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@ -117,7 +117,7 @@
\begin{document} \begin{document}
\title{A Basis Set Correction For Wave Function Theory Based on Density Functional Theory: Application to Coupled Cluster} \title{A Density-Based Basis Set Correction For Wave Function Theory}
\author{Bath\'elemy Pradines} \author{Bath\'elemy Pradines}
\affiliation{\LCT} \affiliation{\LCT}
@ -139,7 +139,7 @@ We report a universal density-based basis set incompleteness correction that can
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. 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 coupled-cluster with single and double substitutions and triple CCSD(T) correlation energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets. As illustrative examples, we show how this density-based correction allows us to obtain coupled-cluster with single and double substitutions and triple CCSD(T) correlation energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets.
\titou{For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 6.06 kcal/mol compared to CCSD(T)/CBS atomization energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 1.19 and 0.85 kcal/mol (respectively) with the same basis.} \titou{For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 6.06 kcal/mol compared to CCSD(T)/CBS correlation energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 1.19 and 0.85 kcal/mol (respectively) with the same basis.}
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -158,29 +158,21 @@ This undesirable feature was put into light by Kutzelnigg more than thirty years
To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12} The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12}
For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17} For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
%\trashPFL{Except for these computational considerations, a possible drawback of F12 theory is its quite complicated formulation which requires a deep knowledge in this field in order to adapt F12 theory to a new WFT model.}
To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019} To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65} Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
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}
In the \manu{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}
%especially in the range-separated (RS) context where the WFT method is relieved from describing the short-range part of the correlation hole. \cite{TouColSav-PRA-04, FraMusLupTou-JCP-15}
%To obtain accurate results within DFT, one must develop the art of selecting the adequate exchange-correlation functional, which can be classified in various families depending on their physical input quantities. \cite{Bec-JCP-14}
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}
%The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $n$. \cite{Dir-PCPRS-30, VosWilNus-CJP-80} 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}
%The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla n$ as an extra ingredient.\cite{Bec-PRA-88, PerWan-PRA-91, PerBurErn-PRL-96}
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.
As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15} As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
Therefore, a number of approximate RS-DFT schemes have been developed using either 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 using either 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 using either single-reference WFT approaches (such as M{\o}ller-Plesset perturbation theory\cite{AngGerSavTou-PRA-05}, coupled cluster\cite{GolWerSto-PCCP-05}, random-phase approximations\cite{TouGerJanSavAng-PRL-09,JanHenScu-JCP-09}) or multi-reference WFT approaches (such as multi-reference CI\cite{LeiStoWerSav-CPL-97}, multiconfiguration self-consistent field\cite{FroTouJen-JCP-07}, multi-reference perturbation theory\cite{FroCimJen-PRA-10}, density-matrix renormalization group\cite{HedKneKieJenRei-JCP-15}, selected CI\cite{FerGinTou-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 these new methodological development together with the first numerical tests on molecular systems.
\manu{Very recently, a step forward has been performed by some of the present authors thanks to a density-based basis set correction which merges WFT and RS-DFT\cite{GinPraFerAssSavTou-JCP-18}. }
The present work proposes an extension of \manu{this new theory with an application to CCSD(T)} together with the first numerical tests on molecular systems.
Unless otherwise stated, atomic units are used.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory} \section{Theory}
@ -220,11 +212,11 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 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 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 cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) 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 \trashMG{Coulomb} \manu{two-electron} interaction at $r_{12} = 0$. 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 $r_{12} = 0$.
Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction. Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$. Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
The first step of the present basis set correction consists of obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ ``mimicking'' the \manu{effect of the basis set incompleteness of $\Bas$ on the }Coulomb operator \trashMG{in a finite basis $\Bas$}. The first step of the present basis set correction consists of obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in a incomplete basis $\Bas$.
%The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$. %The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$. In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation. In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
@ -266,8 +258,7 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
\iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation} \end{equation}
it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 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} \manu{does not affect \eqref{eq:int_eq_wee} and} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originates from the e-e cusp. 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 originates from the e-e cusp.
\trashMG{A similar correction for the electron-nucleus cusp is currently under active development.}
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries. As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons
@ -306,9 +297,8 @@ coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\B
%================================================================= %=================================================================
%\subsection{Short-range correlation functionals} %\subsection{Short-range correlation functionals}
%================================================================= %=================================================================
\manu{Once defined a range separation function $\rsmu{\Bas}{}(\br{})$, we can use RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$, and Once defined a range separation function $\rsmu{\Bas}{}(\br{})$, we can use RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
} As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
as in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}]
@ -321,7 +311,7 @@ where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
\label{eq:argmin} \label{eq:argmin}
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \manu{\hat{\w{}{\lr,\rsmu{}{}}}(\hat{r}_{ij})}$. with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%\begin{multline} %\begin{multline}
% \label{eq:ec_md_mu} % \label{eq:ec_md_mu}
% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} % \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
@ -359,15 +349,15 @@ Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluate
The LDA version of the ECMD complementary functional is defined as The LDA version of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{\manu{\Bas}}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}, \bE{\LDA}{\Bas}[\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})] = \int \be{\LDA}{\sr}\qty(\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})) \n{}{}(\br{}) \dbr{},
\end{equation} \end{equation}
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the \trashMG{short-range} reduced (i.e.~per electron) ECMD of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}. where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the reduced (i.e.~per electron) ECMD functional of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized 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 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 PBE ECMD functional In order to correct such a defect, we propose here a new PBE ECMD functional
\begin{equation} \begin{equation}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBE}{\manu{\Bas}}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{} \bE{\PBE}{\Bas}[\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})] = \int \be{\PBE}{\sr}\qty(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{\Bas}{}(\br{})) \n{}{}(\br{}) \dbr{}
\end{equation} \end{equation}
inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
\begin{subequations} \begin{subequations}
@ -379,17 +369,16 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}. \beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
The difference between the ECMD \trashMG{PBE} functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{})) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}))$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found 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} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{})) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}))$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$. This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\manu{\Bas}}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\manu{\Bas}}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
%================================================================= %=================================================================
%\subsection{Valence approximation} %\subsection{Valence approximation}
%================================================================= %=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of spinorbitals. As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of spinorbitals.
\manu{I like the idea of defining the $\BasFC$ as the complementary of $\Cor$, but the line over $\Bas$ is barely visible ... :(}
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core \trashMG{spinorbitals} \manu{spatial orbitals}, and define the FC version of the effective interaction as We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core \trashMG{spinorbitals} \manu{spatial orbitals}, and define the FC version of the effective interaction as
\begin{equation} \begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) = \W{\Bas}{\FC}(\br{1},\br{2}) =
@ -415,26 +404,26 @@ and the corresponding FC range-separation function
\label{eq:muval} \label{eq:muval}
\rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}). \rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}).
\end{equation} \end{equation}
It is worth not\manu{ic}ing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still \trashMG{satisfies Eq.~\eqref{eq:lim_W}} \manu{tends to the regular Coulomb interaction when $\Bas \to \infty$}. It is worth noticing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with a model $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\manu{\Bas}}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\manu{\Bas}}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$. Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with a model $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\Bas}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
%================================================================= %=================================================================
%\subsection{Computational considerations} %\subsection{Computational considerations}
%================================================================= %=================================================================
One of the most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. One of the most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is \manu{strongly} 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 (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.
\manu{As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a single Slater determinant wave function to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems, and therefore we use this framework all through this work. } \titou{As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a single Slater determinant wave function to compute $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems, and therefore we use this framework all through this work. }
%\begin{equation} %\begin{equation}
% \label{eq:fcoal} % \label{eq:fcoal}
% \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}, % \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
%\end{equation} %\end{equation}
In our current implementation, the \manu{computational} bottleneck \manu{of the basis set correction} is the four-index transformation to get the two-electron integrals in the molecular orbital basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. In our current implementation, the computational bottleneck of the present basis set correction is the four-index transformation to get the two-electron integrals in the molecular orbital 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.
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.
%When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations. %When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations.
To conclude this section, we point out that \manu{because of the definitions \eqref{eq:def_weebasis}, \eqref{eq:mu_of_r} and 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 of 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 CBS limit for a given WFT model. iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model.
@ -503,8 +492,8 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
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 sets (X $=$ D, T, Q and 5).
\titou{In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.} \titou{In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.}
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2-1 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11} \ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2-1 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 \trashMG{entire} correlation energies of the \manu{entire} G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets. In a second time, we compute the correlation energies of the entire G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
This molecular set has been exhausively 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}) \manu{ and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}. This molecular set has been exhausively 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}) \titou{and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}.
%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and 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 ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$. %The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and 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 ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
As a method $\modY$ we employ either CCSD(T) or exFCI. 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} Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
@ -552,9 +541,8 @@ 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}.
\titou{Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.} \titou{Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
\manu{Encouraged by these results for weakly correlated ground states molecules, we are developing this theory towards the treatment of the basis set error for strongly correlated systems, excited states and the treatment of the one-electron error in the basis set incompleteness. } Encouraged by these results for weakly correlated ground states molecules, we are developing this theory towards the treatment of the basis set error for strongly correlated systems, excited states and the treatment of the one-electron error in the basis set incompleteness.}
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\section*{Supporting information} \section*{Supporting information}
@ -569,7 +557,6 @@ The authors would like to thank the \textit{Centre National de la Recherche Scie
\end{acknowledgements} \end{acknowledgements}
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