updated manuscript

Manuscript/G2-srDFT.tex

@@ -56,6 +56,7 @@
 \newcommand{\EexDMC}{E_\text{exDMC}}
 \newcommand{\Ead}{\Delta E_\text{ad}}
 \newcommand{\efci}[0]{E_{\text{FCI}}^{\basis}}
+\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
 \newcommand{\efuncbasisfci}[0]{\bar{E}^\basis[\denfci]}
 \newcommand{\efuncbasis}[0]{\bar{E}^\basis[\den]}
 \newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
@@ -185,11 +186,10 @@
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{The DFT basis-set correction in a nutshell}
 The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing from the description of the WFT in a finite basis set. 
 Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in \cite{GinPraFerAssSavTou-JCP-18}.  
 
-\subsection{The basic concepts}
+\subsection{Correcting the basis set error of any wave function method}
 Consider an incomplete basis-set $\basis$ for which we assume to have accurate approximations of both the FCI density $\denfci$ and energy $\efci$. According to equation (15) of \cite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
 \begin{equation}
   \label{eq:e0basis}
@@ -204,9 +204,26 @@ where $\efuncbasis$ is the complementary density functional defined in equation
 \end{equation}
 where $\Psi$ is a general wave function being obtained in a complete basis. Provided that functional $\efuncbasis$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\denfci$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set. 
 
+An important aspect of such a theory is that, in the limit of a complete basis set $\basis$ (which we refer as $\basis \rightarrow \infty$), the functional $\efuncbasisfci$ tends to zero
+\begin{equation}
+  \label{eq:limitfunc}
+        \lim_{\basis \rightarrow \infty} \efuncbasisfci = 0
+\end{equation}
+which implies that the exact ground state energy coincides with the FCI energy in a computed in complete basis set (which we refer as $\efcicomplete$)
+\begin{equation}
+  \label{eq:limitfunc}
+        \lim_{\basis \rightarrow \infty} \efci + \efuncbasisfci = \efcicomplete\,\,.
+\end{equation}
+
+We propose here to generalize this procedure to any WFT approach. 
+
 The functional $\efuncbasisfci$ is not universal as it depends on the basis set $\basis$ used. A simple analytical form for such a functional is of course not known and we approximate it in two-steps. First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \ref{sec:weff}). 
 Then, 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{Toulouse2005_ecmd}, that we evaluate at the FCI density $\denfci$ (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.   
 
+\subsection{Generalization to any single-reference WFT method}
+The theory provided in \cite{GinPraFerAssSavTou-JCP-18} proposes a basis set correction to a FCI or selected CI wave function while ensuring the correct limit for a complete basis set. We propose here to generalize it to any single-reference WFT. 
+
+
 \subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$}
 One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also come from a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point. 
 
@@ -375,6 +392,17 @@ A general scheme to approximate  $\efuncbasisfci$ is to use $\ecmuapprox$ with t
  \efuncbasisfci \approx \ecmuapproxmurfci
 \end{equation}
 Therefore, any approximated ECMD can be used to estimate $\efuncbasisfci$. 
+It is important to notice that in the limit of a complete basis set, as 
+\begin{equation}
+        \lim_{\basis \rightarrow \infty} \wbasiscoalval{} = +\infty \quad \forall\,\, \psibasis\,\,\text{and}\,\,\,{\bf r}\,,
+\end{equation}
+the local range separation parameter $\murpsi$ (or $\murpsival$) tends to infinity and therefore 
+\begin{equation}
+ \lim_{\basis \rightarrow \infty} \ecmuapproxmurfci = 0 \quad ,
+\end{equation}
+which is a condition required by the exact theory (see \eqref{eq:limitfunc}). 
+Also, it means that one recovers a WFT model in the limit of a complete basis set, whatever the choice of $\psibasis$, functional ECMD or density used. 
+
 \subsubsection{LDA approximation for the complementary functional}
 Therefore, one can define an LDA-like functional for $\efuncbasis$ as
 \begin{equation}