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Emmanuel Giner 2019-04-06 16:40:57 +02:00
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Manuscript/G2-srDFT.tex
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@ -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} %\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 $\n{}{}$, respectively, one may write According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ {\color{red} are reasonable approximations of the FCI energy and density within $\Bas$ } \sout{\textit{exact} ground state energy }, the exact ground state energy $\E{}{}$ \sout{and density $\n{}{}$, respectively, one may write} may be written as
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
@ -304,8 +304,8 @@ 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{\modY}{\Bas}] ) = \E{\modX}{} \approx E, \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\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 $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$. In the case $\modX = \FCI$, we have as strict equality as $E_{\FCI}^\infty = E$.
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 $\modY$ {\color{red} not clear to my eyes ... I think that one should say in what sence these are approximations in terms of the density and energy}.
%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 $\emodel$.
%As any wave function model is necessary an approximation to the FCI model, one can write %As any wave function model is necessary an approximation to the FCI model, one can write