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Manuscript/G2-srDFT.tex
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@ -272,15 +272,15 @@ Unless otherwise stated, atomic used are used.
\section{Theory} \section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set. The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set.
Here, we provide the main working equations. Here, we only provide the main working equations.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about the formal derivation of the theory. We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about its formal derivation.
%================================================================= %=================================================================
%\subsection{Correcting the basis set error of a general WFT model} %\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 $\Bas$. 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 $\n{}{}$, one may write 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
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
@ -294,7 +294,7 @@ where
= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} = \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation} \end{equation}
is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively. is the basis-dependent complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
Both wave functions yield the same target density $\n{}{}$. Both wave functions yield the same target density $\n{}{}$.
@ -306,7 +306,7 @@ An important aspect of such theory is that, in the limit of a complete basis set
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}] ) = \E{\modX}{} \approx E, \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}] ) = \E{\modX}{} \approx E,
\end{equation} \end{equation}
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set. where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$. In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$. Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$.
%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$. %Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
@ -405,6 +405,8 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12
Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals. Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a range-separated \textit{function} $\rsmu{}{}(\br{})$. Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a range-separated \textit{function} $\rsmu{}{}(\br{})$.
The first step of our basis set correction consists in obtaining the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$.
%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. %Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
%First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . %First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). %Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}).
@ -422,7 +424,6 @@ Contrary to conventional RS-DFT schemes which require a range-separated paramete
%================================================================= %=================================================================
\subsection{Effective Coulomb operator} \subsection{Effective Coulomb operator}
%================================================================= %=================================================================
%The present section briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that
\begin{equation} \begin{equation}
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
@ -435,7 +436,7 @@ In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ de
which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$ which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
\begin{equation} \begin{equation}
\label{eq:WeeB} \label{eq:WeeB}
\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i} \hWee{\Bas} = \alert{\frac{1}{2} \sum_{ijkl \in \Bas}\vijkl \aic{k} \aic{l} \ai{j} \ai{i}}
\end{equation} \end{equation}
over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals. over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that