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Cover_Letter/CoverLetter.tex
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@ -1,6 +1,8 @@
\documentclass[10pt]{letter} \documentclass[10pt]{letter}
\usepackage{UPS_letterhead,color,mhchem,mathpazo,ragged2e} \usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e}
\newcommand{\alert}[1]{\textcolor{red}{#1}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\begin{document} \begin{document}
@ -15,7 +17,7 @@ Please find enclosed our manuscript entitled
\textit{``A Density-based Basis Set Correction for Wave function Theory''}, \textit{``A Density-based Basis Set Correction for Wave function Theory''},
\end{quote} \end{quote}
which we would like you to consider as a Letter in the \textit{Journal of Physical Chemistry Letters}. 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''}. This contribution fits nicely in the section \textit{``Spectroscopy and Photochemistry; General theory''} {\color{darkgreen}MG: are we sure of the section ?}.
One of the most fundamental drawbacks of conventional wave function methods is the slow convergence of energies and properties with respect to the one-electron basis set. One of the most fundamental drawbacks of conventional wave function methods is the slow convergence of energies and properties with respect to the one-electron basis set.
As proposed by Kutzelnigg more than thirty years ago, one can introduce explicitly the interelectronic distance $r_{12}$ to significantly speed up the convergence. As proposed by Kutzelnigg more than thirty years ago, one can introduce explicitly the interelectronic distance $r_{12}$ to significantly speed up the convergence.

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Manuscript/G2-srDFT.tex
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@ -168,7 +168,7 @@ Present-day DFT calculations are almost exclusively done within the so-called Ko
DFT's attractiveness originates from its very favorable cost/efficiency ratio as it can provide accurate energies and properties at a relatively low computational cost. DFT's attractiveness originates from its very favorable cost/efficiency ratio as it can provide accurate energies and properties at a relatively low computational cost.
Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89} Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05} Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15} In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the \trashMG{size} \manu{quality} of the basis set. \cite{FraMusLupTou-JCP-15}
Progress toward unifying WFT and DFT are on-going. Progress toward unifying WFT and DFT are on-going.
In particular, range-separated DFT (RS-DFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a smooth long-range part and a (complementary) short-range part treated with WFT and DFT, respectively. In particular, range-separated DFT (RS-DFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a smooth long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
@ -176,7 +176,7 @@ As the WFT method is relieved from describing the short-range part of the correl
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, FerGinTou-JCP-18} WFT approaches. Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, FerGinTou-JCP-18} WFT approaches.
Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18} Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
The present work proposes an extension of this new methodological development together alongside the first numerical tests on molecular systems. The present work proposes an extension of this new methodological development \trashMG{together} alongside the first numerical tests on molecular systems.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%\section{Theory} %\section{Theory}
@ -201,10 +201,10 @@ where
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation} \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. 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 wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set \manu{(CBS)}, respectively.
Both wave functions yield the same target density $\n{}{}$. Both wave functions yield the same target density $\n{}{}$.
Importantly, in the limit of a complete basis set (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$. Importantly, \trashMG{in the limit of a complete basis set} \manu{when reaching 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 This implies that
\begin{equation} \begin{equation}
\label{eq:limitfunc} \label{eq:limitfunc}
@ -250,7 +250,7 @@ and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\
\f{\Bas}{}(\br{1},\br{2}) \f{\Bas}{}(\br{1},\br{2})
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals. \trashMG{and} $\V{pq}{rs}$ are the usual two-electron Coulomb integrals \manu{and $\wf{}{\Bas}$ is a wave function belonging to the $N-$electron Hilbert space of $\Bas$ and which will be defined later on. }
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
@ -412,8 +412,8 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
%\subsection{Computational considerations} %\subsection{Computational considerations}
%================================================================= %=================================================================
The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point. The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant. Yet embarrassingly parallel, this step scales, in the general (\manu{\textit{i.e.} $\wf{}{\Bas}$ is multi-determinantal in \eqref{eq:n2basis} and \eqref{eq:fbasis} }) case, as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant \manu{for $\wf{}{\Bas}$}.
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this simple choice already provides, for weakly correlated systems, a quantitative representation of the basis set incompleteness. As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this \trashMG{simple choice} \manu{simplification for $\wf{}{\Bas}$} already provides, for weakly correlated systems, a quantitative representation of the \trashMG{basis set} incompleteness \manu{of the basis set $\Bas$}.
Hence, unless otherwise stated, we will stick to this choice throughout the current study. Hence, unless otherwise stated, we will stick to this choice throughout the current 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}. 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. Nevertheless, this step usually has to be performed for most correlated WFT calculations.
@ -422,7 +422,7 @@ Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-
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 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 model that provides an energy and a density, i) can be applied to any WFT model that provides an energy and a density,
ii) does not correct one-electron systems, and ii) does not correct one-electron systems, and
iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered complete basis set (CBS) limit for a given WFT model. iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered \trashMG{complete basis set} CBS limit for a given WFT model.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%\section{Results} %\section{Results}
@ -514,7 +514,7 @@ Nevertheless, the deviation observed for the largest basis set is typically with
In most cases, the basis set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones. 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 SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. Importantly, the sensitivity with respect to the SR-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}. However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
Such weak sensitivity when reaching large basis sets shows the robustness of the approach. Such weak sensitivity \manu{to the approximated DFT functional} 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.
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. 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.
@ -530,7 +530,7 @@ Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is re
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. 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}.
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. Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction provides significant basis set \manu{error} 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. Encouraged by these promising results, we are currently pursuing various avenues toward basis set reduction for strongly correlated systems and electronically excited states.
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