From bfa23f48da6af852ec9454ca7a442d4e941dff52 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 6 Apr 2019 15:10:49 +0200 Subject: [PATCH] theory --- Manuscript/G2-srDFT.tex | 64 ++++++++++++++++++++++------------------- 1 file changed, 34 insertions(+), 30 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 46864a9..6efdeea 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -278,7 +278,7 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo %\subsection{Correcting the basis set error of a general WFT model} %================================================================= Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. -According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \textit{exact} ground state energy $\E{}{}$ and density $\n{}{}$, respectively, one may write +According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and \alert{$\n{\modY}{\Bas}$} are reasonable approximations of the \textit{exact} ground state energy $\E{}{}$ and density $\n{}{}$, respectively, one may write \begin{equation} \label{eq:e0basis} \E{}{} @@ -480,9 +480,9 @@ Although this choice is not unique, the long-range interaction we have chosen is and ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of same-spin electron pairs yields \begin{equation} \label{eq:mu_of_r} - \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}). + \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\bx{},\Bar{\bx{}}), \end{equation} - +where $(\bx{},\Bar{\bx{}})$ represents a couple of same-spin electrons at the same position $\br{}$. %More precisely, if we define the value of the interaction at coalescence as %\begin{equation} % \label{eq:def_wcoal} @@ -592,7 +592,7 @@ Therefore, the PBE complementary function reads \label{eq:def_lda_tot} \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{} \end{equation} - +Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. %The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$. %Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$. @@ -625,7 +625,7 @@ where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar exp %\begin{equation} % \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}, %\end{equation} -Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as +Following the spirit of Eq.~\eqref{eq:fbasis}, we have \begin{multline} \label{eq:fbasisval} \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) @@ -633,38 +633,42 @@ Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Va = \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}. \end{multline} and, the valence part of the effective interaction is -\begin{equation} - \label{eq:def_weebasis} +\begin{subequations} +\begin{gather} \W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})/\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2}), -\end{equation} + \\ + \rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\bx{},\Bar{\bx{}}), +\end{gather} +\end{subequations} where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons. %\begin{equation} % \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . %\end{equation} %It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$. -It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfies Eq.~\eqref{eq:lim_W}. +It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ still satisfies Eq.~\eqref{eq:lim_W}. -We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. -Defining the valence one-body spin density matrix as -\begin{equation} - \begin{aligned} - \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\ - & = 0 \qquad \text{in other cases} - \end{aligned} -\end{equation} -then one can define the valence density as: -\begin{equation} - \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) -\end{equation} -Therefore, we propose the following valence-only approximations for the complementary functional -\begin{equation} - \label{eq:def_lda_tot} - \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,, -\end{equation} -\begin{equation} - \label{eq:def_lda_tot} - \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval) -\end{equation} +%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. +%Defining the valence one-body spin density matrix as +%\begin{equation} +% \begin{aligned} +% \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\ +% & = 0 \qquad \text{in other cases} +% \end{aligned} +%\end{equation} +%then one can define the valence density as: +%\begin{equation} +% \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) +%\end{equation} +%Therefore, we propose the following valence-only approximations for the complementary functional +%\begin{equation} +% \label{eq:def_lda_tot} +% \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,, +%\end{equation} +%\begin{equation} +% \label{eq:def_lda_tot} +% \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval) +%\end{equation} +Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluate as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$. %%%%%%%%%%%%%%%%%%%%%%%% %\section{Results}