From b308434029e8cc3e91f4d13aa73431e05d2e78a1 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Mon, 22 Apr 2019 22:00:45 +0200
Subject: [PATCH] first screening

---
 Cover_Letter/CoverLetter.tex  |  2 +-
 Manuscript/G2-srDFT.tex       | 81 +++++++++++++++++------------------
 Manuscript/SI/G2_srDFT-SI.tex |  2 +-
 3 files changed, 42 insertions(+), 43 deletions(-)

diff --git a/Cover_Letter/CoverLetter.tex b/Cover_Letter/CoverLetter.tex
index 4b226ba..d59fda9 100644
--- a/Cover_Letter/CoverLetter.tex
+++ b/Cover_Letter/CoverLetter.tex
@@ -14,7 +14,7 @@
 \justifying
 Please find enclosed our manuscript entitled
 \begin{quote}
-\textit{``A Density-Based Basis-Set Correction For Wave-Function Theory''}, 
+\textit{``A Density-Based Basis Set Correction For Wave Function Theory''}, 
 \end{quote}
 which we would like you to consider as a Letter in the \textit{Journal of Physical Chemistry Letters}.
 This contribution fits nicely in the section \textit{``Spectroscopy and Photochemistry; General theory''}. 
diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex
index c60b81d..e6c75a8 100644
--- a/Manuscript/G2-srDFT.tex
+++ b/Manuscript/G2-srDFT.tex
@@ -118,7 +118,7 @@
 
 \begin{document}	
 
-\title{A Density-Based Basis-Set Correction For Wave-Function Theory}
+\title{A Density-Based Basis-Set Correction For Wave Function Theory}
 
 \author{Pierre-Fran\c{c}ois Loos}
 \email{loos@irsamc.ups-tlse.fr}
@@ -141,7 +141,7 @@
 	\includegraphics[width=\linewidth]{TOC}
 \end{wrapfigure}
 We report a universal density-based basis set incompleteness correction that can be applied to any wave function method.
-The present correction, which appropriately vanishes in the complete-basis-set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
+The present correction, which appropriately vanishes in the complete basis set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
 Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separation \textit{parameter} $\mu$, the key ingredient here is a range-separation \textit{function} $\mu(\bf{r})$ which automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
 As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization and correlation energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets. 
 \end{abstract}
@@ -186,7 +186,7 @@ Here, we only provide the main working equations.
 We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.  
 
 Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$. 
-According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be \titou{approximated} as
 \begin{equation}
 	\label{eq:e0basis}
 	\E{}{} 
@@ -202,7 +202,7 @@ where
 \end{equation}
 is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
 In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
-Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in the basis set $\Bas$). 
+Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in \trashPFL{the basis set} $\Bas$). 
 
 Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$. 
 This implies that
@@ -212,11 +212,11 @@ This implies that
 \end{equation}
 where  $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
 In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a 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 $\modY$ and $\modZ$.
+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 $\modY$ and $\modZ$, \titou{and the lack of self-consistency.}
 
 The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
 Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct 
-for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions. 
+for the lack of \titou{cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$} at the e-e coalescence points, a universal condition of exact wave functions. 
 Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence. 
 Therefore, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
 Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.  
@@ -233,7 +233,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
 	\label{eq:def_weebasis}
 	\W{\Bas}{}(\br{1},\br{2})   = 
  	\begin{cases}
-       \f{\Bas}{}(\br{1},\br{2})/\n{2,\Bas}{}(\br{1},\br{2}), 	& \text{if $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$,}
+       \f{\Bas}{}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), 	& \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
        \\
        \infty, 												& \text{otherwise,}
     \end{cases}
@@ -241,7 +241,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
 where
 \begin{equation}
 	\label{eq:n2basis}
-	\n{2,\Bas}{}(\br{1},\br{2})
+	\n{2}{\Bas}(\br{1},\br{2})
 	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
 \end{equation}
 and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
@@ -254,15 +254,14 @@ and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb inte
 With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
 \begin{equation}
 	\label{eq:int_eq_wee}
-        \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
+        \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
 where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
-Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
-\begin{eqnarray}
-        \frac{1}{2}\iint \frac{1}{r_{12}} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx}
-\nonumber\\
- \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
-\end{eqnarray}
+Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
+\begin{equation}
+	\alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
+	\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
+\end{equation}
 it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
 Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp.
 
@@ -270,9 +269,9 @@ As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}
 Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) 
 \begin{equation}
 	\label{eq:lim_W}
-        \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \frac{1}{r_{12}}   
+        \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \titou{r_{12}^{-1}   }
 \end{equation}
-for any $(\br{1},\br{2})$ such that $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. 
+for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. 
 
 %=================================================================
 %\subsection{Range-separation function}
@@ -283,7 +282,7 @@ A key quantity is the value of the effective interaction at coalescence of oppos
 %	\label{eq:wcoal}
 %	\W{\Bas}{}(\br{}) =  \W{\Bas}{}(\br{},{\br{}}),
 %\end{equation}
-which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2,\Bas}{}(\br{},\br{})$ is non vanishing. 
+which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing. 
 Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
 Although this choice is not unique, we choose here the range-separation function
 \begin{equation}
@@ -341,10 +340,10 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
 \begin{subequations}
 \begin{align}
 	\label{eq:large_mu_ecmd}
-	\lim_{\mu \to \infty}  \bE{}{\sr}[\n{}{},\rsmu{}{}] & = 0,
+	\lim_{\mu \to \infty}  \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
 	\\
 	\label{eq:small_mu_ecmd}
-	\lim_{\mu \to 0}  \bE{}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
+	\lim_{\mu \to 0}  \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
 \end{align}
 \end{subequations}
 where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. 
@@ -358,29 +357,29 @@ Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated wi
 The local-density approximation (LDA) of the ECMD complementary functional is defined as 
 \begin{equation}
 	\label{eq:def_lda_tot}
-        \bE{\LDA}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\LDA}\qty(\{\n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
+        \bE{\LDA}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{},
 \end{equation}
-where $\be{\text{c,md}}{\sr,\LDA}(\{\n{\sigma}{}\},\rsmu{}{})$ is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\{\n{\sigma}{}\}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
+where $\be{\text{c,md}}{\sr,\LDA}(\qty{\n{\sigma}{}},\rsmu{}{})$ is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
 The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. 
 In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional 
-\begin{eqnarray}
+\begin{multline}
 	\label{eq:def_pbe_tot}
-        \bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxx}
-\nonumber\\
- \int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\PBE}\qty(\{\n{\sigma}{}(\br{})\},\{\nabla \n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
-\end{eqnarray}
-inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional~\cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
+	\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = 
+	\\
+	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{},
+\end{multline}
+inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional~\cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
 \begin{subequations}
 \begin{gather}
 	\label{eq:epsilon_cmdpbe}
-	\be{\text{c,md}}{\sr,\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{1 + \beta(\{n_\sigma\},\{\nabla n_\sigma\})\; \rsmu{}{3} },
+	\be{\text{c,md}}{\sr,\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})}{1 + \beta(\qty{n_\sigma},\qty{\nabla n_\sigma}) \rsmu{}{3} },
 	\\
 	\label{eq:beta_cmdpbe}
-	\beta(\{n_\sigma\},\{\nabla n_\sigma\}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(0,\{\n{\sigma}{}\})}.
+	\beta(\qty{n_\sigma},\qty{\nabla n_\sigma}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})}{\n{2}{\UEG}(0,\qty{\n{\sigma}{}})}.
 \end{gather}
 \end{subequations}
-The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2,\Bas}{}(\br{},\br{})  \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, where $0$ refers to $r_{12}=0$ and $\n{2}{\UEG}(0,\{n_\sigma\}) \approx 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
-This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2,\Bas}{}(\br{},\br{})$.
+The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(0,\qty{\n{\sigma}{}(\br{})})$, where $0$ refers to $r_{12}=0$ and $\n{2}{\UEG}(0,\qty{n_\sigma}) \approx 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
+This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
 
 Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
 
@@ -391,9 +390,9 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
 As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs. 
 We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
  \begin{equation}
-	\W{\Bas}{\FC}(\br{1},\br{2})  \! = \!
+	\W{\Bas}{\FC}(\br{1},\br{2})  = 
     \begin{cases}
-	\f{\Bas}{\FC}(\br{1},\br{2})/\n{2,\Bas}{\FC}(\br{1},\br{2}),\!  & \!\!\! \text{if $\n{2,\Bas}{\FC}(\br{1},\br{2}) \!\ne \! 0$},
+	\f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\Bas,\FC}(\br{1},\br{2}),  & \text{if $\n{2}{\Bas,\FC}(\br{1},\br{2}) \ne 0$},
 	\\
 	\infty, \! & \!\!\! \text{otherwise,}
     \end{cases}
@@ -405,7 +404,7 @@ with
 	\f{\Bas}{\FC}(\br{1},\br{2})
 	=  \sum_{pq \in \Bas} \sum_{rstu \in \BasFC}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
 	\\
-	\n{2,\Bas}{\FC}(\br{1},\br{2})
+	\n{2}{\Bas,\FC}(\br{1},\br{2})
 	= \sum_{pqrs \in \BasFC}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
 \end{gather}
 \end{subequations}
@@ -422,9 +421,9 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
 %\subsection{Computational considerations}
 %=================================================================
 The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{},\br{})$ at each quadrature grid point. 
-Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.  
+Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a \titou{multi}-determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.  
 As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
-Hence, we will stick to this choice throughout the current study.
+Hence, we will stick to this choice throughout the \titou{present} study.
 In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. 
 Nevertheless, this step usually has to be performed for most correlated WFT calculations. 
 Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.
@@ -494,15 +493,15 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
 	\label{fig:G2_Ec}}
 \end{figure*}
 
-We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
+We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis \trashPFL{sets} (X $=$ D, T, Q and 5).
 \ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11} 
-In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
+In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ \titou{basis set family}.
 This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
 As a method $\modY$ we employ either CCSD(T) or exFCI.
 Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
 We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
 In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy. 
-In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several millions of determinants.
+In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several millions \trashPFL{of} determinants.
 CCSD(T) energies are computed with Gaussian09 using standard threshold values, \cite{g09} while RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
 For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
 Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
@@ -511,7 +510,7 @@ In the context of the basis-set correction, the set of active MOs $\BasFC$ invol
 The FC density-based correction is used consistently when the FC approximation was applied in WFT methods.
 To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}, we perform a two-point X$^{-3}$ extrapolation of the correlation energies using the quadruple- and quintuple-$\zeta$ data that we add up to the HF energies obtained in the largest (i.e.~quintuple-$\zeta$) basis.
 
-As the exFCI calculations are converged with a precision of about 0.1 {\kcal} on atomization energies, we can label those as near-FCI. 
+As the exFCI \titou{atomization energies} are converged with a precision of about 0.1 {\kcal} \trashPFL{on atomization energies}, we can label \titou{these} as near-FCI. 
 Hence, they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}. 
 The results for these diatomic molecules are reported in Fig.~\ref{fig:diatomics}. 
 The corresponding numerical data can be found in the {\SI}.
diff --git a/Manuscript/SI/G2_srDFT-SI.tex b/Manuscript/SI/G2_srDFT-SI.tex
index 4bc3390..17c9197 100644
--- a/Manuscript/SI/G2_srDFT-SI.tex
+++ b/Manuscript/SI/G2_srDFT-SI.tex
@@ -116,7 +116,7 @@
 
 \begin{document}	
 
-\title{Supplementary Information for ``A Density-Based Basis-Set Correction For Wave-Function Theory''}
+\title{Supplementary Information for ``A Density-Based Basis Set Correction For Wave Function Theory''}
 
 \author{Pierre-Fran\c{c}ois Loos}
 \email{loos@irsamc.ups-tlse.fr}