theory

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}
+\begin{subequations}
-	\label{eq:def_weebasis}
+\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. 
+%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
+%Defining the valence one-body spin density matrix as
-\begin{equation}
+%\begin{equation}
- \begin{aligned}
+% \begin{aligned}
- \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
+% \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}
+%                      & = 0 \qquad \text{in other cases}
- \end{aligned}
+% \end{aligned}
-\end{equation}
+%\end{equation}
-then one can define the valence density as:
+%then one can define the valence density as:
-\begin{equation}
+%\begin{equation}
-  \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) 
+%  \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) 
-\end{equation}
+%\end{equation}
-Therefore, we propose the following valence-only approximations for the complementary functional 
+%Therefore, we propose the following valence-only approximations for the complementary functional 
-\begin{equation}
+%\begin{equation}
- \label{eq:def_lda_tot}
+% \label{eq:def_lda_tot}
-  \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
+%  \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
-\end{equation}
+%\end{equation}
-\begin{equation}
+%\begin{equation}
- \label{eq:def_lda_tot}
+% \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)
+%  \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
-\end{equation}
+%\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}