From 68ddb0431a8903ed0090d8c04adc82bb6b5fab3a Mon Sep 17 00:00:00 2001 From: eginer Date: Wed, 2 Oct 2019 21:04:58 +0200 Subject: [PATCH] working on the theory section --- Manuscript/srDFT_SC.aux | 50 ++++++++++++++++++-------------- Manuscript/srDFT_SC.out | 3 +- Manuscript/srDFT_SC.tex | 63 ++++++++++++++++++++++++++++++++++------- 3 files changed, 83 insertions(+), 33 deletions(-) diff --git a/Manuscript/srDFT_SC.aux b/Manuscript/srDFT_SC.aux index 3c361ec..100aedf 100644 --- a/Manuscript/srDFT_SC.aux +++ b/Manuscript/srDFT_SC.aux @@ -30,29 +30,37 @@ \citation{G2,excited} \citation{kato} \citation{GinPraFerAssSavTou-JCP-18} -\bibdata{srDFT_SCNotes,srDFT_SC} -\bibstyle{aipnum4-1} -\citation{REVTEX41Control} -\citation{aip41Control} +\citation{GinPraFerAssSavTou-JCP-18} \@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{2}{section*.5}} \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}{}{}{}} +\newlabel{eq:wbasis}{{6}{2}{}{equation.2.6}{}} +\newlabel{eq:fbasis}{{8}{2}{}{equation.2.8}{}} +\newlabel{eq:cbs_wbasis}{{10}{2}{}{equation.2.10}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{2}{section*.7}} +\citation{GinPraFerAssSavTou-JCP-18} +\bibdata{srDFT_SCNotes,srDFT_SC} +\bibstyle{aipnum4-1} +\citation{REVTEX41Control} +\citation{aip41Control} +\newlabel{eq:weelr}{{11}{3}{}{equation.2.11}{}} +\newlabel{eq:cbs_mu}{{14}{3}{}{equation.2.14}{}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{3}{section*.8}} +\newlabel{LastBibItem}{{0}{3}{}{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. }}{4}{figure.1}} +\newlabel{fig:N2_avdz}{{1}{4}{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. }}{5}{figure.2}} +\newlabel{fig:N2_avtz}{{2}{5}{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. }}{6}{figure.3}} +\newlabel{fig:F2_avdz}{{3}{6}{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. }}{7}{figure.4}} +\newlabel{fig:F2_avtz}{{4}{7}{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}{}} +\@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{LastPage}{{}{9}{}{}{}} diff --git a/Manuscript/srDFT_SC.out b/Manuscript/srDFT_SC.out index d97d6fe..18dd645 100644 --- a/Manuscript/srDFT_SC.out +++ b/Manuscript/srDFT_SC.out @@ -4,4 +4,5 @@ \BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4 \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 +\BOOKMARK [2][-]{section*.7}{Definition of an range-separation parameter varying in real space}{section*.4}% 7 +\BOOKMARK [1][-]{section*.8}{Results}{section*.2}% 8 diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 09463d3..aede7a6 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -105,6 +105,7 @@ \newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} \newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} \newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} \newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} @@ -115,7 +116,7 @@ \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} -\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} \newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} \newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})} \newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} @@ -170,6 +171,8 @@ \newcommand{\sr}{\text{sr}} \newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + \newcommand{\n}[2]{n_{#1}^{#2}} \newcommand{\E}[2]{E_{#1}^{#2}} @@ -183,7 +186,7 @@ \newcommand{\w}[2]{w_{#1}^{#2}} \newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} \newcommand{\rsmu}[2]{\mu_{#1}^{#2}} -\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} \newcommand{\modX}{\text{X}} \newcommand{\modY}{\text{Y}} @@ -280,7 +283,7 @@ The exact ground state energy $E_0$ of a $N-$electron system can be obtained by \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 +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}. @@ -292,17 +295,17 @@ Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ \begin{equation} F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr} \end{equation} +where $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$, and 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}}, + &- \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 +Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:e0approx} E_0 = \efci + \efuncbasisFCI @@ -317,17 +320,55 @@ As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as \begin{equation} - \wbasis = \fbasis/\twodmrdiagpsi + \label{eq:wbasis} + \wbasis = + \begin{cases} + \fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} \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), + \twodmrdiagpsi = \sum_{pqrs} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \end{equation} -$\Gam{pq}{rs}$ +$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals, \begin{equation} - \int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \elemm{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}}, + \label{eq:fbasis} + \fbasis + = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \end{equation} -where $\twodmrdiagpsi$ is the two-body density of +and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. +With such a definition, one can show that $\wbasis$ satisfies +\begin{equation} + \int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. +\end{equation} +As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set, that is +\begin{equation} + \label{eq:cbs_wbasis} + \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}. +\end{equation} +The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set. +\subsection{Definition of an range-separation parameter varying in real space} +As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$ +\begin{equation} + \label{eq:weelr} + w_{ee}^{\lr}(\mu;\br{1},\br{2}) = \frac{\text{erf}\big(\mu \,|\br{1}-\br{2}| \big)}{|\br{1}-\br{2}|}. +\end{equation} +As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space +\begin{equation} + \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal +\end{equation} +such that +\begin{equation} + w_{ee}^{\lr}(\murpsi;\br{ },\br{ }) = \wbasiscoal. +\end{equation} +Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis}) +\begin{equation} + \label{eq:cbs_mu} + \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, +\end{equation} +which is fundamental to guarantee the good behaviour of the theory at the CBS limit. %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} %%%%%%%%%%%%%%%%%%%%%%%%