baby is awake

JPCL_revision/G2-srDFT.tex

@@ -316,7 +316,6 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
 	\label{eq:large_mu_ecmd}
 	\lim_{\mu \to \infty}  \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
 	&
-%	\label{eq:small_mu_ecmd}
 	\lim_{\mu \to 0}  \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
 \end{align}
 where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. 
@@ -349,7 +348,7 @@ $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ 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 \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}. 
+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 \cite{LooGil-WIRES-16} (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 $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.}
@@ -395,7 +394,7 @@ As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{
 Hence, we will stick to this choice throughout the 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. 
-\trashPFL{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.}
+%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.
 
 To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
 i) can be applied to any WFT method that provides an energy and a density, 
@@ -418,7 +417,7 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
 	\caption{
 	Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
 	The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
-	See {\SI} for raw data.
+	See {\SI} for raw data \titou{and the corresponding LDA results}.
 	\label{fig:diatomics}}
 \end{figure*}
 
@@ -428,7 +427,7 @@ 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 \titou{and the definition of the LDA ECMD functional}.
+	See {\SI} for raw data \titou{and the corresponding LDA results}.
 	\label{tab:stats}}
 	\begin{ruledtabular}
 		\begin{tabular}{ldddd}
@@ -438,10 +437,10 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
 			CCSD(T)/cc-pVTZ			&	6.06		&	6.84			&	14.25		&	2			\\
 			CCSD(T)/cc-pVQZ			&	2.50		&	2.86			&	6.75		&	9			\\
 			CCSD(T)/cc-pV5Z			&	1.28		&	1.46			&	3.46		&	21			\\
-			\\
-			CCSD(T)+LDA/cc-pVDZ		&	3.24		&	3.67			&	8.13		&	7			\\
-			CCSD(T)+LDA/cc-pVTZ		&	1.19		&	1.49			&	4.67		&	27			\\
-			CCSD(T)+LDA/cc-pVQZ		&	0.33		&	0.44			&	1.32		&	53			\\
+%			\\
+%			CCSD(T)+LDA/cc-pVDZ		&	3.24		&	3.67			&	8.13		&	7			\\
+%			CCSD(T)+LDA/cc-pVTZ		&	1.19		&	1.49			&	4.67		&	27			\\
+%			CCSD(T)+LDA/cc-pVQZ		&	0.33		&	0.44			&	1.32		&	53			\\
 			\\
 			CCSD(T)+PBE/cc-pVDZ		&	1.96		&	2.59			&	7.33		&	19			\\
 			CCSD(T)+PBE/cc-pVTZ		&	0.85		&	1.11			&	2.64		&	36			\\
@@ -454,11 +453,13 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
 \begin{figure*}
 	\includegraphics[width=\linewidth]{fig2a}
 	\includegraphics[width=\linewidth]{fig2b}
-	\includegraphics[width=\linewidth]{fig2c}
+%	\includegraphics[width=\linewidth]{fig2c}
 	\caption{
-	Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
+	Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).}
+%	Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
 	The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
-	See {\SI} for raw data.
+	\titou{Note the difference in scaling of the vertical axes.}
+	See {\SI} for raw data \titou{and the corresponding LDA results}.
 	\label{fig:G2_Ec}}
 \end{figure*}
 
@@ -470,7 +471,8 @@ This molecular set has been intensively studied in the last 20 years (see, for e
 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, \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 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, \titou{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}
@@ -483,24 +485,25 @@ To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKl
 As the exFCI atomization energies are converged with a precision of about 0.1 {\kcal}, we can label 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}.
+The corresponding numerical data \titou{(as well as the corresponding LDA results)} 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 \trashPFL{(both at the LDA and PBE levels)}.
+First, in a given basis set, the basis-set correction systematically improves the atomization energies.
+%(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.
-\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.}
+%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.
 
 %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 \titou{CCSD(T)+PBE} values are depicted in Fig.~\ref{fig:G2_Ec}.
-The raw data can be found in the {\SI}.
+The raw data \titou{(as well as the corresponding LDA results)} 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}.
@@ -516,9 +519,9 @@ Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely cla
 Encouraged by these promising results, we are currently pursuing various avenues toward basis-set reduction for strongly correlated systems and electronically excited states.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section*{Supporting information}
+\section*{Supporting Information Available}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set.
+See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set \titou{as well as the definition of the LDA ECMD functionals (and the corresponding numerical results).}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}

JPCL_revision/SI/G2_srDFT-SI.tex

@@ -153,6 +153,58 @@ The sensitivity with respect to the RS-DFT functional is quite large for the dou
 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.
 
+%%% FIGURE 1 %%%
+\begin{figure*}
+	\includegraphics[width=0.30\linewidth]{fig1a}
+	\hspace{1cm}
+	\includegraphics[width=0.30\linewidth]{fig1b}
+	\\
+	\includegraphics[width=0.30\linewidth]{fig1c}
+	\hspace{1cm}
+	\includegraphics[width=0.30\linewidth]{fig1d}
+	\caption{
+	Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
+	The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
+	\label{fig:diatomics}}
+\end{figure*}
+
+%%% TABLE II %%%
+\begin{table}
+	\caption{
+	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.
+	\label{tab:stats}}
+	\begin{ruledtabular}
+		\begin{tabular}{ldddd}
+			Method					&	\tabc{MAD}	&	\tabc{RMSD}		&	\tabc{MAX}	&	\tabc{CA}	\\
+			\hline
+			CCSD(T)/cc-pVDZ			&	14.29		&	16.21			&	36.95		&	2			\\
+			CCSD(T)/cc-pVTZ			&	6.06		&	6.84			&	14.25		&	2			\\
+			CCSD(T)/cc-pVQZ			&	2.50		&	2.86			&	6.75		&	9			\\
+			CCSD(T)/cc-pV5Z			&	1.28		&	1.46			&	3.46		&	21			\\
+			\\
+			CCSD(T)+LDA/cc-pVDZ		&	3.24		&	3.67			&	8.13		&	7			\\
+			CCSD(T)+LDA/cc-pVTZ		&	1.19		&	1.49			&	4.67		&	27			\\
+			CCSD(T)+LDA/cc-pVQZ		&	0.33		&	0.44			&	1.32		&	53			\\
+			\\
+			CCSD(T)+PBE/cc-pVDZ		&	1.96		&	2.59			&	7.33		&	19			\\
+			CCSD(T)+PBE/cc-pVTZ		&	0.85		&	1.11			&	2.64		&	36			\\
+			CCSD(T)+PBE/cc-pVQZ		&	0.31		&	0.42			&	1.16		&	53			\\
+		\end{tabular}
+	\end{ruledtabular}
+\end{table}
+
+%%% FIGURE 2 %%%
+\begin{figure*}
+	\includegraphics[width=\linewidth]{fig2a}
+	\includegraphics[width=\linewidth]{fig2b}
+	\includegraphics[width=\linewidth]{fig2c}
+	\caption{
+	Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
+	The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
+	\label{fig:G2_Ec}}
+\end{figure*}
 
 %%% TABLE I %%%
 \begin{table*}
@@ -288,7 +340,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
 \end{squeezetable}
 \end{turnpage}
 
-\bibliography{../G2_srDFT,../G2_srDFT-control}
+\bibliography{../G2-srDFT,../G2-srDFT-control}