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Cover_Letter/CoverLetter.tex
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@ -14,7 +14,7 @@
\justifying \justifying
Please find enclosed our manuscript entitled Please find enclosed our manuscript entitled
\begin{quote} \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} \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''}.

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Manuscript/G2-srDFT.tex
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@ -118,7 +118,7 @@
\begin{document} \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} \author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
@ -141,7 +141,7 @@
\includegraphics[width=\linewidth]{TOC} \includegraphics[width=\linewidth]{TOC}
\end{wrapfigure} \end{wrapfigure}
We report a universal density-based basis set incompleteness correction that can be applied to any wave function method. 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. 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. 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} \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. 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 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} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
@ -202,7 +202,7 @@ where
\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 normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively. 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 $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$. 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 This implies that
@ -212,11 +212,11 @@ This implies that
\end{equation} \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 $\E{\FCI}{} = \E{}{}$. 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 $\Bas$. 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 \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. 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. 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 $\mathbb{R}^3$. 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} \label{eq:def_weebasis}
\W{\Bas}{}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2}) =
\begin{cases} \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,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
@ -241,7 +241,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
where where
\begin{equation} \begin{equation}
\label{eq:n2basis} \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}, = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation} \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), 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}) 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}
\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} \end{equation}
where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
\begin{eqnarray} \begin{equation}
\frac{1}{2}\iint \frac{1}{r_{12}} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx} \alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
\nonumber\\ \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
\frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \end{equation}
\end{eqnarray}
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} 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. 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}) Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:lim_W} \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} \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} %\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} % \label{eq:wcoal}
% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), % \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
%\end{equation} %\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. 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 Although this choice is not unique, we choose here the range-separation function
\begin{equation} \begin{equation}
@ -341,10 +340,10 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \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} \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{align}
\end{subequations} \end{subequations}
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. 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 The local-density approximation (LDA) of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \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} \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 $\mu$. 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 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} \label{eq:def_pbe_tot}
\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxx} \bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] =
\nonumber\\ \\
\int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\PBE}\qty(\{\n{\sigma}{}(\br{})\},\{\nabla \n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{},
\end{eqnarray} \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} $\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding 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{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \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} \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{gather}
\end{subequations} \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}. 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 $\n{2,\Bas}{}(\br{},\br{})$. 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}. 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. 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 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} \begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) \! = \! \W{\Bas}{\FC}(\br{1},\br{2}) =
\begin{cases} \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,} \infty, \! & \!\!\! \text{otherwise,}
\end{cases} \end{cases}
@ -405,7 +404,7 @@ with
\f{\Bas}{\FC}(\br{1},\br{2}) \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}, = \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}, = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather} \end{gather}
\end{subequations} \end{subequations}
@ -422,9 +421,9 @@ 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 $\W{\Bas}{}(\br{},\br{})$ at each quadrature grid point. 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 $\Bas$. 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}. 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.
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.
@ -494,15 +493,15 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
\label{fig:G2_Ec}} \label{fig:G2_Ec}}
\end{figure*} \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} \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. 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.
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}
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. 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, 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} 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} 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. 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. 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. 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}. 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 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 can be found in the {\SI}.

2
Manuscript/SI/G2_srDFT-SI.tex
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@ -116,7 +116,7 @@
\begin{document} \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} \author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}