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added some more stuffs for extensivity

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Emmanuel Giner 2019-10-14 00:39:15 +08:00
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Manuscript/srDFT_SC.aux
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@ -82,10 +82,10 @@
\newlabel{eq:lim_n2}{{20}{4}{}{equation.2.20}{}}
\newlabel{eq:lim_muinf}{{21}{4}{}{equation.2.21}{}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Properties of approximated functionals}{4}{section*.10}}
\newlabel{eq:lim_ebasis}{{22}{4}{}{equation.2.22}{}}
\citation{GarBulHenScu-PCCP-15}
\citation{LooPraSceTouGin-JCPL-19}
\citation{GorSav-PRA-06}
\newlabel{eq:lim_ebasis}{{22}{5}{}{equation.2.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E}Requirements for the approximated functionals in the strong correlation }{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 {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}}
@ -95,6 +95,8 @@
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Definition of the different types of functionals}{5}{section*.16}}
\newlabel{eq:def_pbeueg}{{24}{5}{}{equation.2.24}{}}
\newlabel{eq:def_n2ueg}{{25}{5}{}{equation.2.25}{}}
\citation{FerGinTou-JCP-18,excited}
\citation{GorSav-PRA-06}
\bibdata{srDFT_SCNotes,srDFT_SC}
\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}}
@ -113,6 +115,15 @@
\bibcite{EvaDauMal-ChemPhys-83}{{15}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}}
\bibcite{Cim-JCP-1985}{{16}{1985}{{Cimiraglia}}{{}}}
\bibcite{Cim-JCC-1987}{{17}{1987}{{Cimiraglia\ and\ Persico}}{{}}}
\newlabel{eq:def_pbeueg}{{26}{6}{}{equation.2.26}{}}
\newlabel{eq:def_pbeueg}{{28}{6}{}{equation.2.28}{}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{6}{section*.17}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Numerical tests of extensivity}{6}{section*.18}}
\newlabel{sec:results}{{III\tmspace +\thinmuskip {.1667em}A}{6}{}{table.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{6}{section*.19}}
\newlabel{sec:conclusion}{{IV}{6}{}{section*.19}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.1}}
\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{PovRubIll-TCA-92}{{19}{1992}{{Povill, Rubio,\ and\ Illas}}{{}}}
\bibcite{BunCarRam-JCP-06}{{20}{2006}{{Bunge\ and\ Carb{\'o}-Dorca}}{{}}}
@ -123,6 +134,12 @@
\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{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}}
\newlabel{fig:N2_avtz}{{2}{7}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
\@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{SchEva-JCP-16}{{29}{2016}{{Schriber\ and\ Evangelista}}{{}}}
\bibcite{LiuHofJCTC-16}{{30}{2016}{{Liu\ and\ Hoffmann}}{{}}}
@ -134,16 +151,12 @@
\bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}}
\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}}}
\@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$_2$. }}{6}{table.1}}
\newlabel{conv_He_table}{{I}{6}{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{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{6}{section*.17}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{6}{section*.18}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{6}{section*.19}}
\newlabel{sec:results}{{III}{6}{}{section*.19}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{6}{section*.20}}
\newlabel{sec:conclusion}{{IV}{6}{}{section*.20}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.1}}
\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}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.4}}
\newlabel{fig:F2_avtz}{{4}{8}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}}
\@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. }}{8}{figure.5}}
\newlabel{fig:H10_vdz}{{5}{8}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.6}}
\newlabel{fig:H10_vtz}{{6}{8}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
\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{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
@ -166,10 +179,6 @@
\bibcite{TouColSav-PRA-04}{{58}{2004}{{Toulouse, Colonna,\ and\ Savin}}{{}}}
\bibcite{FraMusLupTou-JCP-15}{{59}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}}
\bibcite{AngGerSavTou-PRA-05}{{60}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}}
\@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}}
\newlabel{fig:N2_avtz}{{2}{7}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
\@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{GolWerSto-PCCP-05}{{61}{2005}{{Goll, Werner,\ and\ Stoll}}{{}}}
\bibcite{TouGerJanSavAng-PRL-09}{{62}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}}
\bibcite{JanHenScu-JCP-09}{{63}{2009}{{Janesko, Henderson,\ and\ Scuseria}}{{}}}
@ -179,12 +188,6 @@
\bibcite{FroTouJen-JCP-07}{{67}{2007}{{Fromager, Toulouse,\ and\ Jensen}}{{}}}
\bibcite{FroCimJen-PRA-10}{{68}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}}
\bibcite{HedKneKieJenRei-JCP-15}{{69}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.4}}
\newlabel{fig:F2_avtz}{{4}{8}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}}
\@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. }}{8}{figure.5}}
\newlabel{fig:H10_vdz}{{5}{8}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.6}}
\newlabel{fig:H10_vtz}{{6}{8}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
\bibcite{HedTouJen-JCP-18}{{70}{2018}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}{{}}}
\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}}}
@ -202,5 +205,5 @@
\citation{aip41Control}
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{9}{figure.7}}
\newlabel{fig:H10_vqz}{{7}{9}{H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.7}{}}
\newlabel{LastBibItem}{{81}{9}{}{section*.20}{}}
\newlabel{LastBibItem}{{81}{9}{}{section*.19}{}}
\newlabel{LastPage}{{}{9}{}{}{}}

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Manuscript/srDFT_SC.out
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@ -14,7 +14,6 @@
\BOOKMARK [2][-]{section*.14}{Requirement on B for the extensivity}{section*.4}% 14
\BOOKMARK [2][-]{section*.15}{Approximations for the strong correlation regime}{section*.4}% 15
\BOOKMARK [3][-]{section*.16}{Definition of the different types of functionals}{section*.15}% 16
\BOOKMARK [3][-]{section*.17}{Introduction of the effective spin-density}{section*.15}% 17
\BOOKMARK [3][-]{section*.18}{Requirement for B for size extensivity}{section*.15}% 18
\BOOKMARK [1][-]{section*.19}{Results}{section*.2}% 19
\BOOKMARK [1][-]{section*.20}{Conclusion}{section*.2}% 20
\BOOKMARK [1][-]{section*.17}{Results}{section*.2}% 17
\BOOKMARK [2][-]{section*.18}{Numerical tests of extensivity}{section*.17}% 18
\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 19

59
Manuscript/srDFT_SC.tex
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@ -73,7 +73,8 @@
\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{\argepbeuegspin}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\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{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
@ -107,6 +108,7 @@
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
\newcommand{\ntwo}[0]{n^{(2)}}
\newcommand{\ntwoextrap}[0]{\tilde{n}^{(2)}_{\psibasis}}
\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
@ -121,9 +123,9 @@
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ n^{2}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ n^{2}_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ n^{2}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
@ -404,7 +406,7 @@ which is fundamental to guarantee the good behaviour of the theory at the CBS li
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
\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}.
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 $n^{2}(\br{})$. In the present work, the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $n^{2}(\br{})$ 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, the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $\ntwo(\br{})$ are be computed from the same wave function $\psibasis$ used to define $\murpsi$.
The generic form for the approximations to $\efuncden{\denr}$ proposed here reads
\begin{equation}
\begin{aligned}
@ -420,7 +422,7 @@ where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as
with
\begin{equation}
\label{eq:def_beta}
\beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{n^{2}/\den},
\beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den},
\end{equation}
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}.
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
@ -430,9 +432,9 @@ The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by
which can be qualified as the weak correlation regime, and the large $\mu$ limit
\begin{equation}
\label{eq:lim_mularge}
\ecmd(\argecmd) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5}),
\ecmd(\argecmd) = \frac{1}{\mu^3} \ntwo + o(\frac{1}{\mu^5}),
\end{equation}
which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$, provided that $n^{2}$ is the \textit{exact} on-top pair density of the system.
which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$, provided that $\ntwo$ is the \textit{exact} on-top pair density of the system.
In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching the strong correlation regime. The importance of the on-top pair density in the strong correlation regime have been also acknowledged by Pernal and co-workers\cite{GritMeePer-PRA-18} and Gagliardi and co-workers\cite{CarTruGag-JPCA-17}.
Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes
\begin{equation}
@ -445,7 +447,6 @@ Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when
\label{eq:lim_muinf}
\lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0.
\end{equation}
%For equation \eqref{eq:lim_muinf} to be exact, the \textit{exact} on-top pair density $n^{2}$ of the physical system is needed, which is of course rarely affordable and therefore, in the present work, it will be approximated by that computed by an approximated wave function $\psibasis$.
\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.
@ -503,12 +504,39 @@ where $\ntwo_{\text{UEG}}$ is an approximation of the on-top pair density of the
\label{eq:def_n2ueg}
\ntwo_{\text{UEG}}(n,\xi) = n^2(1-\xi)g_0(n)
\end{equation}
using the pair-distribution function $g_0(n)$ of equation (46) of Ref. \onlinecite{GorSav-PRA-06}.
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.
using the pair-distribution function $g_0(n)$ of equation (46) of Ref. \onlinecite{GorSav-PRA-06}.
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.
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}
\label{eq:def_pbeueg}
\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}
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".
Therefore, one can define the PBE-ont-$\xi$-CAS as
\begin{equation}
\label{eq:def_pbeueg}
\begin{aligned}
\efuncdenpbe{\argepbeontxihf} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeontxihf).
\end{aligned}
\end{equation}
Such a functional can be further improved by using the effective spin density $\Xi$ to give the PBE-ont-$\Xi$-CAS.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\subsection{Numerical tests of extensivity}
\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$_2$. }
\begin{tabular}{lcc}
@ -524,13 +552,6 @@ Error to additivity & 1.2 $\times 10^{-12}$ & 7 $\times 10^{-15}$
\label{conv_He_table}
\end{table*}
\subsubsection{Introduction of the effective spin-density}
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}

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@ -0,0 +1,272 @@
Date: 13/10/2019 23:41:45
===============
Quantum Package
===============
Git Commit: trying to fix the casscf
Git Date : Wed Sep 18 13:55:16 2019 +0200
Git SHA1 : c8cd1611626d12424fa9776ad42e17a0ce2ce228
EZFIO Dir : n_sz_0.ezfio
Task server running : tcp://127.0.1.1:41279
.. >>>>> [ IO READ: no_core_density ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000204 s ] [ CPU TIME: 0.002093 s ] <<<<< ..
.. >>>>> [ IO READ: on_top_from_cas ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000476 s ] [ CPU TIME: 0.002304 s ] <<<<< ..
.. >>>>> [ IO READ: mu_of_r_potential ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000680 s ] [ CPU TIME: 0.002458 s ] <<<<< ..
.. >>>>> [ IO READ: read_wf ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000892 s ] [ CPU TIME: 0.002610 s ] <<<<< ..
.. >>>>> [ IO READ: mu_of_r_functional ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001139 s ] [ CPU TIME: 0.002795 s ] <<<<< ..
LDA, PBE and PBE-on-top / mu(r) PSI coallescence with frozen core interaction
****************************************
Functional used = basis_set_on_top_PBE
****************************************
mu_of_r_potential = psi_cas_ful
MR DFT energy with pure correlation part for the DFT
.. >>>>> [ IO READ: grid_type_sgn ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001353 s ] [ CPU TIME: 0.002949 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.004738 GB ] [ VIRT MEM : 0.041679 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001565 s ] [ CPU TIME: 0.003100 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_charge ] <<<<< ..
.. >>>>> [ RES MEM : 0.005466 GB ] [ VIRT MEM : 0.065128 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001984 s ] [ CPU TIME: 0.003592 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_label ] <<<<< ..
.. >>>>> [ RES MEM : 0.005466 GB ] [ VIRT MEM : 0.065128 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.002240 s ] [ CPU TIME: 0.004262 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.005466 GB ] [ VIRT MEM : 0.190128 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.002518 s ] [ CPU TIME: 0.005586 s ] <<<<< ..
Nuclear Coordinates (Angstroms)
===============================
================ ============ ============ ============ ============
Atom Charge X Y Z
================ ============ ============ ============ ============
N 7.000000 0.000000 0.000000 0.000000
================ ============ ============ ============ ============
.. >>>>> [ IO READ: thresh_grid ] <<<<< ..
.. >>>>> [ RES MEM : 0.006691 GB ] [ VIRT MEM : 0.190979 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.005062 s ] [ CPU TIME: 0.015260 s ] <<<<< ..
n_points_final_grid = 22046
n max point = 22348
.. >>>>> [ IO READ: n_states ] <<<<< ..
.. >>>>> [ RES MEM : 0.006691 GB ] [ VIRT MEM : 0.190979 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.005366 s ] [ CPU TIME: 0.015909 s ] <<<<< ..
providing the mu_of_r ...
* mo_num 23
.. >>>>> [ IO READ: mo_class ] <<<<< ..
.. >>>>> [ RES MEM : 0.007698 GB ] [ VIRT MEM : 0.254852 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.006370 s ] [ CPU TIME: 0.018624 s ] <<<<< ..
* Number of active MOs 4
* Number of core MOs 1
* Number of inactive MOs 0
* mo_label Canonical
* Number of determinants 9
* Dimension of the psi arrays 100000
Read psi_coef 9 1
providing core_inact_act_two_bod_alpha_beta_mo ...
* N_int 1
.. >>>>> [ IO READ: ao_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.008564 GB ] [ VIRT MEM : 0.255600 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.008946 s ] [ CPU TIME: 0.027502 s ] <<<<< ..
Read mo_coef
.. >>>>> [ IO READ: elec_beta_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.008564 GB ] [ VIRT MEM : 0.255600 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.009826 s ] [ CPU TIME: 0.029359 s ] <<<<< ..
.. >>>>> [ IO READ: elec_alpha_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.008564 GB ] [ VIRT MEM : 0.255600 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.010048 s ] [ CPU TIME: 0.029827 s ] <<<<< ..
Read psi_det
* Number of unique alpha determinants 3
* Number of unique beta determinants 3
core_inact_act_two_bod_alpha_beta_mo provided in 6.2838260055286810E-003
Core MOs:
1
USING THE VALENCE ONLY TWO BODY DENSITY
providing core_inact_act_two_bod_alpha_beta_mo_physicist ...
core_inact_act_two_bod_alpha_beta_mo_physicist provided in 1.1119991540908813E-006
providing the core_inact_act_on_top_of_r
.. >>>>> [ IO READ: ao_prim_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.010189 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.015404 s ] [ CPU TIME: 0.050392 s ] <<<<< ..
.. >>>>> [ IO READ: ao_expo ] <<<<< ..
.. >>>>> [ RES MEM : 0.010189 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.015826 s ] [ CPU TIME: 0.051451 s ] <<<<< ..
.. >>>>> [ IO READ: ao_coef ] <<<<< ..
.. >>>>> [ RES MEM : 0.010189 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.016234 s ] [ CPU TIME: 0.052473 s ] <<<<< ..
.. >>>>> [ IO READ: ao_power ] <<<<< ..
.. >>>>> [ RES MEM : 0.010189 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.016584 s ] [ CPU TIME: 0.054533 s ] <<<<< ..
.. >>>>> [ IO READ: ao_nucl ] <<<<< ..
.. >>>>> [ RES MEM : 0.010723 GB ] [ VIRT MEM : 0.275307 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.017108 s ] [ CPU TIME: 0.055801 s ] <<<<< ..
mo_num,n_points_final_grid 23 22046
* Number of virtual MOs 18
* Number of deleted MOs 0
Active MOs:
2 3 4 5
0 1 2 3
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:
1 2 3 4 5
provided the core_inact_act_on_top_of_r
Time to provide : 4.2960599996149540E-002
MO map initialized: 38226
.. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.057812 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191514 s ] [ CPU TIME: 0.395640 s ] <<<<< ..
.. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.057812 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191746 s ] [ CPU TIME: 0.396157 s ] <<<<< ..
AO map initialized : 52975
.. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.057812 GB ] [ VIRT MEM : 0.313763 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191996 s ] [ CPU TIME: 0.396700 s ] <<<<< ..
Providing the AO integrals
Sorting the map
AO integrals provided:
Size of AO map : 8.5845947265625000E-002 MB
Number of AO integrals : 7966
cpu time : 0.14260299999999998 s
wall time : 8.4920340996177401E-002 s ( x 1.6792560925587794 )
AO -> MO integrals transformation
---------------------------------
.. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.060402 GB ] [ VIRT MEM : 0.524967 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.277423 s ] [ CPU TIME: 0.541045 s ] <<<<< ..
Buffers : 0.459625244 MB / core
Molecular integrals provided:
Size of MO map 0.36048507690429688 MB
Number of MO integrals: 18894
cpu time : 0.18409100000000000 s
wall time : 5.7356710996828042E-002 s ( x 3.2095808284784786 )
Providing core_inact_act_V_kl_contracted_transposed .....
Time to provide core_inact_act_V_kl_contracted_transposed = 0.10440728099638363
Providing core_inact_act_rho2_kl_contracted_transposed .....
Time to provide core_inact_act_rho2_kl_contracted_transposed = 1.4405565998458769E-002
Providing core_inact_act_f_psi_ab .....
Time to provide core_inact_act_f_psi_ab = 4.5104700038791634E-003
providing the cas_full_mu_of_r_psi_coal_vector ...
Time to provide cas_full_mu_of_r_psi_coal_vector = 4.5001399848842993E-004
Time to provide mu_of_r = 0.47207160700054374
Providing Energy_c_md_n_and_PBE_mu_of_r ...
.. >>>>> [ IO READ: density_for_dft ] <<<<< ..
.. >>>>> [ RES MEM : 0.083344 GB ] [ VIRT MEM : 0.658817 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.479092 s ] [ CPU TIME: 1.195428 s ] <<<<< ..
.. >>>>> [ IO READ: normalize_dm ] <<<<< ..
.. >>>>> [ RES MEM : 0.083344 GB ] [ VIRT MEM : 0.658817 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.479350 s ] [ CPU TIME: 1.196356 s ] <<<<< ..
Time for the Energy_c_md_n_and_PBE_mu_of_r : 0.12119978699774947
Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_LDA_mu_of_r : 2.9739354999037459E-002
Providing Energy_c_md_LDA_mu_of_r ...
Time for Energy_c_md_n_and_LDA_mu_of_r : 2.8482141999120358E-002
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 : 6.0877834002894815E-002
.. >>>>> [ IO READ: ontop_approx ] <<<<< ..
.. >>>>> [ RES MEM : 0.084461 GB ] [ VIRT MEM : 0.659637 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.764506 s ] [ CPU TIME: 1.523235 s ] <<<<< ..
Inactive MOs:
providing the core_inact_act_on_top_of_r_new
providint all_states_act_two_rdm_alpha_beta_mo
ispin = 3
USING THE VALENCE ONLY TWO BODY DENSITY
provided the core_inact_act_on_top_of_r_new
Time to provide : 2.7583030023379251E-003
Providing Energy_c_md_mu_of_r_PBE_on_top ...
Time for the Energy_c_md_on_top_PBE_mu_of_r: 0.27808773300057510
Providing Energy_c_md_PBE_mu_of_r ...
Time for the Energy_c_md_PBE_mu_of_r: 6.8387204999453388E-002
Corrections using Multi determinant mu
Functionals with UEG ontop pair density at large mu
ECMD LDA regular spin dens = -0.0331999682056341
ECMD LDA effective spin dens = -0.0265407099448326
ECMD PBE regular spin dens = -0.0333659332905280
ECMD PBE effective spin dens = -0.0230740500348705
Functionals with extrapolated exact ontop based on current wave function
ECMD PBE/ontop regular spin dens = -0.0254042078502341
ECMD PBE/ontop effective spin dens = -0.0247392466968251
mu_average for basis set = 0.9116337460
Wall time: 0:00:03

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Date: 13/10/2019 23:40:44
===============
Quantum Package
===============
Git Commit: trying to fix the casscf
Git Date : Wed Sep 18 13:55:16 2019 +0200
Git SHA1 : c8cd1611626d12424fa9776ad42e17a0ce2ce228
EZFIO Dir : n_sz_0.ezfio
Task server running : tcp://127.0.1.1:41279
* mo_num 23
.. >>>>> [ IO READ: mo_class ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000262 s ] [ CPU TIME: 0.011449 s ] <<<<< ..
.. >>>>> [ IO READ: io_mo_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000547 s ] [ CPU TIME: 0.012076 s ] <<<<< ..
* N_int 1
MO map initialized: 38226
.. >>>>> [ IO READ: io_ao_two_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.000886 s ] [ CPU TIME: 0.012793 s ] <<<<< ..
.. >>>>> [ IO READ: ao_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001135 s ] [ CPU TIME: 0.013345 s ] <<<<< ..
AO map initialized : 52975
.. >>>>> [ IO READ: ao_power ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001530 s ] [ CPU TIME: 0.014290 s ] <<<<< ..
.. >>>>> [ IO READ: ao_prim_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.001812 s ] [ CPU TIME: 0.016728 s ] <<<<< ..
.. >>>>> [ IO READ: ao_expo ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.002234 s ] [ CPU TIME: 0.017788 s ] <<<<< ..
.. >>>>> [ IO READ: ao_coef ] <<<<< ..
.. >>>>> [ RES MEM : 0.004498 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.002668 s ] [ CPU TIME: 0.018724 s ] <<<<< ..
.. >>>>> [ IO READ: ao_nucl ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.004192 s ] [ CPU TIME: 0.023193 s ] <<<<< ..
.. >>>>> [ IO READ: ao_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.251705 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.004428 s ] [ CPU TIME: 0.024228 s ] <<<<< ..
Providing the AO integrals
.. >>>>> [ IO READ: nucl_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.400280 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.014788 s ] [ CPU TIME: 0.054298 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_label ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.400280 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.015033 s ] [ CPU TIME: 0.054484 s ] <<<<< ..
.. >>>>> [ IO READ: nucl_charge ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.400280 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.015240 s ] [ CPU TIME: 0.054630 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.005478 GB ] [ VIRT MEM : 0.400280 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.015444 s ] [ CPU TIME: 0.054773 s ] <<<<< ..
Nuclear Coordinates (Angstroms)
===============================
================ ============ ============ ============ ============
Atom Charge X Y Z
================ ============ ============ ============ ============
N 7.000000 0.000000 0.000000 0.000000
================ ============ ============ ============ ============
Sorting the map
AO integrals provided:
Size of AO map : 8.5845947265625000E-002 MB
Number of AO integrals : 7966
cpu time : 0.12584999999999999 s
wall time : 7.7138761000242084E-002 s ( x 1.6314755172124820 )
AO -> MO integrals transformation
---------------------------------
Read mo_coef
.. >>>>> [ IO READ: mo_integrals_threshold ] <<<<< ..
.. >>>>> [ RES MEM : 0.006847 GB ] [ VIRT MEM : 0.462940 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.086319 s ] [ CPU TIME: 0.161404 s ] <<<<< ..
Buffers : 0.368026733 MB / core
Molecular integrals provided:
Size of MO map 1.3504028320312500E-003 MB
Number of MO integrals: 65
cpu time : 0.12866600000000000 s
wall time : 4.1548692002834287E-002 s ( x 3.0967521189649707 )
.. >>>>> [ IO READ: n_states ] <<<<< ..
.. >>>>> [ RES MEM : 0.017288 GB ] [ VIRT MEM : 0.587971 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.123858 s ] [ CPU TIME: 0.285098 s ] <<<<< ..
* mo_label Canonical
.. >>>>> [ IO READ: read_wf ] <<<<< ..
.. >>>>> [ RES MEM : 0.017288 GB ] [ VIRT MEM : 0.587971 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.124249 s ] [ CPU TIME: 0.285934 s ] <<<<< ..
* Number of determinants 1
* Dimension of the psi arrays 100000
.. >>>>> [ IO READ: elec_beta_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.017788 GB ] [ VIRT MEM : 0.588718 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.125553 s ] [ CPU TIME: 0.289305 s ] <<<<< ..
.. >>>>> [ IO READ: elec_alpha_num ] <<<<< ..
.. >>>>> [ RES MEM : 0.017788 GB ] [ VIRT MEM : 0.588718 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.126080 s ] [ CPU TIME: 0.292590 s ] <<<<< ..
.. >>>>> [ IO READ: do_pt2 ] <<<<< ..
.. >>>>> [ RES MEM : 0.019299 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.128884 s ] [ CPU TIME: 0.299232 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.019585 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.129989 s ] [ CPU TIME: 0.307920 s ] <<<<< ..
.. >>>>> [ IO READ: io_nuclear_repulsion ] <<<<< ..
.. >>>>> [ RES MEM : 0.019585 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.130234 s ] [ CPU TIME: 0.308257 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.019585 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.130417 s ] [ CPU TIME: 0.308504 s ] <<<<< ..
* Nuclear repulsion energy 0.000000000000000
.. >>>>> [ IO READ: distributed_davidson ] <<<<< ..
.. >>>>> [ RES MEM : 0.019585 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.130727 s ] [ CPU TIME: 0.308919 s ] <<<<< ..
.. >>>>> [ IO READ: n_det_max_full ] <<<<< ..
.. >>>>> [ RES MEM : 0.019585 GB ] [ VIRT MEM : 0.590210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.130984 s ] [ CPU TIME: 0.309260 s ] <<<<< ..
.. >>>>> [ IO READ: io_mo_integrals_e_n ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.137271 s ] [ CPU TIME: 0.331178 s ] <<<<< ..
.. >>>>> [ IO READ: io_ao_integrals_e_n ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.137554 s ] [ CPU TIME: 0.331547 s ] <<<<< ..
.. >>>>> [ IO READ: io_mo_integrals_kinetic ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.137933 s ] [ CPU TIME: 0.337984 s ] <<<<< ..
.. >>>>> [ IO READ: io_ao_integrals_kinetic ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.138191 s ] [ CPU TIME: 0.338325 s ] <<<<< ..
.. >>>>> [ IO READ: io_mo_one_e_integrals ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.139874 s ] [ CPU TIME: 0.341512 s ] <<<<< ..
Providing the one-electron integrals
.. >>>>> [ IO READ: do_pseudo ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.140165 s ] [ CPU TIME: 0.341896 s ] <<<<< ..
.. >>>>> [ IO READ: selection_factor ] <<<<< ..
.. >>>>> [ RES MEM : 0.021740 GB ] [ VIRT MEM : 2.465210 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.140523 s ] [ CPU TIME: 0.342436 s ] <<<<< ..
.. >>>>> [ IO READ: pt2_max ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.141140 s ] [ CPU TIME: 0.349290 s ] <<<<< ..
.. >>>>> [ IO READ: correlation_energy_ratio_max ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.141476 s ] [ CPU TIME: 0.349711 s ] <<<<< ..
.. >>>>> [ IO READ: s2_eig ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.141781 s ] [ CPU TIME: 0.352244 s ] <<<<< ..
.. >>>>> [ IO READ: variance_max ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.142051 s ] [ CPU TIME: 0.352598 s ] <<<<< ..
.. >>>>> [ IO READ: pt2_relative_error ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.142307 s ] [ CPU TIME: 0.352936 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.142535 s ] [ CPU TIME: 0.353247 s ] <<<<< ..
Read n_states_diag
* N_generators_bitmask 1
.. >>>>> [ IO READ: weight_one_e_dm ] <<<<< ..
.. >>>>> [ RES MEM : 0.023380 GB ] [ VIRT MEM : 2.467476 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.142886 s ] [ CPU TIME: 0.353733 s ] <<<<< ..
.. >>>>> [ IO READ: n_det_max ] <<<<< ..
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.471584 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.146889 s ] [ CPU TIME: 0.370070 s ] <<<<< ..
.. >>>>> [ IO READ: threshold_generators ] <<<<< ..
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.471584 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.147095 s ] [ CPU TIME: 0.370370 s ] <<<<< ..
* Target maximum memory (GB) 2000
* Number of occupation patterns 1
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.473824 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.147674 s ] [ CPU TIME: 0.371477 s ] <<<<< ..
.. >>>>> [ IO READ: threshold_davidson ] <<<<< ..
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.473824 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.147947 s ] [ CPU TIME: 0.371845 s ] <<<<< ..
Diagonalization of H using Lapack
.. >>>>> [ IO READ: only_expected_s2 ] <<<<< ..
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.473824 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.148271 s ] [ CPU TIME: 0.372279 s ] <<<<< ..
!!!!!!!! WARNING !!!!!!!!!
Within the 1 determinants selected
and the 4 states requested
We did not find any state with S^2 values close to 3.7500000000000000
We will then set the first N_states eigenvectors of the H matrix
as the CI_eigenvectors
You should consider more states and maybe ask for s2_eig to be .True. or just enlarge the CI space
.. >>>>> [ RES MEM : 0.027409 GB ] [ VIRT MEM : 2.473824 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.148940 s ] [ CPU TIME: 0.375430 s ] <<<<< ..
* Energy of state 1 -54.24886715796833
* S^2 of state 1 0.7500000000000000
* Saved determinants 1
--------------------------------------------------------------------------------
* Number of unique beta determinants 1
* Number of unique alpha determinants 1
.. >>>>> [ IO READ: weight_selection ] <<<<< ..
.. >>>>> [ RES MEM : 0.027653 GB ] [ VIRT MEM : 2.477074 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.157984 s ] [ CPU TIME: 0.406242 s ] <<<<< ..
Using pt2-matching weight in selection
.. >>>>> [ IO READ: pseudo_sym ] <<<<< ..
.. >>>>> [ RES MEM : 0.027653 GB ] [ VIRT MEM : 2.477081 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.158328 s ] [ CPU TIME: 0.406695 s ] <<<<< ..
.. >>>>> [ IO READ: h0_type ] <<<<< ..
.. >>>>> [ RES MEM : 0.027653 GB ] [ VIRT MEM : 2.477081 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.158616 s ] [ CPU TIME: 0.407095 s ] <<<<< ..
.. >>>>> [ RES MEM : 0.027653 GB ] [ VIRT MEM : 2.477081 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.158826 s ] [ CPU TIME: 0.407373 s ] <<<<< ..
* Number of generators 1
.. >>>>> [ RES MEM : 0.027653 GB ] [ VIRT MEM : 2.477081 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.159038 s ] [ CPU TIME: 0.407650 s ] <<<<< ..
* Number of selectors 1
* Number of comb teeth 1
* Number of core MOs 2
* pt2_n_tasks_max 1
* PT2 Energy denominator -54.24886715796833
* Estimated memory/thread (Gb) 0.1043081283569336E-06
1 -3.5250892786480123E-002 4 1
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483562 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.179152 s ] [ CPU TIME: 0.471101 s ] <<<<< ..
* Number of generators 1
* Number of comb teeth 1
* pt2_n_tasks_max 1
* Saved determinants 7
# PT2 weight 1.00000000
# var weight 1.00000000
* Number of unique beta determinants 3
* Number of unique alpha determinants 3
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.190725 s ] [ CPU TIME: 0.505886 s ] <<<<< ..
* Number of selectors 1
# PT2 weight 1.00000000
# var weight 1.00000000
Using pt2-matching weight in selection
* Correlation ratio 1.000000000000000
Summary at N_det = 7
-----------------------------------
# ============ =============================
State 1
# ============ =============================
# E -54.24886716
# PT2 -0.03525089 0.00000000
# rPT2 -0.01740119 0.00000000
#
# E+PT2 -54.28411805 0.00000000
# E+rPT2 -54.26626835 0.00000000
# ============ =============================
N_det = 7
N_states = 1
N_sop = 5
* State 1
< S^2 > = 1.0000000000000000
E = -54.248867157968334
Variance = 1.2426254422441883E-003
PT norm = 1.0128053134583239
PT2 = -3.5250892786480123E-002
rPT2 = -1.7401191985924193E-002
E+PT2 = -54.284118050754813 +/- 0.0000000000000000
E+rPT2 = -54.266268349954260 +/- 0.0000000000000000
-----
.. >>>>> [ IO READ: io_mo_integrals_pseudo ] <<<<< ..
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.191427 s ] [ CPU TIME: 0.508023 s ] <<<<< ..
Energy components
=================
Vnn : Nucleus-Nucleus potential energy
Ven : Electron-Nucleus potential energy
Vee : Electron-Electron potential energy
Vecp : Potential energy of the pseudo-potentials
T : Electronic kinetic energy
State 1
---------
Vnn = 0.0000000000000000
Ven = -128.26859672094469
Vee = 19.656246836147339
Vecp = 0.0000000000000000
T = 54.363482726829012
.. >>>>> [ IO READ: energy_iterations ] <<<<< ..
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.192359 s ] [ CPU TIME: 0.510083 s ] <<<<< ..
.. >>>>> [ IO READ: pt2_iterations ] <<<<< ..
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.192718 s ] [ CPU TIME: 0.510635 s ] <<<<< ..
.. >>>>> [ IO READ: n_det_iterations ] <<<<< ..
.. >>>>> [ RES MEM : 0.030998 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.193061 s ] [ CPU TIME: 0.511167 s ] <<<<< ..
* Saved determinants 7
Diagonalization of H using Lapack
.. >>>>> [ RES MEM : 0.031528 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.197929 s ] [ CPU TIME: 0.533248 s ] <<<<< ..
* Energy of state 1 -54.38987072911426
* S^2 of state 1 3.750000000000000
* Saved determinants 7
--------------------------------------------------------------------------------
* Number of unique beta determinants 3
* Number of unique alpha determinants 3
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.031528 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.205060 s ] [ CPU TIME: 0.552810 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.031528 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.205268 s ] [ CPU TIME: 0.553097 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
* PT2 Energy denominator -54.38987072911426
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.031528 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.208460 s ] [ CPU TIME: 0.564509 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.031528 GB ] [ VIRT MEM : 2.483574 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.208671 s ] [ CPU TIME: 0.564790 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
* Number of tasks 3
* Number of fragmented tasks 0
* Number of threads for PT2 4
* Memory (Gb) 0.2079386264085770E-04
========== ================= =========== =============== =============== =================
Samples Energy Stat. Err Variance Norm Seconds
========== ================= =========== =============== =============== =================
========== ================= =========== =============== =============== =================
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.229260 s ] [ CPU TIME: 0.626846 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.229473 s ] [ CPU TIME: 0.627132 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
# PT2 weight 1.50000000
# var weight 1.00000000
Using pt2-matching weight in selection
* Correlation ratio 1.000000000000000
Summary at N_det = 7
-----------------------------------
# ============ =============================
State 1
# ============ =============================
# E -54.38987073
# PT2 0.00000000 **************
# rPT2 0.00000000 **************
#
# E+PT2 -54.38987073 **************
# E+rPT2 -54.38987073 **************
# ============ =============================
N_det = 7
N_states = 1
N_sop = 5
* State 1
< S^2 > = 3.7500000000000000
E = -54.389870729114257
Variance = 3.3330920638927635E-031
PT norm = 0.0000000000000000
PT2 = 0.0000000000000000
rPT2 = 0.0000000000000000
E+PT2 = -54.389870729114257 +/- 3.4028234663852886E+038
E+rPT2 = -54.389870729114257 +/- 3.4028234663852886E+038
-----
Energy components
=================
State 1
---------
Vnn = 0.0000000000000000
Ven = -128.26859672094588
Vee = 19.515243265001999
Vecp = 0.0000000000000000
T = 54.363482726829631
Extrapolated energies
------------------------
State 1
=========== ===================
minimum PT2 Extrapolated energy
=========== ===================
-0.0174 -54.38987073
=========== ===================
* Saved determinants 9
Diagonalization of H using Lapack
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.238963 s ] [ CPU TIME: 0.664193 s ] <<<<< ..
* Energy of state 1 -54.38987072911425
* S^2 of state 1 3.750000000000000
* Saved determinants 9
* Saved determinants 9
* Number of unique beta determinants 3
* Number of unique alpha determinants 3
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.243211 s ] [ CPU TIME: 0.680518 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.244483 s ] [ CPU TIME: 0.681280 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
* PT2 Energy denominator -54.38987072911425
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.254950 s ] [ CPU TIME: 0.716323 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.032478 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.255171 s ] [ CPU TIME: 0.716615 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
* Number of tasks 3
* Number of fragmented tasks 0
* Number of threads for PT2 4
* Memory (Gb) 0.2124089747667313E-04
========== ================= =========== =============== =============== =================
Samples Energy Stat. Err Variance Norm Seconds
========== ================= =========== =============== =============== =================
========== ================= =========== =============== =============== =================
Using pt2-matching weight in selection
.. >>>>> [ RES MEM : 0.032486 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.264264 s ] [ CPU TIME: 0.744235 s ] <<<<< ..
* Number of generators 3
.. >>>>> [ RES MEM : 0.032486 GB ] [ VIRT MEM : 2.484348 GB ] <<<<< ..
.. >>>>> [ WALL TIME: 0.264408 s ] [ CPU TIME: 0.744320 s ] <<<<< ..
* Number of selectors 3
* Number of comb teeth 1
* pt2_n_tasks_max 1
# PT2 weight 2.25000000
# var weight 1.00000000
Using pt2-matching weight in selection
Summary at N_det = 9
-----------------------------------
# ============ =============================
State 1
# ============ =============================
# E -54.38987073
# PT2 0.00000000 **************
# rPT2 0.00000000 **************
#
# E+PT2 -54.38987073 **************
# E+rPT2 -54.38987073 **************
# ============ =============================
N_det = 9
N_states = 1
N_sop = 7
* State 1
< S^2 > = 3.7499999999999996
E = -54.389870729114250
Variance = 1.7889867724903257E-031
PT norm = 0.0000000000000000
PT2 = 0.0000000000000000
rPT2 = 0.0000000000000000
E+PT2 = -54.389870729114250 +/- 3.4028234663852886E+038
E+rPT2 = -54.389870729114250 +/- 3.4028234663852886E+038
-----
Energy components
=================
State 1
---------
Vnn = 0.0000000000000000
Ven = -128.26859672094585
Vee = 19.515243265001970
Vecp = 0.0000000000000000
T = 54.363482726829631
Extrapolated energies
------------------------
State 1
=========== ===================
minimum PT2 Extrapolated energy
=========== ===================
0.0000 -54.38987073
-0.0174 -54.38987073
=========== ===================
Wall time: 0:00:01