starting revision

JPCL_revision/G2-srDFT.tex

@@ -193,13 +193,15 @@ The present basis-set correction relies on the RS-DFT formalism to capture the m
 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 $\Bas$. 
+%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 approximated 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 approximated as
+Let us assume we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.}
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
 \begin{equation}
 	\label{eq:e0basis}
-	\E{}{} 
+	\titou{\E{}{} 
-	\approx \E{\modY}{\Bas} 
+	\approx \E{\CCSDT}{\Bas} 
-	+ \bE{}{\Bas}[\n{\modZ}{\Bas}],
+	+ \bE{}{\Bas}[\n{\HF}{\Bas}],}
 \end{equation}
 where 
 \begin{equation}
@@ -215,11 +217,15 @@ Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have,
 This implies that
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
+        \titou{\lim_{\Bas \to \infty} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{} \approx \E{}{},}
 \end{equation}
-where  $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
+%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 $\E{\FCI}{} = \E{}{}$.
+where \titou{$\E{\CCSDT}{}$ is the $\CCSDT$ energy} in the CBS limit.
-Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
+\titou{Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.}
+%In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
+In the case where \titou{$\CCSDT$ is replaced by $\FCI$} 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 sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
+Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, and the lack of self-consistency in the present scheme.
 
 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 
@@ -343,12 +349,11 @@ $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between
 	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)}.
 \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.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
+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 \titou{uniform electron gas (UEG)} version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ 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 $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
-\titou{The complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.}
+\titou{The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.}
-
+\titou{The local-density approximation (LDA) version of the ECMD functional is discussed in the {\SI}.}
-
 %=================================================================
 %\subsection{Frozen-core approximation}
 %=================================================================
@@ -378,7 +383,7 @@ with
 and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
 It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
 %Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
-\titou{Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$}.
+\titou{Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$}.
 
 %=================================================================
 %\subsection{Computational considerations}
@@ -423,7 +428,7 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
 	Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}. 
 	Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
 	CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
-	See {\SI} for raw data.
+	See {\SI} for raw data \titou{and the definition of the LDA ECMD functional}.
 	\label{tab:stats}}
 	\begin{ruledtabular}
 		\begin{tabular}{ldddd}
@@ -461,10 +466,11 @@ We begin our investigation of the performance of the basis-set correction by com
 \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 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.
+%As a method $\modY$ we employ either CCSD(T) or exFCI.
+As \titou{a ``reference'' method}, 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 the CCSD(T) calculations, \trashPFL{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 million 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}
@@ -481,27 +487,30 @@ The corresponding numerical data can be found in the {\SI}.
 As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with the cc-pV5Z basis set, and the atomization energies are consistently underestimated.
 A similar trend holds for CCSD(T).
 Regarding the effect of the basis-set correction, several general observations can be made for both exFCI and CCSD(T). 
-First, in a given basis set, the basis-set correction systematically improves the atomization energies (both at the LDA and PBE levels).
+First, in a given basis set, the basis-set correction systematically improves the atomization energies \trashPFL{(both at the LDA and PBE levels)}.
 A small overestimation can occur compared to the CBS value by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level). 
 Nevertheless, the deviation observed for the largest basis set is typically within the CBS extrapolation error, which is highly satisfactory knowing the marginal computational cost of the present correction.
 In most cases, the basis-set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones.
-Importantly, the sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. 
+\trashPFL{Importantly, the sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. 
 However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}. 
-Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach. 
+Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.}
 
-As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
+%As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
+As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set \titou{with $\CCSDT$} and the cc-pVXZ basis sets.
 Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.
 Investigating the convergence of correlation energies (or difference of such quantities) is commonly done to appreciate the performance of basis-set corrections aiming at correcting two-electron effects. \cite{Tenno-CPL-04, TewKloNeiHat-PCCP-07, IrmGru-arXiv-2019}
-The ``plain'' CCSD(T) atomization energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
+The ``plain'' CCSD(T) atomization energies as well as the corrected \titou{CCSD(T)+PBE} values are depicted in Fig.~\ref{fig:G2_Ec}.
 The raw data can be found in the {\SI}.
 A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies.
 Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
 From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
 For a commonly used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
 Applying the basis-set correction drastically reduces the basis-set incompleteness error.
-Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
+%Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ level, the MAD is reduced to 3.24 and 1.96 {\kcal}.
+Already at the \titou{CCSD(T)+PBE/cc-pVDZ level}, the MAD is reduced to 1.96 {\kcal}.
 With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
-CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
+%CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
+\titou{CCSD(T)+PBE/cc-pVQZ returns a MAD of 0.31 kcal/mol} while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
 
 Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis-set correction provides significant basis-set reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
 Encouraged by these promising results, we are currently pursuing various avenues toward basis-set reduction for strongly correlated systems and electronically excited states.

JPCL_revision/SI/G2_srDFT-SI.tex

@@ -149,6 +149,11 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
 where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
 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 $\mu$. 
 
+The sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. 
+However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}. 
+Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
+
+
 %%% TABLE I %%%
 \begin{table*}
 	\caption{

Response_Letter/ResponseLetter.tex

@@ -33,6 +33,9 @@ We look forward to hearing from you.
 	I expect it to have immediate applications among users, certainly once user friendly code is released. 
 	The work is definitely worthy of being published in JPCL, after changes. 
 	That said, for the reasons explained below, I think the current manuscript requires changes somewhere between major and minor.}
+	\\
+	\alert{We thank the reviewer for his/her support.
+	We also believe that the present contribution is a major advance in electronic structure theory.}
 	
 	\item 
 	\textit{My main issue with the manuscript is that it is not sufficiently well self-contained or explained. 
@@ -41,18 +44,23 @@ We look forward to hearing from you.
 	For example, the authors refer several times to work in the Appendix of a previous paper. 
 	The gist of such results should (IMO) be summarised here.}
 	\\
-	\alert{We have added a summary of the different results derived in the previous paper.}
+	\alert{I am not super sure this is worth it.
+	%We have added a summary of the different results derived in the previous paper.
+	}
 	
 	\item 
 	\textit{This readability issue is not made easier by the authors' commendable focus on generality, which leaves the reader carrying a lot of variables and ideas in their head. 
 	Fine for a long paper, not so much for a Letter.}
+	\\
+	\alert{Thank you for pointing that out.
+	We have made the manuscript more explicit.}
 	
 	\item 
 	\textit{My first suggestion to the authors would be to change from describing things in terms of a generic method "Y" to using a specific case [e.g. CCSD(T)] and then generalizing only at the end, e.g., "Of course, the above holds true for any method that provides a good approximation to the energy, not just CCSD(T).". 
 	Other changes along these lines would probably also be useful. 
 	This would help the reader cement the key concept (basis correction) without worrying about quite so many variables.}
 	\\
-	\alert{As proposed by the reviewer, we have explicitly specified the methods X and Y we employ here and left for the end the generalization to any method.}	
+	\alert{As proposed by the reviewer, we have explicitly specified the methods X and Y that we have employed.}	
 		
 	\item 
 	\textit{On a related note, I do not see the benefit of reporting the LDA correction in the main text, although for sure it belongs in the SI. 
@@ -68,7 +76,8 @@ We look forward to hearing from you.
 	If values for multiple basis sets were reported it might also help in understanding how and where larger basis sets help, which might point to how to improve basis sets in a more systematic fashion. 
 	Removing the discussion on LDA would probably free enough space to show this, especially if Figure 2 was condensed into a single figure (which should be feasible sans LDA).}
 	\\
-	\alert{This is for Manu!}	
+	\alert{This is for Manu!
+	We have reported a figure showing $\mu(\bm{r})$ in \ce{} for various basis sets.}	
 		
 	\item 
 	\textit{One final (minor) key point is that the proposed use of density fitting or related time-saving steps seems rather ambitious, given that it necessarily introduces a further basis set dependence.