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added separaility tests

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Emmanuel Giner 2019-10-16 11:38:24 +02:00
parent 90e8a5e47a
commit 584e7da053
9 changed files with 442 additions and 319 deletions
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Manuscript/srDFT_SC.aux
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@ -43,7 +43,7 @@
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\citation{GinPraFerAssSavTou-JCP-18} \citation{GinPraFerAssSavTou-JCP-18}
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@ -81,22 +81,30 @@
\newlabel{eq:lim_mularge}{{19}{4}{}{equation.2.19}{}} \newlabel{eq:lim_mularge}{{19}{4}{}{equation.2.19}{}}
\newlabel{eq:lim_n2}{{20}{4}{}{equation.2.20}{}} \newlabel{eq:lim_n2}{{20}{4}{}{equation.2.20}{}}
\newlabel{eq:lim_muinf}{{21}{4}{}{equation.2.21}{}} \newlabel{eq:lim_muinf}{{21}{4}{}{equation.2.21}{}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Properties of approximated functionals}{4}{section*.10}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {E}Requirements for the approximated functionals in the strong correlation }{5}{section*.11}} \@writefile{toc}{\contentsline {subsection}{\numberline {E}Requirements for the approximated functionals in the strong correlation regime}{5}{section*.11}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Requirements: separability of the energies and $S_z$ invariance}{5}{section*.12}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Requirements: separability of the energies and $S_z$ invariance}{5}{section*.12}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Condition for the functional $\mathaccentV {bar}916{E}_{\text {PBE}}^\mathcal {B}[{n},\xi ,s,n^{(2)},\mu _{\Psi ^{\mathcal {B}}}]$ to obtain $S_z$ invariance}{5}{section*.13}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Condition for the functional $\mathaccentV {bar}916{E}_{\text {X}}^\mathcal {B}[{n},\xi ,s,n^{(2)},\mu _{\Psi ^{\mathcal {B}}}]$ to obtain $S_z$ invariance}{5}{section*.13}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {F}Requirement on $\Psi ^{\mathcal {B}}$ for the extensivity}{5}{section*.14}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Conditions on $\Psi ^{\mathcal {B}}$ for the extensivity}{5}{section*.14}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G}Approximations for the strong correlation regime}{5}{section*.15}} \@writefile{toc}{\contentsline {subsection}{\numberline {F}Different types of approximations for the functional}{5}{section*.15}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Definition of the different types of functionals}{5}{section*.16}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Definition of the protocol to design functionals}{5}{section*.16}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Definition of a hierarchy of functionals}{6}{section*.17}}
\newlabel{eq:def_pbeueg}{{27}{6}{}{equation.2.27}{}}
\newlabel{eq:def_pbeueg}{{28}{6}{}{equation.2.28}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{6}{section*.18}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Numerical tests of extensivity}{6}{section*.19}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Dissociation to closed shell ground states}{6}{section*.20}}
\bibdata{srDFT_SCNotes,srDFT_SC} \bibdata{srDFT_SCNotes,srDFT_SC}
\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}} \bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}} \bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}}
@ -115,15 +123,6 @@
\bibcite{EvaDauMal-ChemPhys-83}{{15}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}} \bibcite{EvaDauMal-ChemPhys-83}{{15}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}}
\bibcite{Cim-JCP-1985}{{16}{1985}{{Cimiraglia}}{{}}} \bibcite{Cim-JCP-1985}{{16}{1985}{{Cimiraglia}}{{}}}
\bibcite{Cim-JCC-1987}{{17}{1987}{{Cimiraglia\ and\ Persico}}{{}}} \bibcite{Cim-JCC-1987}{{17}{1987}{{Cimiraglia\ and\ Persico}}{{}}}
\newlabel{eq:def_pbeueg}{{26}{6}{}{equation.2.26}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{6}{section*.17}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Numerical tests of extensivity}{6}{section*.18}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{6}{section*.19}}
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\newlabel{fig:N2_avdz}{{1}{6}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}}
\bibcite{IllRubRic-JCP-88}{{18}{1988}{{Illas, Rubio,\ and\ Ricart}}{{}}} \bibcite{IllRubRic-JCP-88}{{18}{1988}{{Illas, Rubio,\ and\ Ricart}}{{}}}
\bibcite{PovRubIll-TCA-92}{{19}{1992}{{Povill, Rubio,\ and\ Illas}}{{}}} \bibcite{PovRubIll-TCA-92}{{19}{1992}{{Povill, Rubio,\ and\ Illas}}{{}}}
\bibcite{BunCarRam-JCP-06}{{20}{2006}{{Bunge\ and\ Carb{\'o}-Dorca}}{{}}} \bibcite{BunCarRam-JCP-06}{{20}{2006}{{Bunge\ and\ Carb{\'o}-Dorca}}{{}}}
@ -134,12 +133,6 @@
\bibcite{CafGinScemRam-JCTC-14}{{25}{2014}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Giner, Scemama,\ and\ Ram{\'\i }rez-Sol{\'\i }s}}} \bibcite{CafGinScemRam-JCTC-14}{{25}{2014}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Giner, Scemama,\ and\ Ram{\'\i }rez-Sol{\'\i }s}}}
\bibcite{GinSceCaf-JCP-15}{{26}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}} \bibcite{GinSceCaf-JCP-15}{{26}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
\bibcite{CafAplGinScem-arxiv-16}{{27}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} \bibcite{CafAplGinScem-arxiv-16}{{27}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}}
\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$. }}{7}{table.1}}
\newlabel{conv_He_table}{{I}{7}{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$}{table.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.3}}
\newlabel{fig:F2_avdz}{{3}{7}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}}
\bibcite{CafAplGinSce-JCP-16}{{28}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} \bibcite{CafAplGinSce-JCP-16}{{28}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}}
\bibcite{SchEva-JCP-16}{{29}{2016}{{Schriber\ and\ Evangelista}}{{}}} \bibcite{SchEva-JCP-16}{{29}{2016}{{Schriber\ and\ Evangelista}}{{}}}
\bibcite{LiuHofJCTC-16}{{30}{2016}{{Liu\ and\ Hoffmann}}{{}}} \bibcite{LiuHofJCTC-16}{{30}{2016}{{Liu\ and\ Hoffmann}}{{}}}
@ -148,15 +141,23 @@
\bibcite{SchEva-JCTC-17}{{33}{2017}{{Schriber\ and\ Evangelista}}{{}}} \bibcite{SchEva-JCTC-17}{{33}{2017}{{Schriber\ and\ Evangelista}}{{}}}
\bibcite{PerCle-JCP-17}{{34}{2017}{{Per\ and\ Cleland}}{{}}} \bibcite{PerCle-JCP-17}{{34}{2017}{{Per\ and\ Cleland}}{{}}}
\bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}} \bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Dissociation to open shell ground states}{7}{section*.21}}
\newlabel{sec:results}{{III\tmspace +\thinmuskip {.1667em}A\tmspace +\thinmuskip {.1667em}2}{7}{}{table.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{7}{section*.22}}
\newlabel{sec:conclusion}{{IV}{7}{}{section*.22}{}}
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\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the Ne atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system Ne--F$_2$. }}{8}{table.1}}
\newlabel{tab:extensiv_closed}{{I}{8}{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the Ne atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system Ne--F$_2$}{table.1}{}}
\@writefile{lot}{\contentsline {table}{\numberline {II}{\ignorespaces Total energies (in Hartree) for N$_2$ in the aug-cc-pvdz basis set. }}{8}{table.2}}
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\bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}} \bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}}
\bibcite{LiOttHolShaUmr-JCP-2018}{{37}{2018}{{Li\ \emph {et~al.}}}{{Li, Otten, Holmes, Sharma,\ and\ Umrigar}}} \bibcite{LiOttHolShaUmr-JCP-2018}{{37}{2018}{{Li\ \emph {et~al.}}}{{Li, Otten, Holmes, Sharma,\ and\ Umrigar}}}
\bibcite{ChiHolOttUmrShaZim-JPCA-18}{{38}{2018}{{Chien\ \emph {et~al.}}}{{Chien, Holmes, Otten, Umrigar, Sharma,\ and\ Zimmerman}}} \bibcite{ChiHolOttUmrShaZim-JPCA-18}{{38}{2018}{{Chien\ \emph {et~al.}}}{{Chien, Holmes, Otten, Umrigar, Sharma,\ and\ Zimmerman}}}
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\bibcite{SceBenJacCafLoo-JCP-18}{{39}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Benali, Jacquemin, Caffarel,\ and\ Loos}}} \bibcite{SceBenJacCafLoo-JCP-18}{{39}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Benali, Jacquemin, Caffarel,\ and\ Loos}}}
\bibcite{LooSceBloGarCafJac-JCTC-18}{{40}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}} \bibcite{LooSceBloGarCafJac-JCTC-18}{{40}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
\bibcite{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}} \bibcite{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
@ -164,6 +165,12 @@
\bibcite{GarGinMalSce-JCP-16}{{43}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Giner, Malrieu,\ and\ Scemama}}} \bibcite{GarGinMalSce-JCP-16}{{43}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Giner, Malrieu,\ and\ Scemama}}}
\bibcite{LooBogSceCafJac-JCTC-19}{{44}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}} \bibcite{LooBogSceCafJac-JCTC-19}{{44}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
\bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}} \bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{9}{figure.5}}
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\bibcite{Kut-TCA-85}{{46}{1985}{{Kutzelnigg}}{{}}} \bibcite{Kut-TCA-85}{{46}{1985}{{Kutzelnigg}}{{}}}
\bibcite{KutKlo-JCP-91}{{47}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}} \bibcite{KutKlo-JCP-91}{{47}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
\bibcite{NogKut-JCP-94}{{48}{1994}{{Noga\ and\ Kutzelnigg}}{{}}} \bibcite{NogKut-JCP-94}{{48}{1994}{{Noga\ and\ Kutzelnigg}}{{}}}
@ -192,18 +199,19 @@
\bibcite{FerGinTou-JCP-18}{{71}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}} \bibcite{FerGinTou-JCP-18}{{71}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}}
\bibcite{GinPraFerAssSavTou-JCP-18}{{72}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}} \bibcite{GinPraFerAssSavTou-JCP-18}{{72}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}}
\bibcite{LooPraSceTouGin-JCPL-19}{{73}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}} \bibcite{LooPraSceTouGin-JCPL-19}{{73}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}}
\bibcite{TouGorSav-TCA-05}{{74}{2005}{{Toulouse, Gori-Giorgi,\ and\ Savin}}{{}}} \bibcite{GinSceTouLoo-JCP-19}{{74}{2019}{{Giner\ \emph {et~al.}}}{{Giner, Scemama, Toulouse,\ and\ Loos}}}
\bibcite{PerBurErn-PRL-96}{{75}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}} \bibcite{TouGorSav-TCA-05}{{75}{2005}{{Toulouse, Gori-Giorgi,\ and\ Savin}}{{}}}
\bibcite{GoriSav-PRA-06}{{76}{2006{}}{{Gori-Giorgi\ and\ Savin}}{{}}} \bibcite{PerBurErn-PRL-96}{{76}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}}
\bibcite{PazMorGorBac-PRB-06}{{77}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}} \bibcite{GoriSav-PRA-06}{{77}{2006{}}{{Gori-Giorgi\ and\ Savin}}{{}}}
\bibcite{GritMeePer-PRA-18}{{78}{2018}{{Gritsenko, van Meer,\ and\ Pernal}}{{}}} \bibcite{PazMorGorBac-PRB-06}{{78}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}}
\bibcite{CarTruGag-JPCA-17}{{79}{2017}{{Carlson, Truhlar,\ and\ Gagliardi}}{{}}} \bibcite{GritMeePer-PRA-18}{{79}{2018}{{Gritsenko, van Meer,\ and\ Pernal}}{{}}}
\bibcite{GarBulHenScu-PCCP-15}{{80}{2015}{{Garza\ \emph {et~al.}}}{{Garza, Bulik, Henderson,\ and\ Scuseria}}} \bibcite{CarTruGag-JPCA-17}{{80}{2017}{{Carlson, Truhlar,\ and\ Gagliardi}}{{}}}
\bibcite{GorSav-PRA-06}{{81}{2006{}}{{Gori-Giorgi\ and\ Savin}}{{}}} \bibcite{GarBulHenScu-PCCP-15}{{81}{2015}{{Garza\ \emph {et~al.}}}{{Garza, Bulik, Henderson,\ and\ Scuseria}}}
\bibcite{GorSav-PRA-06}{{82}{2006{}}{{Gori-Giorgi\ and\ Savin}}{{}}}
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Manuscript/srDFT_SC.bbl
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@ -6,7 +6,7 @@
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{note} {pMID: 31090432},\ \Eprint {note} {pMID: 31090432},\ \Eprint
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Giner}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}},
\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}}, \ and\
\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}},\ }\href
{\doibase 10.1063/1.5122976} {\bibfield {journal} {\bibinfo {journal} {The
Journal of Chemical Physics}\ }\textbf {\bibinfo {volume} {151}},\ \bibinfo
{pages} {144118} (\bibinfo {year} {2019})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.5122976}
{https://doi.org/10.1063/1.5122976} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\ \bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\
\citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}% \citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}%
\BibitemOpen \BibitemOpen

13
Manuscript/srDFT_SC.bib
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@ -12657,3 +12657,16 @@ eprint = {https://doi.org/10.1021/acs.jpclett.9b01176}
doi = {10.1103/PhysRevA.98.062510}, doi = {10.1103/PhysRevA.98.062510},
url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510} url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510}
} }
@article{GinSceTouLoo-JCP-19,
author = {Giner,Emmanuel and Scemama,Anthony and Toulouse,Julien and Loos,Pierre-François },
title = {Chemically accurate excitation energies with small basis sets},
journal = {The Journal of Chemical Physics},
volume = {151},
number = {14},
pages = {144118},
year = {2019},
doi = {10.1063/1.5122976},
URL = {https://doi.org/10.1063/1.5122976},
eprint = {https://doi.org/10.1063/1.5122976}
}

67
Manuscript/srDFT_SC.blg
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@ -12,7 +12,6 @@ Reallocated wiz_functions (elt_size=4) to 6000 items from 3000.
Database file #1: srDFT_SCNotes.bib Database file #1: srDFT_SCNotes.bib
Database file #2: srDFT_SC.bib Database file #2: srDFT_SC.bib
Warning--I didn't find a database entry for "exicted" Warning--I didn't find a database entry for "exicted"
Warning--I didn't find a database entry for "excited"
Warning--I didn't find a database entry for "kato" Warning--I didn't find a database entry for "kato"
control{REVTEX41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{N/A}, control.pages{N/A}, control.year{N/A}, control.eprint{N/A}, control{REVTEX41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{N/A}, control.pages{N/A}, control.year{N/A}, control.eprint{N/A},
control{aip41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{}, control.pages{0}, control.year{N/A}, control.eprint{N/A}, control{aip41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{}, control.pages{0}, control.year{N/A}, control.eprint{N/A},
@ -27,45 +26,45 @@ Control: page (0) single
Control: year (1) truncated Control: year (1) truncated
Control: production of eprint (0) enabled Control: production of eprint (0) enabled
Warning--missing journal in CafAplGinScem-arxiv-16 Warning--missing journal in CafAplGinScem-arxiv-16
You've used 83 entries, You've used 84 entries,
5918 wiz_defined-function locations, 5918 wiz_defined-function locations,
2192 strings with 31256 characters, 2198 strings with 31484 characters,
and the built_in function-call counts, 85895 in all, are: and the built_in function-call counts, 87081 in all, are:
= -- 5509 = -- 5583
> -- 2814 > -- 2853
< -- 524 < -- 531
+ -- 874 + -- 886
- -- 715 - -- 725
* -- 13242 * -- 13437
:= -- 8845 := -- 8963
add.period$ -- 82 add.period$ -- 83
call.type$ -- 83 call.type$ -- 84
change.case$ -- 327 change.case$ -- 331
chr.to.int$ -- 79 chr.to.int$ -- 80
cite$ -- 84 cite$ -- 85
duplicate$ -- 7640 duplicate$ -- 7745
empty$ -- 6094 empty$ -- 6179
format.name$ -- 1447 format.name$ -- 1468
if$ -- 17038 if$ -- 17274
int.to.chr$ -- 5 int.to.chr$ -- 5
int.to.str$ -- 90 int.to.str$ -- 91
missing$ -- 1009 missing$ -- 1022
newline$ -- 295 newline$ -- 298
num.names$ -- 243 num.names$ -- 246
pop$ -- 3272 pop$ -- 3312
preamble$ -- 1 preamble$ -- 1
purify$ -- 405 purify$ -- 410
quote$ -- 0 quote$ -- 0
skip$ -- 3031 skip$ -- 3072
stack$ -- 0 stack$ -- 0
substring$ -- 2250 substring$ -- 2282
swap$ -- 7428 swap$ -- 7534
text.length$ -- 261 text.length$ -- 265
text.prefix$ -- 0 text.prefix$ -- 0
top$ -- 10 top$ -- 10
type$ -- 1164 type$ -- 1179
warning$ -- 2 warning$ -- 2
while$ -- 322 while$ -- 326
width$ -- 0 width$ -- 0
write$ -- 710 write$ -- 719
(There were 5 warnings) (There were 4 warnings)

19
Manuscript/srDFT_SC.out
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@ -8,12 +8,15 @@
\BOOKMARK [2][-]{section*.8}{Generic form and properties of the approximations for B[n\(r\)] }{section*.4}% 8 \BOOKMARK [2][-]{section*.8}{Generic form and properties of the approximations for B[n\(r\)] }{section*.4}% 8
\BOOKMARK [3][-]{section*.9}{Generic form of the approximated functionals}{section*.8}% 9 \BOOKMARK [3][-]{section*.9}{Generic form of the approximated functionals}{section*.8}% 9
\BOOKMARK [3][-]{section*.10}{Properties of approximated functionals}{section*.8}% 10 \BOOKMARK [3][-]{section*.10}{Properties of approximated functionals}{section*.8}% 10
\BOOKMARK [2][-]{section*.11}{Requirements for the approximated functionals in the strong correlation }{section*.4}% 11 \BOOKMARK [2][-]{section*.11}{Requirements for the approximated functionals in the strong correlation regime}{section*.4}% 11
\BOOKMARK [3][-]{section*.12}{Requirements: separability of the energies and Sz invariance}{section*.11}% 12 \BOOKMARK [3][-]{section*.12}{Requirements: separability of the energies and Sz invariance}{section*.11}% 12
\BOOKMARK [3][-]{section*.13}{Condition for the functional PBEB[n,,s,n\(2\),B] to obtain Sz invariance}{section*.11}% 13 \BOOKMARK [3][-]{section*.13}{Condition for the functional XB[n,,s,n\(2\),B] to obtain Sz invariance}{section*.11}% 13
\BOOKMARK [2][-]{section*.14}{Requirement on B for the extensivity}{section*.4}% 14 \BOOKMARK [3][-]{section*.14}{Conditions on B for the extensivity}{section*.11}% 14
\BOOKMARK [2][-]{section*.15}{Approximations for the strong correlation regime}{section*.4}% 15 \BOOKMARK [2][-]{section*.15}{Different types of approximations for the functional}{section*.4}% 15
\BOOKMARK [3][-]{section*.16}{Definition of the different types of functionals}{section*.15}% 16 \BOOKMARK [3][-]{section*.16}{Definition of the protocol to design functionals}{section*.15}% 16
\BOOKMARK [1][-]{section*.17}{Results}{section*.2}% 17 \BOOKMARK [3][-]{section*.17}{Definition of a hierarchy of functionals}{section*.15}% 17
\BOOKMARK [2][-]{section*.18}{Numerical tests of extensivity}{section*.17}% 18 \BOOKMARK [1][-]{section*.18}{Results}{section*.2}% 18
\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 19 \BOOKMARK [2][-]{section*.19}{Numerical tests of extensivity}{section*.18}% 19
\BOOKMARK [3][-]{section*.20}{Dissociation to closed shell ground states}{section*.19}% 20
\BOOKMARK [3][-]{section*.21}{Dissociation to open shell ground states}{section*.19}% 21
\BOOKMARK [1][-]{section*.22}{Conclusion}{section*.2}% 22

205
Manuscript/srDFT_SC.tex
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@ -66,16 +66,8 @@
\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} \newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} \newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} \newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{PBE}}^\Bas[#1]} \newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{X}}^\Bas[#1]}
\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} \newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
\newcommand{\argepbe}[0]{\den,\xi,s}
\newcommand{\argebasis}[0]{\den,\xi,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argecmd}[0]{\den,\xi,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeuegxihf}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
\newcommand{\argepbeontxihf}[0]{\den,\xi,s,\ntwoextrap,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} \newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} \newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} \newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]}
@ -92,7 +84,36 @@
\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}} \newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}}
\newcommand{\psibasis}[0]{\Psi^{\basis}} \newcommand{\psibasis}[0]{\Psi^{\basis}}
%pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\xi\text{-HF}^\Bas}
\newcommand{\argpbeuegxihf}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\xi(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
%pbeuegxiCAS
\newcommand{\pbeuegxicas}{\text{PBE-UEG-}\xi\text{-CAS}^\Bas}
\newcommand{\argpbeuegxicas}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\xi(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS
\newcommand{\pbeuegXicas}{\text{PBE-UEG-}\Xi\text{-CAS}^\Bas}
\newcommand{\argpbeuegXicas}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegXicas}[0]{\den(\br{}),\Xi(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontxiCAS
\newcommand{\pbeontxicas}{\text{PBE-ONT-}\xi\text{-CAS}^\Bas}
\newcommand{\argpbeontxicas}[0]{\den,\xi,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeontxicas}[0]{\den(\br{}),\xi(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontXiCAS
\newcommand{\pbeontXicas}{\text{PBE-ONT-}\Xi\text{-CAS}^\Bas}
\newcommand{\argpbeontXicas}[0]{\den,\Xi,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeontXicas}[0]{\den(\br{}),\Xi(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%%%%%% arguments
\newcommand{\argepbe}[0]{\den,\xi,s}
\newcommand{\argebasis}[0]{\den,\xi,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argecmd}[0]{\den,\xi,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\xi,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
% numbers % numbers
@ -108,7 +129,9 @@
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} \newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})} \newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
\newcommand{\ntwo}[0]{n^{(2)}} \newcommand{\ntwo}[0]{n^{(2)}}
\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}}
\newcommand{\ntwoextrap}[0]{\tilde{n}^{(2)}_{\psibasis}} \newcommand{\ntwoextrap}[0]{\tilde{n}^{(2)}_{\psibasis}}
\newcommand{\ntwoextrapcas}[0]{\tilde{n}^{(2)\,\basis}_{\text{CAS}}}
\newcommand{\mur}[0]{\mu({\bf r})} \newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})} \newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} \newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
@ -339,7 +362,7 @@ Assuming that the density $\denFCI$ associated to the ground state FCI wave func
\label{eq:e0approx} \label{eq:e0approx}
E_0 = \efci + \efuncbasisFCI E_0 = \efci + \efuncbasisFCI
\end{equation} \end{equation}
where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,excited}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set. where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set.
The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity. The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity.
\subsection{Definition of an effective interaction within $\Bas$} \subsection{Definition of an effective interaction within $\Bas$}
@ -406,8 +429,8 @@ which is fundamental to guarantee the good behaviour of the theory at the CBS li
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ } \subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
\subsubsection{Generic form of the approximated functionals} \subsubsection{Generic form of the approximated functionals}
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}. As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}.
Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, the spin polarisation $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $\ntwo(\br{})$ are be computed from the same wave function $\psibasis$ used to define $\murpsi$. Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, the spin polarisation $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, all the density-related quantities are computed with the same wave function $\psibasis$ used to define $\murpsi$.
The generic form for the approximations to $\efuncden{\denr}$ proposed here reads Therefore, a given approximation X of $\efuncden{\denr}$ have the following generic form
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\label{eq:def_ecmdpbebasis} \label{eq:def_ecmdpbebasis}
@ -424,7 +447,8 @@ with
\label{eq:def_beta} \label{eq:def_beta}
\beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den}, \beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den},
\end{equation} \end{equation}
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. Before introducing the different flavour of approximated functionals that we will use here (see \ref{sec:def_func}), we would like to give some motivations based on physical requirements for the such a choice of functional form.
The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by some of the present authors in the context of RSDFT~\cite{FerGinTou-JCP-18} in order to fulfill the two following limits The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by some of the present authors in the context of RSDFT~\cite{FerGinTou-JCP-18} in order to fulfill the two following limits
\begin{equation} \begin{equation}
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe), \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe),
@ -441,7 +465,7 @@ Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes
\label{eq:lim_n2} \label{eq:lim_n2}
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0 \lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0
\end{equation} \end{equation}
which is exact for systems with vanishing spin density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems. which is exact for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems.
Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$
\begin{equation} \begin{equation}
\label{eq:lim_muinf} \label{eq:lim_muinf}
@ -449,7 +473,7 @@ Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when
\end{equation} \end{equation}
\subsubsection{Properties of approximated functionals} \subsubsection{Properties of approximated functionals}
Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, the approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties. Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties.
Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$: Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$:
\begin{equation} \begin{equation}
\label{eq:lim_ebasis} \label{eq:lim_ebasis}
@ -459,7 +483,7 @@ which guarantees an unaltered limit when reaching the CBS limit.
Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system. Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}). Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}).
\subsection{Requirements for the approximated functionals in the strong correlation } \subsection{Requirements for the approximated functionals in the strong correlation regime}
\subsubsection{Requirements: separability of the energies and $S_z$ invariance} \subsubsection{Requirements: separability of the energies and $S_z$ invariance}
An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is particularly important to avoid any ambiguity in computing interaction energies. An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is particularly important to avoid any ambiguity in computing interaction energies.
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple HF wave function leads to extensive energies. When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple HF wave function leads to extensive energies.
@ -468,88 +492,141 @@ Another important requirement is the independence of the energy with respect to
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components. Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance} \subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which for the $\ecmd(\argecmd)$, is the spin density $s(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}). A possible way to eliminate the $S_z$ dependency would be to simply set $\xi(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$. Therefore, we use the effective spin density depending on the on-top pair density and the total density introduced by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which reads A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\xi(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}). A possible way to eliminate the $S_z$ dependency would be to simply set $\xi(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$.
Therefore, we use the effective spin polarisation introduced by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which depends on the on-top pair density and the total density of a general multi configurational wave function $\psibasis$:
\begin{equation} \begin{equation}
\label{eq:def_effspin} \label{eq:def_effspin}
\Xi(n,\ntwo) = \Xi(n,\ntwo_{\psibasis}) =
\begin{cases} \begin{cases}
\sqrt{ n^2 - 4 \ntwo }. & \text{if $n^2 - 4 \ntwo > 0$,} \\ \sqrt{ n^2 - 4 \ntwo_{\psibasis} } & \text{if $n^2 - 4 \ntwo_{\psibasis} > 0$,} \\
0 & \text{otherwise.} 0 & \text{otherwise.}
\end{cases} \end{cases}
\end{equation} \end{equation}
Such a definition is inspired by the spin density of a single determinant, which has precisely the form of \eqref{eq:def_effspin}. If the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single HF determinant, the definition \eqref{eq:def_effspin} is equivalent to the usual one
With this definition, the $\Xi(n,\ntwo)$ depends only on $S_z$ invariants quantities, which naturally makes it $S_z$ invariant. \begin{equation}
\Xi(n^{\text{HF}},\ntwohf) = n_{\alpha}^{\text{HF}} - n_{\beta}^{\text{HF}},
\end{equation}
but when $n$ and $\ntwo_{\psibasis}$ are obtained from a general multi configurational wave function $\psibasis$, the definition of the usual spin polarisation and the equation \eqref{eq:def_effspin} do not coincide.
As equation \eqref{eq:def_effspin} depends only on $S_z$ invariants quantities, $\Xi$ is therefore $S_z$ invariant.
\subsection{Requirement on $\psibasis$ for the extensivity} \subsubsection{Conditions on $\psibasis$ for the extensivity}
In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ depends only on local quantities, one sufficient condition for the extensivity is that these quantities must be the same on the system $A$ that in the subsystem $A$ of the super system $A+B$ in the limit of non interacting fragments. In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ depends only on local quantities, one sufficient condition for the extensivity is that these quantities are the same on the system $A$ that in the subsystem $A$ of the super system $A\ldots B$ in the limit of non interacting fragments $A$ and $B$.
As all these quantities are properties of the wave function $\psibasis$, the extensivity requires that the wave function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. As $\efuncdenpbe{\argebasis}$ depends only on quantities which are properties of the wave function $\psibasis$, a sufficient condition for the extensivity of these quantities is that the function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$.
In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property, as will be numerically illustrated in section \ref{sec:separability}. In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property, as will be numerically illustrated in section \ref{sec:separability}.
The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments. The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments.
\subsection{Approximations for the strong correlation regime} \subsection{Different types of approximations for the functional}
\subsubsection{Definition of the different types of functionals} \subsubsection{Definition of the protocol to design functionals}
As the present work proposes to investigate the performance of different flavours of functionals by varying different physical ingredients, we propose here a general nomenclature in order to make things easier. \label{sec:def_func}
The functionals $\efuncdenpbe{\argebasis}$ depends on: i) the wave function $\psibasis$ used to determine the $\murpsi$ and the various density related quantities, ii) the flavour of on-top pair density used, iii) the type of spin density used. As the present work proposes to investigate how different physical quantities impact the description of correlation, we propose here a general protocol and a corresponding nomenclature in order to make things as clear as possible.
Therefore, we propose to use the following notations: PBE-"on-top"-"spin-density"-$\psibasis$. %
For instance, within this convention the PBE-UEG-$\xi$-HF is the functional which was introduced in Ref. \onlinecite{LooPraSceTouGin-JCPL-19} and which reads Here we propose to investigate the dependency of the functionals $\efuncdenpbe{\argebasis}$ on: i) the wave function $\psibasis$ used to determine the $\murpsi$ and the various density related quantities, ii) the flavour of on-top pair density used, iii) the type of spin polarisation used.
\begin{equation} Therefore, we propose to use the following notations: PBE-"on-top"-"spin polarisation"-$\psibasis$.
\label{eq:def_pbeueg}
\begin{aligned} Regarding the spin polarisation, we will use two different types of formula: i) the usual spin polarisation $\xi = n_{\alpha} - n_{\beta}$ which \textit{is not} $S_z$ invariant, ii) $\Xi$ defined in equation \eqref{eq:def_effspin} which \textit{is} $S_z$ invariant.
\efuncdenpbe{\argepbeuegxihf} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeuegxihf)
\end{aligned} For the wave function $\psibasis$, we will use either i) a simple RHF/ROHF wave function, ii) a minimal CASSCF leading to additive energies in the case of dissociated covalent bonds.
\end{equation}
where $\ntwo_{\text{UEG}}$ is an approximation of the on-top pair density of the uniform electron gas (UEG) defined as Regarding the approximation to the \textit{exact} on-top pair density, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
\begin{equation} \begin{equation}
\label{eq:def_n2ueg} \label{eq:def_n2ueg}
\ntwo_{\text{UEG}}(n,\xi) = n^2(1-\xi)g_0(n) \ntwo_{\text{UEG}}(n,\xi,\br{}) = n(\br{})^2\big(1-\xi(\br{})\big)g_0\big(n(\br{})\big)
\end{equation} \end{equation}
using the pair-distribution function $g_0(n)$ of equation (46) of Ref. \onlinecite{GorSav-PRA-06}. where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. The approximation of equation \eqref{eq:def_n2ueg} depends on the density and some spin polarisation. Notice that, when using a CASSCF wave function and $\Xi$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density as $\Xi$ depends on the on-top pair density.
The function $\ntwo_{\text{UEG}}$ in an approximation of the \text{exact} on-top pair density based on informations of the UEG.
Therefore, such a functional uses a HF wave function to define; i) the $\murpsi$, ii) the total density, reduced density gradients, regular spin density $\xi$ and uses the UEG-like on-top pair density. Another approach consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
Of course, because of the use of an HF wave function as $\psibasis$, the density related quantities are extensive only in the case of dissociation in closed shell system.
By changing the definition of $\psibasis=\text{HF}$ to $\psibasis=\text{CAS}$ on obtains the PBE-UEG-$\xi$-CAS where all the quantities are computed from a CASSCF wave function. Therfore, the $\murpsi$, density, reduced density gradient, and on-top pair density are extensive in that functional. Nevertheless, the use of the regular spin density $\xi$ leads to non $S_z$ invariance.
One can change the spin density to the effective spin density $\Xi$ to obtain the PBE-UEG-$\Xi$-CAS which is $S_z$ invariant, and therefore this functional will reads to
\begin{equation} \begin{equation}
\label{eq:def_pbeueg} \ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\begin{aligned}
\efuncdenpbe{\argepbeuegXihf} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeuegXihf).
\end{aligned}
\end{equation}
One can also change the flavour of the on-top pair density by taking using the on-top pair density $\ntwo_{\wf{}{\Bas}}(\br{})$ computed with $\psibasis$.
Following the work of some of the previous authors\cite{FerGinTou-JCP-18,excited} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
\begin{equation}
\ntwoextrap(\ntwo,\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\end{equation} \end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06}. which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06}.
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ont". When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ont".
Therefore, one can define the PBE-ont-$\xi$-CAS as
\subsubsection{Definition of a hierarchy of functionals}
Within the convention proposed in the section \ref{sec:def_func}, the PBE-UEG-$\xi$-HF is the functional which was introduced in Ref. \onlinecite{LooPraSceTouGin-JCPL-19} and which reads
\begin{equation} \begin{equation}
\label{eq:def_pbeueg} \label{eq:def_pbeueg}
\begin{aligned} \begin{aligned}
\efuncdenpbe{\argepbeontxihf} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeontxihf). \pbeuegxihf &\equiv \int d\br{} \,\denr \\ & \ecmd\big(\argrpbeuegxihf\big)
\end{aligned} \end{aligned}
\end{equation} \end{equation}
Such a functional can be further improved by using the effective spin density $\Xi$ to give the PBE-ont-$\Xi$-CAS. Therefore, such a functional uses a HF wave function to define; i) the $\murpsi$, ii) the total density, reduced density gradients, usual spin polarisation $\xi$ and uses the UEG-like on-top pair density with the usual spin polarisation $\xi$.
Of course, because of the use of an HF wave function as $\psibasis$, the density related quantities are extensive only in the case of dissociation in closed shell system. Also, one can notice that changing the spin polarisation from $\xi$ to $\Xi$ does not change the results as by definition, $\Xi = \xi$ for a single Slater determinant.
By changing the definition of $\psibasis=\text{HF}$ to $\psibasis=\text{CASSCF}$ on obtains the PBE-UEG-$\xi$-CAS which reads
\begin{equation}
\label{eq:def_pbeueg}
\begin{aligned}
\pbeuegxicas &\equiv \int d\br{} \,\denr \\ & \ecmd\big(\argrpbeuegxicas\big)
\end{aligned}
\end{equation}
where the density, reduced density gradients, usual spin polarisation and UEG on-top pair density are computed from a CASSCF wave function. Therefore, the $\murpsi$, density, reduced density gradient are extensive in the case of dissociated covalent bonding. Nevertheless, the use of the regular spin polarisation $\xi$ leads to non $S_z$ invariance.
One can change the spin polarisation to the effective spin polarisation $\Xi$ to obtain the PBE-UEG-$\Xi$-CAS which is $S_z$ invariant, and therefore this functional will reads to
\begin{equation}
\label{eq:def_pbeueg}
\begin{aligned}
\pbeuegXicas = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeuegXicas).
\end{aligned}
\end{equation}
One can also change the flavour of the on-top pair density by taking advantage of the on-top pair density $\ntwo_{\wf{}{\Bas}}(\br{})$ computed with $\psibasis$.
Therefore, one can define the PBE-ONT-$\xi$-CAS as
\begin{equation}
\label{eq:def_pbeueg}
\begin{aligned}
\pbeontXicas = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeontXicas).
\end{aligned}
\end{equation}
Such a functional can be further improved by using the $S_z$ invariant effective spin polarisation $\Xi$ to give the PBE-ONT-$\Xi$-CAS.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Results}
\subsection{Numerical tests of extensivity} \subsection{Numerical tests of extensivity}
The first numerical results investigated are the numerical tests of extensivity of the various functionals.
As mentioned before, when considering a super system $A+B$ dissociating into non interacting fragments $A\ldots B$, there are two different situations regarding the extensivity of the energy: when the subsystems $A$ and $B$ dissociate in closed or open shell systems.
Therefore, we shall consider two systems $A$ and $B$ and compare the sum of the energies obtained with the super system $A\ldots B$ in the limit of non interactive fragments. The error to additivity for a given method $Y$ is therefore defined as $E_Y(A) + E_Y(B) - E_Y(A\ldots B)$.
\subsubsection{Dissociation to closed shell ground states}
We begin our study by giving numerical evidence for the extensivity of the present basis set correction for systems dissociating in closed shell systems.
In these cases, the use a HF wave function is sufficient to guarantee the extensivity of the basis set correction, and therefore we use the simple $\pbeuegxihf$ functional. The system under study is $A=\text{F}_2$ at experimental equilibrium geometry (F-F=1.411 angstroms) and $B=\text{Ne}$.
We report in table \ref{tab:extensiv_closed} the error to additivity for the HF energy and for $\pbeuegxihf$ using the aug-cc-pvdz basis set and using a He core to define the $\mu_{\text{HF}}^{\basis}(\br{})$ and the frozen core densities.
The numbers in table \ref{tab:extensiv_closed} clearly show that when HF energies are additive, the $\pbeuegxihf$ is also additive.
Also, the error to additivity using the usual spin polarisation $\xi$ and the extrapolated on-top pair density are much lowered compared to that using UEG on-top pair density, highlighting the important role played by the on-top pair density of the CASSCF wave function.
\begin{table*} \begin{table*}
\caption{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$. } \caption{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the Ne atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system Ne--F$_2$. }
\begin{tabular}{lcc} \begin{tabular}{lcc}
%\hline %\hline
System & HF & $E$ \\ System & HF & $\pbeuegxihf$ \\
\hline \hline
F$_2$ & -2.85570466771188 & -0.0112667838948910 \\ Ne & -128.4963497306184 & -0.1039022285466806 \\
He & -198.698792752661 & -0.1596345827582842 \\ F$_2$ & -198.698792752661 & -0.1596345827582842 \\
He $\ldots$ F$_2$ & -201.554497420371 & -0.1709013666531826 \\ Ne $\ldots$ F$_2$ & -201.554497420371 & -0.2635368113049532 \\
\hline \hline
Error to additivity & 1.2 $\times 10^{-12}$ & 7 $\times 10^{-15}$ \\ Error to additivity & 3.4 $\times 10^{-13}$ & 1.1 $\times 10^{-14}$ \\
\end{tabular} \end{tabular}
\label{conv_He_table} \label{tab:extensiv_closed}
\end{table*}
\subsubsection{Dissociation to open shell ground states}
The system studied to investigate the extensivity in the case of dissociation to open shell systems is the completely dissociated N$_2$ molecule which imply the breaking of three covalent bonds.
As the HF wave function does not lead to extensive energy, it is clear that it cannot be used as $\psibasis$ and therefore for N$_2$ we use a minimal valence CASSCF(6,6) involving the three bonding orbitals ($\sigma$, $\pi_x$, $\pi_y$) and corresponding anti-bonding orbitals and a ROHF wave function for the N atom.
The numerical results for the extensivity of the various flavours of functionals are given in table \ref{tab:extensiv_open}. From these numbers, one can clearly notice that only the functionals using the effective spin polarisation $\Xi$ are size extensive, whatever the type of on-top pair density used.
\begin{table*}
\caption{Total energies (in Hartree) for N$_2$ in the aug-cc-pvdz basis set. }
\begin{tabular}{lccccc}
%\hline
System & ROHF/CASSCF(6,6) & $\pbeuegxicas$ & $\pbeuegXicas$ & $\pbeontxicas$ & $\pbeontXicas$ \\
\hline
N & -128.496349730618 & -0.0230740500348705 & -0.0230740500348705 & -0.0247392466968251 & -0.0247392466968251 \\
N$\ldots$N & -198.698792752661 & -0.0691133629633014 & -0.0461481000697329 & -0.0509457188492165 & -0.0494784933936403 \\
\hline
Error to additivity & 1.0 $\times 10^{-13}$ & 0.02296 & 8.0 $\times 10^{-15}$ & 0.0015 & 9.9 $\times 10^{-15}$ \\
\end{tabular}
\label{tab:extensiv_open}
\end{table*} \end{table*}
\label{sec:results} \label{sec:results}

168
calcs/extensivity/n.ezfio.DFT.out
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@ -1,4 +1,4 @@
Date: 13/10/2019 22:24:33 Date: 16/10/2019 11:59:09
=============== ===============
Quantum Package Quantum Package
=============== ===============
@ -12,28 +12,28 @@ EZFIO Dir : n.ezfio
Task server running : tcp://127.0.1.1:41279 Task server running : tcp://127.0.1.1:41279
.. >>>>> [ IO READ: no_core_density ] <<<<< .. .. >>>>> [ IO READ: no_core_density ] <<<<< ..
.. >>>>> [ RES MEM : 0.004505 GB ] [ VIRT MEM : 0.041687 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004578 GB ] [ VIRT MEM : 0.041687 GB ] <<<<< ..
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.. >>>>> [ IO READ: on_top_from_cas ] <<<<< .. .. >>>>> [ IO READ: on_top_from_cas ] <<<<< ..
.. >>>>> [ RES MEM : 0.004765 GB ] [ VIRT MEM : 0.041687 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004898 GB ] [ VIRT MEM : 0.041687 GB ] <<<<< ..
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.. >>>>> [ IO READ: mu_of_r_potential ] <<<<< .. .. >>>>> [ IO READ: mu_of_r_potential ] <<<<< ..
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.. >>>>> [ WALL TIME: 0.000553 s ] [ CPU TIME: 0.002147 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.000886 s ] [ CPU TIME: 0.002369 s ] <<<<< ..
.. >>>>> [ IO READ: read_wf ] <<<<< .. .. >>>>> [ IO READ: read_wf ] <<<<< ..
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.. >>>>> [ IO READ: mu_of_r_functional ] <<<<< .. .. >>>>> [ IO READ: mu_of_r_functional ] <<<<< ..
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LDA, PBE and PBE-on-top / mu(r) PSI coallescence with frozen core interaction LDA, PBE and PBE-on-top / mu(r) PSI coallescence with frozen core interaction
**************************************** ****************************************
@ -43,27 +43,27 @@ Task server running : tcp://127.0.1.1:41279
MR DFT energy with pure correlation part for the DFT MR DFT energy with pure correlation part for the DFT
.. >>>>> [ IO READ: grid_type_sgn ] <<<<< .. .. >>>>> [ IO READ: grid_type_sgn ] <<<<< ..
.. >>>>> [ RES MEM : 0.004841 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004974 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
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.. >>>>> [ IO READ: nucl_num ] <<<<< .. .. >>>>> [ IO READ: nucl_num ] <<<<< ..
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Nuclear Coordinates (Angstroms) Nuclear Coordinates (Angstroms)
@ -77,24 +77,24 @@ N 7.000000 0.000000 0.000000 0.000000
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n_points_final_grid = 22046 n_points_final_grid = 22046
n max point = 22348 n max point = 22348
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providing the mu_of_r ... providing the mu_of_r ...
* mo_num 23 * mo_num 23
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* Number of active MOs 4 * Number of active MOs 22
* Number of core MOs 1 * Number of core MOs 1
* Number of inactive MOs 0 * Number of inactive MOs 0
* mo_label Canonical * mo_label Canonical
@ -104,136 +104,134 @@ N 7.000000 0.000000 0.000000 0.000000
* N_int 1 * N_int 1
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Read mo_coef Read mo_coef
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* Number of unique alpha determinants 1 * Number of unique alpha determinants 1
* Number of unique beta determinants 1 * Number of unique beta determinants 1
core_inact_act_two_bod_alpha_beta_mo provided in 7.2065409985953011E-003 core_inact_act_two_bod_alpha_beta_mo provided in 3.6840269995082053E-003
Core MOs: Core MOs:
1 1
USING THE VALENCE ONLY TWO BODY DENSITY USING THE VALENCE ONLY TWO BODY DENSITY
providing core_inact_act_two_bod_alpha_beta_mo_physicist ... providing core_inact_act_two_bod_alpha_beta_mo_physicist ...
core_inact_act_two_bod_alpha_beta_mo_physicist provided in 1.6240010154433548E-006 core_inact_act_two_bod_alpha_beta_mo_physicist provided in 9.4613800047227414E-004
providing the core_inact_act_on_top_of_r providing the core_inact_act_on_top_of_r
.. >>>>> [ IO READ: ao_prim_num ] <<<<< .. .. >>>>> [ IO READ: ao_prim_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.010159 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.014290 GB ] [ VIRT MEM : 0.279480 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.018106 s ] [ CPU TIME: 0.057134 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.021762 s ] [ CPU TIME: 0.059488 s ] <<<<< ..
.. >>>>> [ IO READ: ao_expo ] <<<<< .. .. >>>>> [ IO READ: ao_expo ] <<<<< ..
.. >>>>> [ RES MEM : 0.010159 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.014290 GB ] [ VIRT MEM : 0.279480 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.018569 s ] [ CPU TIME: 0.057815 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.022218 s ] [ CPU TIME: 0.068037 s ] <<<<< ..
.. >>>>> [ IO READ: ao_coef ] <<<<< .. .. >>>>> [ IO READ: ao_coef ] <<<<< ..
.. >>>>> [ RES MEM : 0.010159 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.014290 GB ] [ VIRT MEM : 0.279480 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.019022 s ] [ CPU TIME: 0.059222 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.025070 s ] [ CPU TIME: 0.073913 s ] <<<<< ..
.. >>>>> [ IO READ: ao_power ] <<<<< .. .. >>>>> [ IO READ: ao_power ] <<<<< ..
.. >>>>> [ RES MEM : 0.010159 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.014290 GB ] [ VIRT MEM : 0.279480 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.019376 s ] [ CPU TIME: 0.060144 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.029855 s ] [ CPU TIME: 0.086789 s ] <<<<< ..
.. >>>>> [ IO READ: ao_nucl ] <<<<< .. .. >>>>> [ IO READ: ao_nucl ] <<<<< ..
.. >>>>> [ RES MEM : 0.010159 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.014694 GB ] [ VIRT MEM : 0.279480 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.019883 s ] [ CPU TIME: 0.061265 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.032773 s ] [ CPU TIME: 0.100059 s ] <<<<< ..
mo_num,n_points_final_grid 23 22046 mo_num,n_points_final_grid 23 22046
* Number of virtual MOs 18 * Number of virtual MOs 0
* Number of deleted MOs 0 * Number of deleted MOs 0
Active MOs: Active MOs:
2 3 4 5 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Virtual MOs:
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Core, Inactive and Active MOs: Core, Inactive and Active MOs:
1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
provided the core_inact_act_on_top_of_r provided the core_inact_act_on_top_of_r
Time to provide : 4.1155614999297541E-002 Time to provide : 7.5939346880004450
MO map initialized: 38226 MO map initialized: 38226
.. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< .. .. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.057476 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.076382 GB ] [ VIRT MEM : 0.415508 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191075 s ] [ CPU TIME: 0.400365 s ] <<<<< .. .. >>>>> [ WALL TIME: 7.788660 s ] [ CPU TIME: 30.008055 s ] <<<<< ..
.. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< .. .. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.057476 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.076382 GB ] [ VIRT MEM : 0.415508 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191326 s ] [ CPU TIME: 0.400910 s ] <<<<< .. .. >>>>> [ WALL TIME: 7.788863 s ] [ CPU TIME: 30.008745 s ] <<<<< ..
AO map initialized : 52975 AO map initialized : 52975
.. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< .. .. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.057476 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.076382 GB ] [ VIRT MEM : 0.415508 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191596 s ] [ CPU TIME: 0.401487 s ] <<<<< .. .. >>>>> [ WALL TIME: 7.789088 s ] [ CPU TIME: 30.009449 s ] <<<<< ..
Providing the AO integrals Providing the AO integrals
Sorting the map Sorting the map
AO integrals provided: AO integrals provided:
Size of AO map : 8.5845947265625000E-002 MB Size of AO map : 8.5845947265625000E-002 MB
Number of AO integrals : 7966 Number of AO integrals : 7966
cpu time : 0.13432800000000000 s cpu time : 0.10205699999999851 s
wall time : 8.3950192001793766E-002 s ( x 1.6000916352535539 ) wall time : 6.4726441999482631E-002 s ( x 1.5767435509712440 )
AO -> MO integrals transformation AO -> MO integrals transformation
--------------------------------- ---------------------------------
.. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< .. .. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.060123 GB ] [ VIRT MEM : 0.524979 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.078991 GB ] [ VIRT MEM : 0.626755 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.276098 s ] [ CPU TIME: 0.536419 s ] <<<<< .. .. >>>>> [ WALL TIME: 7.854217 s ] [ CPU TIME: 30.112367 s ] <<<<< ..
Buffers : 0.459625244 MB / core Buffers : 0.459625244 MB / core
Molecular integrals provided: Molecular integrals provided:
Size of MO map 0.36037063598632812 MB Size of MO map 0.36027526855468750 MB
Number of MO integrals: 18472 Number of MO integrals: 18894
cpu time : 0.18962299999999999 s cpu time : 0.19265299999999996 s
wall time : 5.7800987000518944E-002 s ( x 3.2806187202010499 ) wall time : 6.1197967000225617E-002 s ( x 3.1480294108346722 )
Providing core_inact_act_V_kl_contracted_transposed ..... Providing core_inact_act_V_kl_contracted_transposed .....
Time to provide core_inact_act_V_kl_contracted_transposed = 0.10545316999923671 Time to provide core_inact_act_V_kl_contracted_transposed = 1.6614129850004247
Providing core_inact_act_rho2_kl_contracted_transposed ..... Providing core_inact_act_rho2_kl_contracted_transposed .....
Time to provide core_inact_act_rho2_kl_contracted_transposed = 1.6968988002190599E-002 Time to provide core_inact_act_rho2_kl_contracted_transposed = 1.6492683180003951
Providing core_inact_act_f_psi_ab ..... Providing core_inact_act_f_psi_ab .....
Time to provide core_inact_act_f_psi_ab = 3.2415269997727592E-003 Time to provide core_inact_act_f_psi_ab = 1.2572189999445982E-002
providing the cas_full_mu_of_r_psi_coal_vector ... providing the cas_full_mu_of_r_psi_coal_vector ...
Time to provide cas_full_mu_of_r_psi_coal_vector = 5.8083600015379488E-004 Time to provide cas_full_mu_of_r_psi_coal_vector = 1.4178099991113413E-004
Time to provide mu_of_r = 0.47625986300045042 Time to provide mu_of_r = 11.600998191999679
Providing Energy_c_md_n_and_PBE_mu_of_r ... Providing Energy_c_md_n_and_PBE_mu_of_r ...
.. >>>>> [ IO READ: density_for_dft ] <<<<< .. .. >>>>> [ IO READ: density_for_dft ] <<<<< ..
.. >>>>> [ RES MEM : 0.083530 GB ] [ VIRT MEM : 0.658264 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.264374 GB ] [ VIRT MEM : 0.841324 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.484660 s ] [ CPU TIME: 1.212247 s ] <<<<< .. .. >>>>> [ WALL TIME: 11.612689 s ] [ CPU TIME: 43.103987 s ] <<<<< ..
.. >>>>> [ IO READ: normalize_dm ] <<<<< .. .. >>>>> [ IO READ: normalize_dm ] <<<<< ..
.. >>>>> [ RES MEM : 0.083530 GB ] [ VIRT MEM : 0.658264 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.264374 GB ] [ VIRT MEM : 0.841324 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.484885 s ] [ CPU TIME: 1.212841 s ] <<<<< .. .. >>>>> [ WALL TIME: 11.613163 s ] [ CPU TIME: 43.107534 s ] <<<<< ..
Time for the Energy_c_md_n_and_PBE_mu_of_r : 0.11964879900187952 Time for the Energy_c_md_n_and_PBE_mu_of_r : 0.12555567500021425
Providing Energy_c_md_LDA_mu_of_r ... Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_LDA_mu_of_r : 2.9454646002704976E-002 Time for Energy_c_md_LDA_mu_of_r : 2.6958546000059869E-002
Providing Energy_c_md_LDA_mu_of_r ... Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_n_and_LDA_mu_of_r : 2.9101256001013098E-002 Time for Energy_c_md_n_and_LDA_mu_of_r : 2.6924847999907797E-002
Providing Energy_c_md_n_and_on_top_PBE_mu_of_r ... Providing Energy_c_md_n_and_on_top_PBE_mu_of_r ...
Time for the Energy_c_md_n_and_on_top_PBE_mu_of_r : 6.3777867999306181E-002 Time for the Energy_c_md_n_and_on_top_PBE_mu_of_r : 6.1856046000684728E-002
.. >>>>> [ IO READ: ontop_approx ] <<<<< .. .. >>>>> [ IO READ: ontop_approx ] <<<<< ..
.. >>>>> [ RES MEM : 0.084538 GB ] [ VIRT MEM : 0.659248 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.268711 GB ] [ VIRT MEM : 0.843632 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.773004 s ] [ CPU TIME: 1.541545 s ] <<<<< .. .. >>>>> [ WALL TIME: 11.897411 s ] [ CPU TIME: 43.435523 s ] <<<<< ..
Inactive MOs: Inactive MOs:
@ -245,11 +243,11 @@ N 7.000000 0.000000 0.000000 0.000000
ispin = 3 ispin = 3
USING THE VALENCE ONLY TWO BODY DENSITY USING THE VALENCE ONLY TWO BODY DENSITY
provided the core_inact_act_on_top_of_r_new provided the core_inact_act_on_top_of_r_new
Time to provide : 3.5796539996226784E-003 Time to provide : 1.2705452380005227
Providing Energy_c_md_mu_of_r_PBE_on_top ... Providing Energy_c_md_mu_of_r_PBE_on_top ...
Time for the Energy_c_md_on_top_PBE_mu_of_r: 0.27439229000083287 Time for the Energy_c_md_on_top_PBE_mu_of_r: 0.27205234399934852
Providing Energy_c_md_PBE_mu_of_r ... Providing Energy_c_md_PBE_mu_of_r ...
Time for the Energy_c_md_PBE_mu_of_r: 6.7258312999911141E-002 Time for the Energy_c_md_PBE_mu_of_r: 6.1262960999556526E-002
Corrections using Multi determinant mu Corrections using Multi determinant mu
@ -266,5 +264,5 @@ N 7.000000 0.000000 0.000000 0.000000
ECMD PBE/ontop effective spin dens = -0.0247392466968251 ECMD PBE/ontop effective spin dens = -0.0247392466968251
mu_average for basis set = 0.9116337460 mu_average for basis set = 0.9116337460
Wall time: 0:00:02 Wall time: 0:00:14

158
calcs/extensivity/n2.ezfio.DFT.out
View File

@ -1,4 +1,4 @@
Date: 13/10/2019 22:24:49 Date: 16/10/2019 11:56:08
=============== ===============
Quantum Package Quantum Package
=============== ===============
@ -12,28 +12,28 @@ EZFIO Dir : n2.ezfio
Task server running : tcp://127.0.1.1:41279 Task server running : tcp://127.0.1.1:41279
.. >>>>> [ IO READ: no_core_density ] <<<<< .. .. >>>>> [ IO READ: no_core_density ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000193 s ] [ CPU TIME: 0.001977 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.002034 s ] [ CPU TIME: 0.003212 s ] <<<<< ..
.. >>>>> [ IO READ: on_top_from_cas ] <<<<< .. .. >>>>> [ IO READ: on_top_from_cas ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000368 s ] [ CPU TIME: 0.002092 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.002577 s ] [ CPU TIME: 0.003531 s ] <<<<< ..
.. >>>>> [ IO READ: mu_of_r_potential ] <<<<< .. .. >>>>> [ IO READ: mu_of_r_potential ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000533 s ] [ CPU TIME: 0.002196 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.003005 s ] [ CPU TIME: 0.003764 s ] <<<<< ..
.. >>>>> [ IO READ: read_wf ] <<<<< .. .. >>>>> [ IO READ: read_wf ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000696 s ] [ CPU TIME: 0.002298 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.003391 s ] [ CPU TIME: 0.003967 s ] <<<<< ..
.. >>>>> [ IO READ: mu_of_r_functional ] <<<<< .. .. >>>>> [ IO READ: mu_of_r_functional ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000880 s ] [ CPU TIME: 0.002422 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.003801 s ] [ CPU TIME: 0.004181 s ] <<<<< ..
LDA, PBE and PBE-on-top / mu(r) PSI coallescence with frozen core interaction LDA, PBE and PBE-on-top / mu(r) PSI coallescence with frozen core interaction
**************************************** ****************************************
@ -43,27 +43,27 @@ Task server running : tcp://127.0.1.1:41279
MR DFT energy with pure correlation part for the DFT MR DFT energy with pure correlation part for the DFT
.. >>>>> [ IO READ: grid_type_sgn ] <<<<< .. .. >>>>> [ IO READ: grid_type_sgn ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001067 s ] [ CPU TIME: 0.002548 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.006199 s ] [ CPU TIME: 0.005540 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_num ] <<<<< .. .. >>>>> [ IO READ: nucl_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.004436 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.004402 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001253 s ] [ CPU TIME: 0.002673 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.008730 s ] [ CPU TIME: 0.007199 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_charge ] <<<<< .. .. >>>>> [ IO READ: nucl_charge ] <<<<< ..
.. >>>>> [ RES MEM : 0.005276 GB ] [ VIRT MEM : 0.127628 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.005180 GB ] [ VIRT MEM : 0.127628 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.006438 s ] [ CPU TIME: 0.014247 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.015577 s ] [ CPU TIME: 0.024745 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_label ] <<<<< .. .. >>>>> [ IO READ: nucl_label ] <<<<< ..
.. >>>>> [ RES MEM : 0.005276 GB ] [ VIRT MEM : 0.127628 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.005180 GB ] [ VIRT MEM : 0.127628 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.006731 s ] [ CPU TIME: 0.015202 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.015887 s ] [ CPU TIME: 0.025467 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.005276 GB ] [ VIRT MEM : 0.252628 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.005180 GB ] [ VIRT MEM : 0.252628 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.007020 s ] [ CPU TIME: 0.015901 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.016192 s ] [ CPU TIME: 0.026211 s ] <<<<< ..
Nuclear Coordinates (Angstroms) Nuclear Coordinates (Angstroms)
@ -78,22 +78,22 @@ N 7.000000 0.000000 0.000000 1000.000072
.. >>>>> [ IO READ: thresh_grid ] <<<<< .. .. >>>>> [ IO READ: thresh_grid ] <<<<< ..
.. >>>>> [ RES MEM : 0.007256 GB ] [ VIRT MEM : 0.254322 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.007378 GB ] [ VIRT MEM : 0.254322 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.018087 s ] [ CPU TIME: 0.054428 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.031278 s ] [ CPU TIME: 0.071872 s ] <<<<< ..
n_points_final_grid = 44092 n_points_final_grid = 44092
n max point = 44998 n max point = 44998
.. >>>>> [ IO READ: n_states ] <<<<< .. .. >>>>> [ IO READ: n_states ] <<<<< ..
.. >>>>> [ RES MEM : 0.007256 GB ] [ VIRT MEM : 0.254322 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.007378 GB ] [ VIRT MEM : 0.254322 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.018397 s ] [ CPU TIME: 0.057538 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.031616 s ] [ CPU TIME: 0.076421 s ] <<<<< ..
providing the mu_of_r ... providing the mu_of_r ...
* mo_num 46 * mo_num 46
.. >>>>> [ IO READ: mo_class ] <<<<< .. .. >>>>> [ IO READ: mo_class ] <<<<< ..
.. >>>>> [ RES MEM : 0.009270 GB ] [ VIRT MEM : 0.256981 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.009392 GB ] [ VIRT MEM : 0.256981 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.019785 s ] [ CPU TIME: 0.061670 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.033638 s ] [ CPU TIME: 0.081802 s ] <<<<< ..
* Number of active MOs 8 * Number of active MOs 8
* Number of core MOs 2 * Number of core MOs 2
@ -106,54 +106,54 @@ N 7.000000 0.000000 0.000000 1000.000072
* N_int 1 * N_int 1
.. >>>>> [ IO READ: ao_num ] <<<<< .. .. >>>>> [ IO READ: ao_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.010387 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.010250 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.020953 s ] [ CPU TIME: 0.064729 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.035617 s ] [ CPU TIME: 0.088608 s ] <<<<< ..
Read mo_coef Read mo_coef
.. >>>>> [ IO READ: elec_beta_num ] <<<<< .. .. >>>>> [ IO READ: elec_beta_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.010387 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.010250 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.022999 s ] [ CPU TIME: 0.074050 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.037877 s ] [ CPU TIME: 0.097141 s ] <<<<< ..
.. >>>>> [ IO READ: elec_alpha_num ] <<<<< .. .. >>>>> [ IO READ: elec_alpha_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.010387 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.010250 GB ] [ VIRT MEM : 0.257874 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.023209 s ] [ CPU TIME: 0.074364 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.038110 s ] [ CPU TIME: 0.097643 s ] <<<<< ..
Read psi_det Read psi_det
* Number of unique alpha determinants 20 * Number of unique alpha determinants 20
* Number of unique beta determinants 20 * Number of unique beta determinants 20
core_inact_act_two_bod_alpha_beta_mo provided in 1.0138563997315941E-002 core_inact_act_two_bod_alpha_beta_mo provided in 8.8323440004387521E-003
Core MOs: Core MOs:
1 2 1 2
USING THE VALENCE ONLY TWO BODY DENSITY USING THE VALENCE ONLY TWO BODY DENSITY
providing core_inact_act_two_bod_alpha_beta_mo_physicist ... providing core_inact_act_two_bod_alpha_beta_mo_physicist ...
core_inact_act_two_bod_alpha_beta_mo_physicist provided in 3.8632999348919839E-005 core_inact_act_two_bod_alpha_beta_mo_physicist provided in 1.4993999684520531E-005
providing the core_inact_act_on_top_of_r providing the core_inact_act_on_top_of_r
.. >>>>> [ IO READ: ao_prim_num ] <<<<< .. .. >>>>> [ IO READ: ao_prim_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.012398 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.012318 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.032296 s ] [ CPU TIME: 0.104271 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.044121 s ] [ CPU TIME: 0.119986 s ] <<<<< ..
.. >>>>> [ IO READ: ao_expo ] <<<<< .. .. >>>>> [ IO READ: ao_expo ] <<<<< ..
.. >>>>> [ RES MEM : 0.012398 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.012318 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.033598 s ] [ CPU TIME: 0.110293 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.044656 s ] [ CPU TIME: 0.121388 s ] <<<<< ..
.. >>>>> [ IO READ: ao_coef ] <<<<< .. .. >>>>> [ IO READ: ao_coef ] <<<<< ..
.. >>>>> [ RES MEM : 0.012398 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.012318 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.034755 s ] [ CPU TIME: 0.114448 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.045176 s ] [ CPU TIME: 0.122740 s ] <<<<< ..
.. >>>>> [ IO READ: ao_power ] <<<<< .. .. >>>>> [ IO READ: ao_power ] <<<<< ..
.. >>>>> [ RES MEM : 0.012398 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.012318 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.035587 s ] [ CPU TIME: 0.117060 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.045528 s ] [ CPU TIME: 0.123594 s ] <<<<< ..
.. >>>>> [ IO READ: ao_nucl ] <<<<< .. .. >>>>> [ IO READ: ao_nucl ] <<<<< ..
.. >>>>> [ RES MEM : 0.012398 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.012318 GB ] [ VIRT MEM : 0.315315 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.036767 s ] [ CPU TIME: 0.120070 s ] <<<<< .. .. >>>>> [ WALL TIME: 0.046367 s ] [ CPU TIME: 0.125622 s ] <<<<< ..
mo_num,n_points_final_grid 46 44092 mo_num,n_points_final_grid 46 44092
* Number of virtual MOs 36 * Number of virtual MOs 36
@ -166,77 +166,77 @@ N 7.000000 0.000000 0.000000 1000.000072
Core, Inactive and Active MOs: Core, Inactive and Active MOs:
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
provided the core_inact_act_on_top_of_r provided the core_inact_act_on_top_of_r
Time to provide : 0.70096241300052498 Time to provide : 0.70333999699960259
MO map initialized: 584821 MO map initialized: 584821
.. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< .. .. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.200268 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.200497 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 1.265357 s ] [ CPU TIME: 3.473769 s ] <<<<< .. .. >>>>> [ WALL TIME: 1.286127 s ] [ CPU TIME: 3.479892 s ] <<<<< ..
.. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< .. .. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.200268 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.200497 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 1.265604 s ] [ CPU TIME: 3.474068 s ] <<<<< .. .. >>>>> [ WALL TIME: 1.289062 s ] [ CPU TIME: 3.489111 s ] <<<<< ..
AO map initialized : 813450 AO map initialized : 813450
.. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< .. .. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.200573 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.200497 GB ] [ VIRT MEM : 0.485504 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 1.265888 s ] [ CPU TIME: 3.474393 s ] <<<<< .. .. >>>>> [ WALL TIME: 1.289537 s ] [ CPU TIME: 3.489272 s ] <<<<< ..
Providing the AO integrals Providing the AO integrals
Sorting the map Sorting the map
AO integrals provided: AO integrals provided:
Size of AO map : 0.44347000122070312 MB Size of AO map : 0.44347000122070312 MB
Number of AO integrals : 38257 Number of AO integrals : 38257
cpu time : 1.0639589999999997 s cpu time : 1.0522440000000004 s
wall time : 0.41615004599952954 s ( x 2.5566715905185853 ) wall time : 0.40554916499968385 s ( x 2.5946151313141543 )
AO -> MO integrals transformation AO -> MO integrals transformation
--------------------------------- ---------------------------------
.. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< .. .. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.203552 GB ] [ VIRT MEM : 0.697113 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.203278 GB ] [ VIRT MEM : 0.697235 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 1.682451 s ] [ CPU TIME: 4.539011 s ] <<<<< .. .. >>>>> [ WALL TIME: 1.695592 s ] [ CPU TIME: 4.542076 s ] <<<<< ..
Buffers : 3.63958740 MB / core Buffers : 3.63958740 MB / core
Molecular integrals provided: Molecular integrals provided:
Size of MO map 9.6878700256347656 MB Size of MO map 9.6795730590820312 MB
Number of MO integrals: 473042 Number of MO integrals: 469580
cpu time : 2.2190269999999996 s cpu time : 2.2609360000000001 s
wall time : 0.80212339799982146 s ( x 2.7664409310754126 ) wall time : 0.81856668300042656 s ( x 2.7620669726168439 )
Providing core_inact_act_V_kl_contracted_transposed ..... Providing core_inact_act_V_kl_contracted_transposed .....
Time to provide core_inact_act_V_kl_contracted_transposed = 1.7762000410002656 Time to provide core_inact_act_V_kl_contracted_transposed = 1.7736809010002617
Providing core_inact_act_rho2_kl_contracted_transposed ..... Providing core_inact_act_rho2_kl_contracted_transposed .....
Time to provide core_inact_act_rho2_kl_contracted_transposed = 0.22741315399980522 Time to provide core_inact_act_rho2_kl_contracted_transposed = 0.25144742500015127
Providing core_inact_act_f_psi_ab ..... Providing core_inact_act_f_psi_ab .....
Time to provide core_inact_act_f_psi_ab = 4.9875350014190190E-003 Time to provide core_inact_act_f_psi_ab = 4.9547449998499360E-003
providing the cas_full_mu_of_r_psi_coal_vector ... providing the cas_full_mu_of_r_psi_coal_vector ...
Time to provide cas_full_mu_of_r_psi_coal_vector = 1.9037699894397520E-004 Time to provide cas_full_mu_of_r_psi_coal_vector = 2.6400199931231327E-004
Time to provide mu_of_r = 4.6907960499993351 Time to provide mu_of_r = 4.7140897310000582
Providing Energy_c_md_n_and_PBE_mu_of_r ... Providing Energy_c_md_n_and_PBE_mu_of_r ...
.. >>>>> [ IO READ: density_for_dft ] <<<<< .. .. >>>>> [ IO READ: density_for_dft ] <<<<< ..
.. >>>>> [ RES MEM : 0.352638 GB ] [ VIRT MEM : 0.860363 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.349529 GB ] [ VIRT MEM : 0.868500 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 4.710455 s ] [ CPU TIME: 14.149410 s ] <<<<< .. .. >>>>> [ WALL TIME: 4.746935 s ] [ CPU TIME: 14.221149 s ] <<<<< ..
.. >>>>> [ IO READ: normalize_dm ] <<<<< .. .. >>>>> [ IO READ: normalize_dm ] <<<<< ..
.. >>>>> [ RES MEM : 0.352638 GB ] [ VIRT MEM : 0.860363 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.349529 GB ] [ VIRT MEM : 0.868500 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 4.710757 s ] [ CPU TIME: 14.150161 s ] <<<<< .. .. >>>>> [ WALL TIME: 4.747532 s ] [ CPU TIME: 14.229911 s ] <<<<< ..
Time for the Energy_c_md_n_and_PBE_mu_of_r : 0.37609190000148374 Time for the Energy_c_md_n_and_PBE_mu_of_r : 0.38293596199946478
Providing Energy_c_md_LDA_mu_of_r ... Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_LDA_mu_of_r : 5.3449625997018302E-002 Time for Energy_c_md_LDA_mu_of_r : 5.3972198999872489E-002
Providing Energy_c_md_LDA_mu_of_r ... Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_n_and_LDA_mu_of_r : 5.2238627999031451E-002 Time for Energy_c_md_n_and_LDA_mu_of_r : 5.2259051999499206E-002
Providing Energy_c_md_n_and_on_top_PBE_mu_of_r ... Providing Energy_c_md_n_and_on_top_PBE_mu_of_r ...
Time for the Energy_c_md_n_and_on_top_PBE_mu_of_r : 0.12120974099889281 Time for the Energy_c_md_n_and_on_top_PBE_mu_of_r : 0.12262305900003412
.. >>>>> [ IO READ: ontop_approx ] <<<<< .. .. >>>>> [ IO READ: ontop_approx ] <<<<< ..
.. >>>>> [ RES MEM : 0.361450 GB ] [ VIRT MEM : 0.864967 GB ] <<<<< .. .. >>>>> [ RES MEM : 0.358109 GB ] [ VIRT MEM : 0.873108 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 5.529283 s ] [ CPU TIME: 15.013616 s ] <<<<< .. .. >>>>> [ WALL TIME: 5.576314 s ] [ CPU TIME: 15.102221 s ] <<<<< ..
Inactive MOs: Inactive MOs:
@ -248,11 +248,11 @@ N 7.000000 0.000000 0.000000 1000.000072
ispin = 3 ispin = 3
USING THE VALENCE ONLY TWO BODY DENSITY USING THE VALENCE ONLY TWO BODY DENSITY
provided the core_inact_act_on_top_of_r_new provided the core_inact_act_on_top_of_r_new
Time to provide : 9.8473186000774149E-002 Time to provide : 0.10430047200043191
Providing Energy_c_md_mu_of_r_PBE_on_top ... Providing Energy_c_md_mu_of_r_PBE_on_top ...
Time for the Energy_c_md_on_top_PBE_mu_of_r: 0.95842244199957349 Time for the Energy_c_md_on_top_PBE_mu_of_r: 0.96463285200024984
Providing Energy_c_md_PBE_mu_of_r ... Providing Energy_c_md_PBE_mu_of_r ...
Time for the Energy_c_md_PBE_mu_of_r: 0.12124202900304226 Time for the Energy_c_md_PBE_mu_of_r: 0.12067553599990788
Corrections using Multi determinant mu Corrections using Multi determinant mu
@ -260,7 +260,7 @@ N 7.000000 0.000000 0.000000 1000.000072
ECMD LDA regular spin dens = -0.0678669003007728 ECMD LDA regular spin dens = -0.0678669003007728
ECMD LDA effective spin dens = -0.0530814198896590 ECMD LDA effective spin dens = -0.0530814198896590
ECMD PBE regular spin dens = -0.0691133629633015 ECMD PBE regular spin dens = -0.0691133629633014
ECMD PBE effective spin dens = -0.0461481000697329 ECMD PBE effective spin dens = -0.0461481000697329
Functionals with extrapolated exact ontop based on current wave function Functionals with extrapolated exact ontop based on current wave function

18
calcs/extensivity/scf_energies
View File

@ -4,6 +4,18 @@
-198.698792752661 -0.1596345827582842 -198.698792752661 -0.1596345827582842
-201.554497420371 -0.1709013666531826 -201.554497420371 -0.1709013666531826
1.9 10^-12 7 10^-15 1.9 10^-12 7 10^-15
# N2 # F2 + Ne
-54.3898707291144 # HF
-108.779741458229 -128.4963497306184 -0.1039022285466806
-198.698792752661 -0.1596345827582842
-327.195142483279 -0.2635368113049532
3.4 10^-13 1.1 10^-14
# N2 ROHF PBE-UEG-xi-ROHF PBE-ONT-xi-ROHF
-54.3898707291144 -0.0230740500348705 -0.0247392466968251
-108.1790794412006 -0.0715505833610001
-0.60066 0.0254024832912591
# CASSCF PBE-UEG-xi-CAS PBE-UEG-Xi-CAS PBE-ONT-xi-CAS PBE-ONT-Xi-CAS
-108.779741458229 -0.0691133629633014 -0.0461481000697329 -0.0509457188492165 -0.0494784933936403
2 10^-13 0.0229652628935603 8. 10^-15 0.0014672254555662 9.9 10^{-15}