersey}, + Author = {A. H. Stroud}, + Date-Added = {2012-12-14 10:36:10 +1100}, + Date-Modified = {2012-12-19 11:04:05 +1100}, + Keywords = {quadrature, maths}, + Publisher = {Prentice Hall}, + Title = {Approximate calculation of multiple integrals}, + Year = {1971}} + +@article{UEGs12, + Author = {P. M. W. Gill and P. F. Loos}, + Date-Added = {2012-12-14 09:43:36 +1100}, + Date-Modified = {2015-06-27 11:28:05 +0000}, + Journal = {Theor. Chem. 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LaTeX info: "xparse/define-command" +. +. Defining command \vnabla with sig. '' on line 25. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \varE with sig. '' on line 29. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ordersymbol with sig. '' on line 30. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \lparen with sig. '' on line 31. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \rparen with sig. '' on line 32. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \quantity with sig. '' on line 35. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. 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" +. +. Defining command \qty with sig. '' on line 63. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pqty with sig. ' l m ' on line 64. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bqty with sig. ' l m ' on line 65. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Bqty with sig. ' l m ' on line 66. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vqty with sig. ' l m ' on line 67. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pmqty with sig. 'm' on line 70. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Pmqty with sig. 'm' on line 71. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bmqty with sig. 'm' on line 72. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vmqty with sig. 'm' on line 73. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixquantity with sig. '' on line 74. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@matrixquantity with sig. ' s g o d() d|| ' on line 100. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mqty with sig. '' on line 101. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixdeterminant with sig. 'm' on line 102. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mdet with sig. '' on line 103. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \spmqty with sig. 'm' on line 105. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sPmqty with sig. 'm' on line 106. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sbmqty with sig. 'm' on line 107. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \svmqty with sig. 'm' on line 108. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smallmatrixquantity with sig. ' s g o d() d|| ' on line +. 133. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smqty with sig. '' on line 134. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smallmatrixdeterminant with sig. 'm' on line 135. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smdet with sig. '' on line 136. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \argopen with sig. 's' on line 138. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \argclose with sig. 's' on line 139. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braces with sig. '' on line 141. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. 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" +. +. Defining command \absolutevalue with sig. ' l m ' on line 176. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \abs with sig. '' on line 177. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \norm with sig. ' l m ' on line 178. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \order with sig. ' l m ' on line 179. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \evaluated with sig. ' s g d[| d(| ' on line 194. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \eval with sig. '' on line 195. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \poissonbracket with sig. ' l m m ' on line 197. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pb with sig. '' on line 198. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \commutator with sig. ' l m m ' on line 201. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \comm with sig. '' on line 202. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \anticommutator with sig. ' l m m ' on line 203. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acommutator with sig. '' on line 204. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acomm with sig. '' on line 205. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorbold with sig. ' s m ' on line 208. +................................................. +................................................. +. 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' g o d() ' on line 256. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \laplacian with sig. ' g o d() ' on line 266. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \principalvalue with sig. 'g' on line 274. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pv with sig. '' on line 275. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \PV with sig. 'g' on line 276. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. 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'' on line 322. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccot with sig. '' on line 324. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asin with sig. '' on line 327. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acos with sig. '' on line 329. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \atan with sig. '' on line 331. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acsc with sig. '' on line 333. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asec with sig. '' on line 335. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acot with sig. '' on line 337. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sinh with sig. '' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cosh with sig. '' on line 340. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tanh with sig. '' on line 341. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \csch with sig. '' on line 343. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sech with sig. '' on line 345. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \coth with sig. '' on line 346. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \exp with sig. '' on line 348. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \log with sig. '' on line 349. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \ln with sig. '' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \det with sig. '' on line 351. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Pr with sig. '' on line 352. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \tr with sig. '' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Tr with sig. '' on line 354. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Res with sig. '' on line 355. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qqtext with sig. ' s m ' on line 377. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qq with sig. '' on line 378. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcomma with sig. '' on line 380. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qc with sig. '' on line 381. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qif with sig. 's' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qthen with sig. 's' on line 384. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qelse with sig. 's' on line 385. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qotherwise with sig. 's' on line 386. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qunless with sig. 's' on line 387. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qgiven with sig. 's' on line 388. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qusing with sig. 's' on line 389. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qassume with sig. 's' on line 390. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qsince with sig. 's' on line 391. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qlet with sig. 's' on line 392. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qfor with sig. 's' on line 393. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qall with sig. 's' on line 394. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qeven with sig. 's' on line 395. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qodd with sig. 's' on line 396. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qinteger with sig. 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LaTeX info: "xparse/define-command" +. +. Defining command \functionalderivative with sig. ' s o m g d() ' on line +. 494. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fderivative with sig. '' on line 495. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fdv with sig. '' on line 496. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. 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" +. +. Defining command \ket with sig. ' s m ' on line 533. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \innerproduct with sig. ' s m g ' on line 548. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braket with sig. '' on line 549. +................................................. +................................................. +. 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Defining command \xmatrix with sig. ' s m m m ' on line 659. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmat with sig. '' on line 660. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zeromatrix with sig. ' m g ' on line 662. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zmat with sig. '' on line 663. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulixmatrix with sig. '' on line 665. +................................................. +................................................. +. 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'O{} >{\SplitList {,}}m ' on line +. 681. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@dmat with sig. 'mmggggggg' on line 714. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dmat with sig. '' on line 715. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \antidiagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on +. line 717. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@admat with sig. 'mmggggggg' on line 750. +................................................. +................................................. +. 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+1,2 @@ +\BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2 +\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 diff --git a/Manuscript/CI-F12.pdf b/Manuscript/CI-F12.pdf new file mode 100644 index 0000000..609a062 Binary files /dev/null and b/Manuscript/CI-F12.pdf differ diff --git a/Manuscript/CI-F12.synctex.gz b/Manuscript/CI-F12.synctex.gz new file mode 100644 index 0000000..bacdf3e Binary files /dev/null and b/Manuscript/CI-F12.synctex.gz differ diff --git a/Manuscript/CI-F12.tex b/Manuscript/CI-F12.tex new file mode 100644 index 0000000..9edae17 --- /dev/null +++ b/Manuscript/CI-F12.tex @@ -0,0 +1,408 @@ +\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}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} + +\newcommand{\eps}{\varepsilon} +\newcommand{\la}{\lambda} + +\newcommand{\br}{\bm{r}} + +\newcommand{\oH}{\mathring{H}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EHF}{E_\text{HF}} +\newcommand{\NGG}{N_\text{GG}} + +\newcommand{\hH}{\Hat{H}} +\newcommand{\hO}{\Hat{O}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hV}{\Hat{V}} +\newcommand{\hU}{\Hat{U}} +\newcommand{\hQ}{\Hat{Q}} +\newcommand{\hS}{\Hat{S}} +\newcommand{\hP}{\Hat{P}} +\newcommand{\hI}{\Hat{I}} + +\newcommand{\mA}{\mathcal{A}} +\newcommand{\mC}{\mathcal{C}} +\newcommand{\mD}{\mathcal{D}} +\newcommand{\mE}{\mathcal{E}} +\newcommand{\mK}{\mathcal{K}} +\newcommand{\mF}{\mathcal{F}} +\newcommand{\mL}{\mathcal{L}} + +\newcommand{\kA}{\ket{A}} +\newcommand{\kD}{\ket{D}} +\newcommand{\kI}{\ket{I}} +\newcommand{\kJ}{\ket{J}} +\newcommand{\kE}{\ket{E}} +\newcommand{\kK}{\ket{K}} +\newcommand{\kL}{\ket{L}} +\newcommand{\kF}{\ket{F}} +\newcommand{\kO}{\ket{0}} + +% addresses +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} + +\begin{document} + +\title{Dressing the configuration interaction matrix with explicit correlation} + +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Michel Caffarel} +\affiliation{\LCPQ} +\author{Pierre-Fran{\c c}ois Loos} +\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} + +\begin{abstract} +We present an explicitly-correlated version of the configuration interaction (CI) method. +An explicitly-correlated term is introduced via dressing of the CI matrix. +The dressing is guided by electron-electron cusp conditions. +This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods. +The performance of the newly-designed explicitly-correlated dressing CI method is illustrated on atoms and molecules. +\end{abstract} + +\keywords{configuration interaction; explicitly-correlated methods; dressed Hamiltonian} + +\maketitle + +%---------------------------------------------------------------- +\textit{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 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}. + +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}. +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}. +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.---} +%---------------------------------------------------------------- +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 +\end{equation} +composed by a linear combination of determinants $\kI$ of coefficients $c_I$ and an ``explicitly-correlated'' part +\begin{equation} +\label{eq:WF-F12-CIPSI} + \kF = \sum_{I} t_I \hQ f \kI +\end{equation} +with coefficients $t_I$. +The projector +\begin{equation} + \hQ = \hI - \sum_{I} \dyad{I} +\end{equation} +ensures the orthogonality between $\kD$ and $\kF$, and +\begin{equation} +\label{eq:Ja} + f = \sum_{i < j} \gamma_{ij} f_{ij} +\end{equation} +is a correlation factor, and +\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}), 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}. +\end{equation} + +%---------------------------------------------------------------- +\textit{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 $\bra{I}$, +\begin{equation} + c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0. +\end{equation} +where $H_{IJ} = \mel{I}{\hH}{J}$. +Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian: +\begin{equation} +\label{eq:DrH} + \oH_{IJ} = + \begin{cases} + H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$}, + \\ + H_{IJ}, & \text{otherwise}, + \end{cases} +\end{equation} +with +\begin{equation} +\label{eq:IHF} + \mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ], +\end{equation} +and $f_{IJ} = \mel{I}{f}{J}$. +It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis. + +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{This yields the following linear system of equations +\begin{equation} +\label{eq:tI} + \sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I, +\end{equation} +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}. + +%%% FIG 1 %%% +%\begin{figure} +% \includegraphics[width=\linewidth]{fig1} +% \caption{ +% \label{fig:CBS} +% Schematic representation of the various orbital spaces and their notation. +% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).} +%\end{figure} +%%% %%% + +%---------------------------------------------------------------- +\textit{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}. +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}. +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}. + +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}). + +In the CBS, one can write +\begin{equation} +\label{eq:RI} + \hI = \sum_{A \in \mA} \dyad{A}{A} +\end{equation} +where $\mA$ is the set of all the determinants $\kA$ corresponding to electronic excitations from occupied orbitals $\qty{i}$ to the extended virtual orbital space $\qty{a} \cup \qty{\alpha}$. +Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets +\begin{equation} +\label{eq:IHF-RI} +\begin{split} + \mel{I}{\hH}{F} + & = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ] + \\ + & = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ}, +\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}. + +%The set $\mC$ is defined by two simple rules. +%First, because $f$ is a two-electron operator (and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}), we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$. +%Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals. +%Three types of determinants have these two properties: +%i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$; +%ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$; +%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}. + +%\begin{gather} +% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} +% \\ +% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta} +% \\ +% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta} +%\end{gather} +% +%\begin{gather} +% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j} +% \\ +% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0 +% \\ +% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} = +%\end{gather} +% +%\begin{gather} +% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} = +% \\ +% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} = +% \\ +% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} = +%\end{gather} +% +%\begin{gather} +% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} = +% \\ +% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} = +% \\ +% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} = +%\end{gather} + +%---------------------------------------------------------------- +\textit{Results.---} +%---------------------------------------------------------------- +In all the calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a} +\begin{equation} + f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la}, +\end{equation} +which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}, i.e. +\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}. + +%%% TABLE 1 %%% +\begin{table} +\caption{ +\label{tab:atoms} +FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set. +The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.} +\begin{ruledtabular} +\begin{tabular}{lcccc} +Atom & cc-pVXZ & FCI-F12 & CIPSI & FCI \\ +\hline +\ce{He} & D & & & $-2.887\,595$\footnotemark[1] \\ + & 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 & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-7.478\,060$\footnotemark[3] \\ +\hline +\ce{Be} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-14.667\,356$\footnotemark[4] \\ +\hline +\ce{B} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-24.653\,866$\footnotemark[5] \\ +\hline +\ce{C} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-37.840\,129$\footnotemark[6] \\ +\hline +\ce{N} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $ 54.588917$\footnotemark[7] \\ +\hline +\ce{O} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-75.066892$\footnotemark[7] \\ +\hline +\ce{F} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-99.733\,424$\footnotemark[7] \\ +\hline +\ce{Ne} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & $-128.937\,274$\footnotemark[7] \\ +\end{tabular} +\end{ruledtabular} +\footnotetext[1]{Reference \onlinecite{Kong12}} +\footnotetext[2]{Reference \onlinecite{Nakashima07}} +\footnotetext[3]{Reference \onlinecite{Puchalski09}} +\footnotetext[4]{Reference \onlinecite{Sharkey11}} +\footnotetext[5]{Reference \onlinecite{Bubin11}} +\footnotetext[6]{Reference \onlinecite{Sharkey10}} +\footnotetext[7]{Reference \onlinecite{Klopper10}} +\end{table} +%%% + +%%% TABLE 2 %%% +\begin{table*} +\caption{ +\label{tab:molecules} +CIPSI, FCI-F12 i-FCIQMC and FCI total ground-state energy of the \ce{H2}, \ce{F2} and \ce{H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set. +The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.} +\begin{ruledtabular} +\begin{tabular}{lccccc} +Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\ +\hline +\ce{H2} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\ +\hline +\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\ + & T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\ +\hline +\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\ + & T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\ + & Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\ + & 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\ +\end{tabular} +\end{ruledtabular} +\footnotetext[1]{Reference \onlinecite{Pachucki10}} +\footnotetext[2]{Reference \onlinecite{Giner15}} +\footnotetext[3]{Reference \onlinecite{Cleland12}} +\footnotetext[4]{Reference \onlinecite{Caffarel16}} +\footnotetext[5]{Reference \onlinecite{AlmoraDiaz14}} +\end{table*} +%%% + +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 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.---} +%---------------------------------------------------------------- +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. +In particular, one eschew the computation of four-electron integrals as well as some types of three-electron integrals. +We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated full CI methods. + +%---------------------------------------------------------------- +\textit{Acknowledgments.---} +This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015). +%---------------------------------------------------------------- + +\bibliography{CI-F12} + +\end{document} diff --git a/Manuscript/CI-F12Notes.bib b/Manuscript/CI-F12Notes.bib new file mode 100644 index 0000000..881f198 --- /dev/null +++ b/Manuscript/CI-F12Notes.bib @@ -0,0 +1,2 @@ +@CONTROL{REVTEX41Control} 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written on fig1.pdf (1 page, 40936 bytes). +PDF statistics: + 36 PDF objects out of 1000 (max. 8388607) + 27 compressed objects within 1 object stream + 2 named destinations out of 1000 (max. 500000) + 4109 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/Manuscript/fig1/fig1.pdf b/Manuscript/fig1/fig1.pdf new file mode 100644 index 0000000..d7016b7 Binary files /dev/null and b/Manuscript/fig1/fig1.pdf differ diff --git a/Manuscript/fig1/fig1.synctex.gz b/Manuscript/fig1/fig1.synctex.gz new file mode 100644 index 0000000..47e4398 Binary files /dev/null and b/Manuscript/fig1/fig1.synctex.gz differ diff --git a/Manuscript/fig1/fig1.tex b/Manuscript/fig1/fig1.tex new file mode 100644 index 0000000..8be31e1 --- /dev/null +++ b/Manuscript/fig1/fig1.tex @@ -0,0 +1,127 @@ +\documentclass{standalone} +\usepackage{adjustbox,graphicx,bm,color,microtype,hyperref,physics,amsmath,amssymb,amsfonts} +\usepackage{tikz} +\usetikzlibrary{decorations.pathreplacing,decorations.pathmorphing,arrows,backgrounds,fit,shapes} + +\tikzset{snake it/.style={ +decoration={snake, + amplitude = .4mm, + segment length = 2mm},decorate} +} + +\begin{document} + +\begin{tikzpicture}[scale=1, + double`1/.style={ellipse,draw,fill opacity=0.5,fill=green!10}, + double2/.style={ellipse,draw,fill opacity=0.5,fill=magenta!10}, + single/.style={ellipse,draw,fill opacity=0.5,fill=orange!10}, + ] +\scriptsize +% Draw windows + % Occupied + \draw[thick,blue,fill=blue!15!white!] (0,0) -- (0,1) -- (2,1) -- (2,0) -- (0,0);% + % Virtual + \draw[thick,yellow,fill=yellow!15!white!] (0,1.1) -- (0,3.1) -- (2,3.1) -- (2,1.1) -- (0,1.1);% + % CABS + \draw[thick,red,fill=red!15!white!] (0,3.2) -- (0,6.2) -- (2,6.2) -- (2,3.2) -- (0,3.2) node[xshift=1cm,yshift=1.5cm]{\textbf{CABS}};% + +% Draw orbitals + % Occupied + \foreach \y in {0.15,0.4,0.8} + \draw[black] + (0.8,\y) -- (1.2,\y); + \foreach \y in {0.6} + \draw[black] + (0.3,\y) -- (0.7,\y) + (1.3,\y) -- (1.7,\y); + % Virtual + \foreach \y in {1.4,1.8,2.3,2.5,2.8,3.0} + \draw[black] + (0.8,\y) -- (1.2,\y); + \foreach \y in {1.25,1.95,2.15,2.4} + \draw[black] + (0.3,\y) -- (0.7,\y) + (1.3,\y) -- (1.7,\y); + +% Draw electrons + \foreach \y in {0.15,0.4,0.8} + \draw[<-,black] + (0.95,\y-0.1) -- (0.95,\y+0.1); + \foreach \y in {0.15,0.4,0.8} + \draw[->,black] + (1.05,\y-0.1) -- (1.05,\y+0.1); + \foreach \y in {0.6} + \draw[<-,black] + (0.45,\y-0.1) -- (0.45,\y+0.1); + \foreach \y in {0.6} + \draw[->,black] + (0.55,\y-0.1) -- (0.55,\y+0.1); + \foreach \y in {0.6} + \draw[<-,black] + (1.45,\y-0.1) -- (1.45,\y+0.1); + \foreach \y in {0.6} + \draw[->,black] + (1.55,\y-0.1) -- (1.55,\y+0.1); + +% Draw curly brackets + % occupied space + \draw[thick,decorate,decoration={brace,amplitude=4pt}] + (2.1,1) -- + node[xshift=0.8cm,yshift=+0.15cm] {occupied} + node[xshift=0.85cm,yshift=-0.15cm] {space $\qty{i}$} (2.1,0);% + % virtual space + \draw[thick,decorate,decoration={brace,amplitude=4pt}] + (2.1,3.1) -- + node[xshift=0.65cm,yshift=+0.15cm] {virtual} + node[xshift=0.85cm,yshift=-0.15cm] {space $\qty{a}$} (2.1,1.1);% + % orbital space + \draw[thick,decorate,decoration={brace,amplitude=4pt}] + (3.6,3.1) -- + node[xshift=1.3cm,yshift=+0.2cm] {orbital space $\qty{p}$} + node[xshift=1.3cm,yshift=-0.2cm] {$\qty{p} = \qty{i} \cup \qty{a}$} + (3.6,0);% + % complementary virtual space + \draw[thick,decorate,decoration={brace,amplitude=4pt}] + (2.1,6.2) -- + node[xshift=1.25cm,yshift=+0.4cm] {complementary} + node[xshift=1.4cm] {virtual space $\qty{\alpha}$} + node[xshift=1.4cm,yshift=-0.4cm] {$\qty{p} \cap \qty{\alpha} = \varnothing$} + (2.1,3.2);% + % complete space + \draw[thick,decorate,decoration={brace,amplitude=4pt}] + (6.1,6.2) -- + node[xshift=0.85cm,yshift=+0.3cm] {complete} + node[xshift=0.7cm] {orbital} + node[xshift=0.65cm,yshift=-0.35cm] {space} + (6.1,0);% + +% Draw dotted lines + \draw[thick,dashed] (2.1,3.1) -- (3.6,3.1); + + \draw[thick,dashed] (2.1,6.2) -- (6.1,6.2); + \draw[thick,dashed] (2.1,0) -- (6.1,0); + +% Draw excitations + + \node [double1, align=center] at (0.45,0.6) (p1) {}; + \node [double1, align=center] at (0.45,6.0) (h1) {}; + \node [double1, align=center] at (0.55,0.6) (p2) {}; + \node [double1, align=center] at (0.55,5.0) (h2) {}; + \node [double2, align=center] at (1.55,0.6) (p3) {}; + \node [double2, align=center] at (1.55,4.0) (h3) {}; + \node [double2, align=center] at (1.45,0.6) (p4) {}; + \node [double2, align=center] at (1.45,1.95)(h4) {}; + \node [single, align=center] at (1.05,0.8) (p5) {}; + \node [single, align=center] at (1.05,3.5) (h5) {}; + \path + (p1) edge [->, green, bend left, in=170, out=10] (h1) + (p2) edge [->, green, bend left, in=170, out=10] (h2) + (p3) edge [->, magenta, bend right, in=190, out=-10] (h3) + (p4) edge [->, magenta, bend right, in=190, out=-10] (h4) + (p5) edge [->, orange, bend left, in=170, out=10] (h5) + ; + + +\end{tikzpicture} + +\end{document}