From d40d34f3b035f1a948d58ee4108d75323debc664 Mon Sep 17 00:00:00 2001 From: eginer Date: Wed, 2 Oct 2019 20:18:19 +0200 Subject: [PATCH] working on the theory section --- Manuscript/srDFT_SC.aux | 52 +++++++++++++++++---------- Manuscript/srDFT_SC.out | 4 ++- Manuscript/srDFT_SC.tex | 78 +++++++++++++++++++++++++++++++++++++---- 3 files changed, 107 insertions(+), 27 deletions(-) diff --git a/Manuscript/srDFT_SC.aux b/Manuscript/srDFT_SC.aux index a650ea0..3c361ec 100644 --- a/Manuscript/srDFT_SC.aux +++ b/Manuscript/srDFT_SC.aux @@ -18,27 +18,41 @@ \providecommand\HyField@AuxAddToCoFields[2]{} \citation{alex_thom,piotr} \citation{scuseria} -\bibdata{srDFT_SCNotes,srDFT_SC} -\bibstyle{aipnum4-1} -\citation{REVTEX41Control} -\citation{aip41Control} +\citation{GinPraFerAssSavTou-JCP-18} \newlabel{FirstPage}{{}{1}{}{section*.1}{}} \@writefile{toc}{\contentsline {title}{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}{1}{section*.2}} \@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}} \@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}} \@writefile{toc}{\contentsline {section}{\numberline {II}Theory}{1}{section*.4}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{1}{section*.5}} -\newlabel{LastBibItem}{{0}{1}{}{figure.6}{}} -\@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. }}{2}{figure.1}} -\newlabel{fig:N2_avdz}{{1}{2}{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 {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{3}{figure.2}} -\newlabel{fig:N2_avtz}{{2}{3}{N$_2$, aug-cc-pvtz: 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+\newlabel{eq:levy}{{1}{2}{}{equation.2.1}{}} +\newlabel{eq:levy_func}{{2}{2}{}{equation.2.2}{}} +\newlabel{eq:e0approx}{{5}{2}{}{equation.2.5}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {B}Definition of an effective interaction within $\mathcal {B}$}{2}{section*.6}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{2}{section*.7}} +\newlabel{LastBibItem}{{0}{2}{}{figure.7}{}} +\@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. }}{3}{figure.1}} +\newlabel{fig:N2_avdz}{{1}{3}{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 {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{4}{figure.2}} +\newlabel{fig:N2_avtz}{{2}{4}{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. }}{5}{figure.3}} +\newlabel{fig:F2_avdz}{{3}{5}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}} +\@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. }}{6}{figure.4}} +\newlabel{fig:F2_avtz}{{4}{6}{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. }}{7}{figure.5}} +\newlabel{fig:H10_vdz}{{5}{7}{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. }}{7}{figure.6}} +\newlabel{fig:H10_vtz}{{6}{7}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}} +\@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. }}{8}{figure.7}} +\newlabel{fig:H10_vqz}{{7}{8}{H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.7}{}} +\newlabel{LastPage}{{}{8}{}{}{}} diff --git a/Manuscript/srDFT_SC.out b/Manuscript/srDFT_SC.out index 3b17b8e..d97d6fe 100644 --- a/Manuscript/srDFT_SC.out +++ b/Manuscript/srDFT_SC.out @@ -2,4 +2,6 @@ \BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 \BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3 \BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4 -\BOOKMARK [1][-]{section*.5}{Results}{section*.2}% 5 +\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5 +\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6 +\BOOKMARK [1][-]{section*.7}{Results}{section*.2}% 7 diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 3233c4c..09463d3 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -62,6 +62,7 @@ \newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} \newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} \newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} \newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} \newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} \newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} @@ -85,8 +86,7 @@ % numbers \newcommand{\rnum}[0]{{\rm I\!R}} -\newcommand{\bfr}[1]{{\bf x}_{#1}} -\newcommand{\bfrb}[1]{{\bf r}_{#1}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} \newcommand{\dr}[1]{\text{d}\bfr{#1}} \newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} \newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } @@ -109,9 +109,9 @@ \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]{ 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{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} @@ -133,6 +133,7 @@ \newcommand{\denmodel}[0]{\den_{\model}^\Bas} \newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} \newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} \newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} \newcommand{\denrfci}[0]{\denr_{\psifci}} \newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} @@ -188,6 +189,7 @@ \newcommand{\modY}{\text{Y}} % basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} \newcommand{\Bas}{\mathcal{B}} \newcommand{\Basval}{\mathcal{B}_\text{val}} \newcommand{\Val}{\mathcal{V}} @@ -200,10 +202,9 @@ \newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} % coordinates -\newcommand{\br}[1]{\mathbf{r}_{#1}} +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} \newcommand{\bx}[1]{\mathbf{x}_{#1}} \newcommand{\dbr}[1]{d\br{#1}} -\newcommand{\dbx}[1]{d\bx{#1}} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} @@ -273,7 +274,60 @@ When the molecular system %%%%%%%%%%%%%%%%%%%%%%%% The theoretical framework of the basis set correction have been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study. +\subsection{Basic formal equations} +The exact ground state energy $E_0$ of a $N-$electron system can be obtained by the Levy-Lieb constrained search formalism which is an elegant mathematical framework connecting WFT and DFT +\begin{equation} + \label{eq:levy} + E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\}, +\end{equation} +where $(v_{ne}(\br)|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional +\begin{equation} + \label{eq:levy_func} + F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}. +\end{equation} +The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density. +As in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$. +Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ as +\begin{equation} + F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr} +\end{equation} +where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as +\begin{equation} + \begin{aligned} + \efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\  + &- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}, + \end{aligned} +\end{equation} +and $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$. +The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$. + +Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18}), one obtains the following approximation for the exact ground state density +\begin{equation} + \label{eq:e0approx} + E_0 = \efci + \efuncbasisFCI +\end{equation} +where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in \cite{GinPraFerAssSavTou-JCP-18} and further emphasized in \cite{G2,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. +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 $\Bas$} +As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, the lack of cusp in any wave function $\wf{}{\Bas}$ could also originate from an effective non-divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence at the electron coalescence point. + +As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction. + +More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as +\begin{equation} + \wbasis = \fbasis/\twodmrdiagpsi +\end{equation} +where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$ +\begin{equation} + \twodmrdiagpsi = \sum_{pqrs} \phi_{p}(\br) \phi_{q}(\br) \Gam{pq}{rs} \phi_{r}(\br) \phi_{s}(\br), +\end{equation} +$\Gam{pq}{rs}$ +\begin{equation} + \int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \elemm{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}}, +\end{equation} +where $\twodmrdiagpsi$ is the two-body density of %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} %%%%%%%%%%%%%%%%%%%%%%%% @@ -338,6 +392,16 @@ The theoretical framework of the basis set correction have been derived in detai \end{figure} +\begin{figure} + \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}\\ + \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_error.eps}\\ +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:H10_vqz}} +\end{figure} + +