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@ -223,7 +223,7 @@ Because the e-e cusp originates from the divergence of the Coulomb operator at $
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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.
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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 $\mathbb{R}^3$.
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The first step of the present basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$.
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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 finite basis $\Bas$.
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%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 $\Bas \to \infty$.
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In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
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In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
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@ -254,9 +254,6 @@ and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
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Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} \titou{ensures} that one-electron systems do not have any basis set correction.
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%\PFL{I don't agree with this. There must be a correction for one-electron system.
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%However, it does not come from the e-e cusp but from the e-n cusp.}
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:int_eq_wee}
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@ -267,7 +264,10 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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\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},
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\end{equation}
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\alert{it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.}
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it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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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 two-electron part of the basis set incompleteness error.
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A one-electron correction is currently under active development.
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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.
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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons
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\begin{equation}
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@ -289,7 +289,7 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
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%\subsection{Range-separation function}
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%=================================================================
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Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT \titou{which employs smooth long-range operators.}
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Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs smooth long-range operators.
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Although this choice is not unique, we choose here the range-separation function
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\begin{equation}
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\label{eq:mu_of_r}
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@ -306,7 +306,7 @@ coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\B
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%=================================================================
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%\subsection{Short-range correlation functionals}
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%=================================================================
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Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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%Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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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}
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\begin{multline}
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\label{eq:ec_md_mu}
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@ -351,11 +351,11 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the following two l
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \titou{=} \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
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%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
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%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
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Therefore, following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
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The LDA version of $\bE{}{\Bas}[\n{}{}]$ is defined as
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Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
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The LDA version of the ECMD complementary functional is defined as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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@ -375,7 +375,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
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\end{subequations}
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The difference between the ECMD 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}.
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This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
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Therefore, the PBE complementary functional reads
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Therefore, the ECMD PBE complementary functional reads
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\begin{equation}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
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@ -386,9 +386,8 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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%\subsection{Valence approximation}
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%=================================================================
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As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
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We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core spinorbitals.
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Therefore, we define the FC version of the effective interaction as
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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.
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We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core spinorbitals, and define the FC version of the effective interaction as
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\begin{equation}
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\W{\Bas}{\FC}(\br{1},\br{2}) =
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\begin{cases}
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@ -415,20 +414,20 @@ and the corresponding FC range-separation function
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\end{equation}
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It is worth noting that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
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Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with the model $\modZ$, the FC contribution of the complementary functional $\bE{}{\FC}[\n{\modZ}{\FC}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
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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 $\bE{\LDA}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
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%=================================================================
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%\subsection{Computational considerations}
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%=================================================================
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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.
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This embarrassingly parallel 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.
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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.
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%\begin{equation}
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% \label{eq:fcoal}
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% \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
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%\end{equation}
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In our current implementation, the bottleneck 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}.
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Nevertheless, this step usually has to be performed for most correlated WFT calculations.
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Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) could be employed to significantly speed up this step.
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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.
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%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.
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To conclude this section, we point out that, independently of the DFT functional, the present basis set correction
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@ -459,9 +458,9 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
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%%% TABLE II %%%
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\begin{table}
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\caption{
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Statistical analysis (in \kcal) of the G2 correlation energies depicted in Fig.~\ref{fig:G2_Ec}.
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Statistical analysis (in \kcal) of the G2-1 correlation energies depicted in Fig.~\ref{fig:G2_Ec}.
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Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference correlation energies.
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CA corresponds to the number of correlation energies (out of 55) obtained with chemical accuracy.
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CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
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See {\SI} for raw data.
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\label{tab:stats}}
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\begin{ruledtabular}
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