First page done

Manuscript/G2-srDFT.tex

@@ -25,6 +25,7 @@
 \newcommand{\mr}{\multirow}
 \newcommand{\SI}{\textcolor{blue}{supporting information}}
 
+% Titou's macros
 \newcommand{\br}{\mathbf{r}}
 
 % second quantized operators
@@ -48,37 +49,35 @@
 
 	
 % energies
-\newcommand{\EHF}{E_\text{HF}}
 \newcommand{\Ec}{E_\text{c}}
 \newcommand{\EPT}{E_\text{PT2}}
-\newcommand{\EFCI}{E_\text{FCI}}
 \newcommand{\EsCI}{E_\text{sCI}}
 \newcommand{\EDMC}{E_\text{DMC}}
 \newcommand{\EexFCI}{E_\text{exFCI}}
-\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\basis}}
+\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}}
 \newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
 \newcommand{\EexDMC}{E_\text{exDMC}}
 \newcommand{\Ead}{\Delta E_\text{ad}}
-\newcommand{\efci}[0]{E_{\text{FCI}}^{\basis}}
+\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
-\newcommand{\emodel}[0]{E_{\model}^{\basis}}
+\newcommand{\emodel}[0]{E_{\model}^{\Bas}}
 \newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}}
 \newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
 \newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
-\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\basis}}
+\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}}
-\newcommand{\efuncbasisfci}[0]{\bar{E}^\basis[\denfci]}
+\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
-\newcommand{\efuncbasis}[0]{\bar{E}^\basis[\den]}
+\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
-\newcommand{\efuncden}[1]{\bar{E}^\basis[#1]}
+\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
-\newcommand{\ecompmodel}[0]{\bar{E}^\basis[\denmodel]}
+\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
 \newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
 \newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
 \newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]}
 \newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
 \newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
 \newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
-\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\basis,\psibasis}[\denmodel]}
+\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\psibasis}[\denmodel]}
-\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\basis,\psibasis}[\den]}
+\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\psibasis}[\den]}
-\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\basis,\psibasis}[\den]}
+\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\psibasis}[\den]}
-\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\basis,\psibasis}[\den]}
+\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\psibasis}[\den]}
 \newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)}
 \newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\psibasis)\right)}
 \newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)}
@@ -129,35 +128,31 @@
 \newcommand{\ra}{\rightarrow}
 \newcommand{\De}{D_\text{e}}
 
-% Basis sets 
-\newcommand{\basis}[0]{\mathcal{B}}
-\newcommand{\basisval}[0]{\mathcal{B}_\text{val}}
-
 % MODEL 
 \newcommand{\model}[0]{\mathcal{Y}}
 
 % densities 
-\newcommand{\denmodel}[0]{\den_{\model}^\basis}
+\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
-\newcommand{\denmodelr}[0]{\den_{\model}^\basis ({\bf r})}
+\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
 \newcommand{\denfci}[0]{\den_{\psifci}}
-\newcommand{\denhf}[0]{\den_{\text{HF}}^\basis}
+\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
 \newcommand{\denrfci}[0]{\denr_{\psifci}}
-\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\basis({\bf r})}
+\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
-\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\basis}
+\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas}
 \newcommand{\den}[0]{{n}}
 \newcommand{\denval}[0]{{n}^{\text{val}}}
 \newcommand{\denr}[0]{{n}({\bf r})}
 \newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
 
 % wave functions
-\newcommand{\psifci}[0]{\Psi^{\basis}_{\text{FCI}}}
+\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
-\newcommand{\psibasis}[0]{\Psi^{\basis}}
+\newcommand{\psibasis}[0]{\Psi^{\Bas}}
 \newcommand{\psimu}[0]{\Psi^{\mu}}
 % operators
-\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\basis}
+\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
 \newcommand{\kinop}[0]{\hat{T}}
 
-\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\basisval}}
+\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}}
 \newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
 
 
@@ -234,147 +229,187 @@ Unless otherwise stated, atomic used are used.
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
-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. 
+The basis set correction investigated here uses the RS-DFT formalism to capture the 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 \onlinecite{GinPraFerAssSavTou-JCP-18}.  
+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 Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}.  
 
-\subsection{Correcting the basis set error of a general WFT model}
+\newcommand{\FCI}{\text{FCI}}
-Consider a $N-$electron physical system described in an incomplete basis-set $\basis$ and for which we assume to have both the FCI density $\denfci$ and energy $\efci$. Assuming that $\denfci$ is a good approximation of the \textit{exact} ground state density, according to equation (15) of \onlinecite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E$ as
+\newcommand{\CCSDT}{\text{CCSD(T)}}
+
+\newcommand{\Nel}{N}
+
+\newcommand{\n}[2]{n_{#1}^{#2}}
+\newcommand{\E}[2]{E_{#1}^{#2}}
+\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
+\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
+
+\newcommand{\modX}{\text{X}}
+\newcommand{\modY}{Y}
+
+% basis sets 
+\newcommand{\Bas}{\mathcal{B}}
+\newcommand{\Basval}{\mathcal{B}_\text{val}}
+
+% operators
+\newcommand{\hT}{\Hat{T}}
+\newcommand{\hWee}{\Hat{W}_\text{ee}}
+
+%=================================================================
+%\subsection{Correcting the basis set error of a general WFT model}
+%=================================================================
+Let us assume we have both the density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$. 
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a good approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
 \begin{equation}
 	\label{eq:e0basis}
-  E \approx \efci + \efuncbasisfci
+	\E{}{} 
+	\approx \E{\modX}{\Bas} 
+	+ \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}],
 \end{equation}
-where $\efuncbasis$ is the complementary density functional defined in equation (8) of \onlinecite{GinPraFerAssSavTou-JCP-18}  
+where
 \begin{equation}
-  \begin{aligned}
 	\label{eq:E_funcbasis}
-    \efuncbasis = & \min_{\Psi \rightarrow \den}  \elemm{\Psi}{\kinop + \weeop}{\Psi} \\&- \min_{\psibasis  \rightarrow \den}  \elemm{\psibasis}{\kinop + \weeop}{\psibasis},
+	 \bE{}{\Bas}[\n{}{}]
-  \end{aligned}
+	= \min_{\wf{}{} \to \n{}{}}  \mel*{\wf{}{}}{\hT + \hWee}{\wf{}{}} 
+	- \min_{\wf{}{\Bas} \to \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee}{\wf{}{\Bas}}
 \end{equation}
-$\psibasis$ is a wave function obtained from the $N-$electron Hilbert space spanned by $\basis$, $\Psi$ is a general $N-$electron wave function being obtained in a complete basis, and both wave functions $\psibasis$ and $\Psi$ yield the same target density $\den$. 
+is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}.
+In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ is the wave function obtained from the $\Nel$-electron Hilbert space spanned by $\Bas$, and $\wf{}{}$ is a general $\Nel$-electron wave function being obtained in a complete basis. 
+Both wave functions yield the same target density $\n{}{}$. 
 
-Provided that the 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. 
+\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ 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 $\efuncbasis$ tends to zero
+An important aspect of such a theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$,
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\basis \rightarrow \infty} \efuncbasis = 0\qquad \forall \,\, \den\,\, ,
+        \lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0,
 \end{equation}
-which implies that the exact ground state energy coincides with the FCI energy in complete basis set (which we refer as $\efcicomplete$)
+which implies that
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\basis \rightarrow \infty} \efci + \efuncbasisfci = E \,\,.
+        \lim_{\Bas \rightarrow \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{\infty} \approx E,
 \end{equation}
+where $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in complete basis set.
+In the case of $\modX = \FCI$, we $\E{\FCI}{\infty} = E$.
+\alert{T2 stopped here.}
 
-Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\basis$ 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 
-\begin{equation}
+%\begin{equation}
- \efci \approx \emodel
+% \efci \approx \emodel
-\end{equation}
+%\end{equation}
-and 
+%and 
-\begin{equation}
+%\begin{equation}
- \denfci \approx \denmodel
+% \denfci \approx \denmodel
-\end{equation}
+%\end{equation}
-and by defining the energy provided by the model $\model$ in the complete basis set 
+%and by defining the energy provided by the model $\model$ in the complete basis set 
-\begin{equation}
+%\begin{equation}
- \emodelcomplete = \lim_{\basis \rightarrow \infty} \emodel\,\, , 
+% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, , 
-\end{equation}
+%\end{equation}
-we can then write 
+%we can then write 
-\begin{equation}
+%\begin{equation}
- \emodelcomplete \approx \emodel + \ecompmodel
+% \emodelcomplete \approx \emodel + \ecompmodel
-\end{equation}
+%\end{equation}
-which verifies the correct limit since 
+%which verifies the correct limit since 
-\begin{equation}
+%\begin{equation}
- \lim_{\basis \rightarrow \infty} \ecompmodel = 0\,\, .
+% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
-\end{equation}
+%\end{equation}
 
 
-\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
+%=================================================================
-In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in order to speed-up the basis set convergence of these models. 
+%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
+%=================================================================
+%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models. 
 
-\subsubsection{Basis set correction for the CCSD(T) energy}
+%=================================================================
-The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant. 
+%\subsubsection{Basis set correction for the CCSD(T) energy}
-Defining $\ecc$ as the CCSD(T) energy obtained in $\basis$, in the present notations we have 
+%=================================================================
-\begin{equation}
+%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant. 
- \emodel = \ecc \,\, .
+%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have 
-\end{equation}
+%\begin{equation}
-In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
+% \emodel = \ecc \,\, .
-\begin{equation}
+%\end{equation}
- \denmodel = \denhf \,\, .
+%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
-\end{equation}
+%\begin{equation}
-Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory. 
+% \denmodel = \denhf \,\, .
-Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
+%\end{equation}
-\begin{equation}
+%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory. 
- \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
+%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
-\end{equation}
+%\begin{equation}
+% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
+%\end{equation}
 
-\subsubsection{Correction of the CIPSI algorithm}
+%=================================================================
-The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
+%\subsubsection{Correction of the CIPSI algorithm}
-The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
+%=================================================================
-which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
+%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
-The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
+%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
-Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\basis$, the CIPSI energy is
+%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
-\begin{align}
+%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
-  E_\mathrm{CIPSI}^{\basis} &= E_\text{v} + E^{(2)} \,\,,
+%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
-\end{align}
+%\begin{align}
-where $E_\text{v}$ is the variational energy
+%  E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
-\begin{align}
+%\end{align}
- E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}}  }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
+%where $E_\text{v}$ is the variational energy
-\end{align}
+%\begin{align}
-where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
+% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}}  }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
-\begin{align}
+%\end{align}
-  E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
+%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
-\end{align}
+%\begin{align}
-where $\kappa$ denotes a determinant outside $\mathcal{R}$.
+%  E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
-To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
+%\end{align}
-of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
+%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
-The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
+%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
-the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$. 
+%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
-In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$. 
+%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
-
+%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$. 
-In the context of the basis set correction, we use the following conventions
+%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$. 
-\begin{equation}
+%
- \emodel = \EexFCIbasis
+%In the context of the basis set correction, we use the following conventions
-\end{equation}
+%\begin{equation}
-\begin{equation}
+% \emodel = \EexFCIbasis
- \denmodelr = \dencipsir
+%\end{equation}
-\end{equation}
+%\begin{equation}
-where the density $\dencipsir$ is defined as
+% \denmodelr = \dencipsir
-\begin{equation}
+%\end{equation}
- \dencipsi = \sum_{ij \in \basis} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
+%where the density $\dencipsir$ is defined as
-\end{equation}
+%\begin{equation}
-and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$. 
+% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
-
+%\end{equation}
-Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
+%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$. 
-\begin{equation}
+%
- \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
+%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
-\end{equation}
+%\begin{equation}
+% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
+%\end{equation}
 
 
 
-\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
+%=================================================================
-The functional $\efuncbasis$ is not universal as it depends on the basis set $\basis$ used and a simple analytical form for such a functional is of course not known. 
+%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
-Following the work of \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\efuncbasis$ in two-steps which grantee the correct behaviour in the limit of a complete basis set (see \eqref{eq:limitfunc}). 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{TouGorSav-TCA-05}, that we evaluate at the density $\denmodel$ provided by the model (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.   
+The functional $\bE{}{\Bas}[\n{}{}]$ is not universal as it depends on the basis set $\Bas$ used and a simple analytical form for such a functional is of course not known. 
+Following the work of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ in two steps which guarantee the correct behaviour in the limit of a complete basis set [see Eq.~\eqref{eq:limitfunc}]. 
+First, we define a real-space representation of the coulomb interaction projected in $\Bas$, 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{TouGorSav-TCA-05}, that we evaluate at the density $\denmodel$ provided by the model (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.   
 
 
-\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$}
+\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
 \label{sec:weff}
-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 originate from an Hamiltonian with 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. 
+One of the consequences of the use of an incomplete basis-set $\Bas$ 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 originate from an Hamiltonian with a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\Bas$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point. 
 
 The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
 \begin{itemize}
- \item is non-divergent at the electron coalescence point as long as an incomplete basis set $\basis$ is used, 
+ \item is non-divergent at the electron coalescence point as long as an incomplete basis set $\Bas$ is used, 
- \item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\basis$. 
+ \item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\Bas$. 
 \end{itemize}
-\subsubsection{General definition of an effective interaction for the basis set $\basis$}
+\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
-Consider the coulomb operator projected in the basis-set $\basis$ 
+Consider the coulomb operator projected in the basis-set $\Bas$ 
  \begin{equation}
     \begin{aligned}
-      \weeopbasis =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basis} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
+      \weeopbasis =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\Bas} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
     \end{aligned}
  \end{equation}
-where the indices run over all orthonormal spin-orbitals in $\basis$ and $\vijkl$ are the usual coulomb two-electron integrals. 
+where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual coulomb two-electron integrals. 
-Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\basis$. 
+Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\Bas$. 
 After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates: 
 \begin{equation}
    \label{eq:expectweeb}
@@ -384,7 +419,7 @@ where the function $\fbasis$ is
  \begin{equation}
    \label{eq:fbasis}
    \begin{aligned}
-   \fbasis =  \sum_{ijklmn\,\,\in\,\,\basis} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}\,\,,
+   \fbasis =  \sum_{ijklmn\,\,\in\,\,\Bas} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}\,\,,
    \end{aligned}
  \end{equation}
 $\gammamnpq{\psibasis}$ is the two-body density tensor of $\psibasis$
@@ -406,7 +441,7 @@ Then, consider the expectation value of the exact coulomb operator over $\psibas
          \elemm{\psibasis}{\weeop}{\psibasis} =  \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \frac{1}{r_{12}} \twodmrdiagpsi\,\, 
  \end{equation}
 where $\twodmrdiagpsi$  is the two-body density associated to $\psibasis$. 
-Because $\psibasis$ belongs to $\basis$, such an expectation value coincides with the expectation value of $\weeopbasis$
+Because $\psibasis$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
  \begin{equation}
    \elemm{\psibasis}{\weeopbasis}{\psibasis} = \elemm{\psibasis}{\weeop}{\psibasis},
  \end{equation}
@@ -422,14 +457,14 @@ where we introduced $\wbasis$
    \label{eq:def_weebasis}
    \wbasis = \frac{\fbasis}{\twodmrdiagpsi},
  \end{equation}
- which is  the effective interaction in the basis set $\basis$.  
+ which is  the effective interaction in the basis set $\Bas$.  
 
-As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\basis$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence  point for an incomplete basis set $\basis$. 
+As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\Bas$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence  point for an incomplete basis set $\Bas$. 
-Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\basis$. 
+Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\Bas$. 
 
 
 \subsubsection{Definition of a valence effective interaction}
-As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\basisval$. 
+As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\Basval$. 
 
 According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying 
 \begin{equation}
@@ -439,15 +474,15 @@ According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective in
 where $\weeopbasisval$ is the valence coulomb operator defined as
  \begin{equation}
     \begin{aligned}
-      \weeopbasisval =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basisval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i}\,\,\,,
+      \weeopbasisval =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\Basval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i}\,\,\,,
     \end{aligned}
  \end{equation}
-and $\basisval$ is the subset of molecular orbitals for which we want to define the expectation value, which will be typically the all MOs except those frozen. 
+and $\Basval$ is the subset of molecular orbitals for which we want to define the expectation value, which will be typically the all MOs except those frozen. 
 Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
  \begin{equation}
    \label{eq:fbasisval}
    \begin{aligned}
-   \fbasisval =  \sum_{ij\,\,\in\,\,\basis}  \,\, \sum_{klmn\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}.
+   \fbasisval =  \sum_{ij\,\,\in\,\,\Bas}  \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}.
    \end{aligned}
  \end{equation}
 
@@ -458,9 +493,9 @@ Then, the effective interaction associated to the valence $\wbasisval$ is simply
  \end{equation}
 where $\twodmrdiagpsival$ is the two body density associated to the valence electrons:
 \begin{equation}
- \twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\basisval} \gammamnkl[\psibasis] \,\, \phix{m}{1} \phix{n}{2}  \phix{k}{1} \phix{l}{2} .
+ \twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\psibasis] \,\, \phix{m}{1} \phix{n}{2}  \phix{k}{1} \phix{l}{2} .
 \end{equation}
-It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\basis$,  and the $(k,l,m,n)$, which span only the valence space $\basisval$. 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 $\basis$, whatever the choice of subset $\basisval$. 
+It is important to notice in \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 $\Basval$. 
 
 
 \subsubsection{Definition of a range-separation parameter varying in space}
@@ -489,26 +524,26 @@ As we defined an effective interaction for the valence electrons, we also introd
  \label{eq:mu_of_r_val}
  \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
 \end{equation}
-An important point to notice is that, in the limit of a complete basis set $\basis$, as 
+An important point to notice is that, in the limit of a complete basis set $\Bas$, as 
 \begin{equation}
  \begin{aligned}
- &\lim_{\basis \rightarrow \infty}\wbasis = 1/r_{12}  \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
+ &\lim_{\Bas \rightarrow \infty}\wbasis = 1/r_{12}  \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
- &\lim_{\basis \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
+ &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
  \end{aligned}
 \end{equation}
 one has
 \begin{equation}
  \begin{aligned}
- &\lim_{\basis \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
+ &\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
- &\lim_{\basis \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
+ &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
  \end{aligned}
 \end{equation}
 and therefore 
 \begin{equation}
  \label{eq:lim_mur}
  \begin{aligned}
- &\lim_{\basis \rightarrow \infty} \murpsi = +\infty \,\, \\
+ &\lim_{\Bas \rightarrow \infty} \murpsi = +\infty \,\, \\
- &\lim_{\basis \rightarrow \infty} \murpsival = +\infty \,\, .
+ &\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
  \end{aligned}
 \end{equation}
 
@@ -560,7 +595,7 @@ A general scheme to approximate  $\ecompmodel$ is to use $\ecmuapprox$ with the
 Therefore, any approximated ECMD can be used to estimate $\ecompmodel$. 
 It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has 
 \begin{equation}
- \lim_{\basis \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
+ \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
 \end{equation}
 for whatever choice of density $\denmodel$, wave function $\psibasis$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. 
 
@@ -621,7 +656,7 @@ We now introduce a valence-only approximation for the complementary functional w
 Defining the valence one-body spin density matrix as
 \begin{equation}
  \begin{aligned}
- \onedmval[\psibasis] & = \elemm{\psibasis}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\psibasis} \qquad \text{if }(i,j)\in \basisval \\
+ \onedmval[\psibasis] & = \elemm{\psibasis}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\psibasis} \qquad \text{if }(i,j)\in \Basval \\
                       & = 0 \qquad \text{in other cases}
  \end{aligned}
 \end{equation}
@@ -644,7 +679,7 @@ Therefore, we propose the following valence-only approximations for the compleme
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
-We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\basis$, the set of valence orbitals $\basisval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core. 
+We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core. 
 %\subsubsection{CIPSI calculations and the basis-set correction}
 %All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$. 
 %Regarding the wave function $\psibasis$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.