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Emmanuel Giner
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@ -126,10 +126,10 @@
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\affiliation{\LCT}
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\begin{abstract}
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We report a universal density-based basis set incompleteness correction that can be applied to any wave function method.
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We report a universal density-based basis set incompleteness correction that can be applied to any wave function method while keeping the correct limit when reaching the complete basis set (CBS).
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The present correction relies on short-range correlation functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
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Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separated \textit{parameter} $\mu$, the key ingredient here is a basis-dependent, range-separated \textit{function} $\mu(\bf{r})$ which is dynamically determined to catch the missing short-range correlation due to the lack of electron-electron cusp in standard wave function methods.
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As illustrative examples, we show how this density-based correction allows to obtain near-complete basis set CCSD(T) atomization energies for the G2 set of molecules with compact Gaussian basis sets.
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Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separated \textit{parameter} $\mu$, the key ingredient here is a range-separated \textit{function} $\mu(\bf{r})$ which automatically adapts to the basis set to represent the non homogeneity of the incompleteness in real space.
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As illustrative examples, we show how this density-based correction allows to obtain CCSD(T) atomization energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets.
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For example, CCSD(T)+LDA/cc-pVTZ and CCSD(T)+PBE/cc-pVTZ return mean absolute deviations of \titou{X.XX} and \titou{X.XX} kcal/mol, respectively, compared to CBS atomization energies.
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\end{abstract}
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@ -195,7 +195,7 @@ where
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- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
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\end{equation}
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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.
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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.
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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 a complete basis, respectively.
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Both wave functions yield the same target density $\n{}{}$.
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%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
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@ -300,15 +300,14 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
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%\end{equation}
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Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Nevertheless, from the physical point of view $\bE{}{\Bas}[\n{}{}]$ plays a quite universal role as it aims at fixing the main
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consequence of the incompleteness of $\Bas$, which is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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Nevertheless, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct for the lack of electronic cups in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points, which is universal.
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As 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 Coulomb interaction at $r_{12} = 0$.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, the spatial inhomogeneity of $\Bas$ forces us to define a range-separated \textit{function} $\rsmu{}{}(\br{})$ as the value of $\rsmu{}{}$ must be known at any point in space.
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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 non divergent two-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} $\rsmu{}{}(\br{})$ which quantifies the incompleteness of a basis set $\Bas$ for each point in ${\rm I\!R}^3$.
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The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which i) is finite at the e-e coalescence point as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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The first step of the basis set correction consists in obtaining an effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which i) is finite at the e-e coalescence point as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction in the limit of a complete basis set.
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In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
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In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
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In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} with $\rsmu{}{}(\br{})$ as the range separation.
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%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
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%First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}).
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@ -326,7 +325,7 @@ In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density funct
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
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@ -345,12 +344,12 @@ $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\w
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
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\end{multline}
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and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
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With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies
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With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies (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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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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where the $\hWee{}$ contains only the alpha-beta component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
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where the $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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\iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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@ -368,7 +367,9 @@ Of course, there exists \textit{a priori} an infinite set of functions in ${\rm
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}),
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\end{equation}
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which therefore guarantees a physically satisfying limit for all $\wf{}{\Bas}$. An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
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and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
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An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
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As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems.
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%=================================================================
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%\subsection{Range-separation function}
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@ -465,13 +466,15 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
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\end{align}
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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 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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The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}].
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Indeed, provided that $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) \approx \W{\wf{}{\Bas}}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\wf{}{\Bas}}{}(\br{})}[\n{}{}(\br{})]$ coincides with $\wf{}{\Bas}$.
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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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%--------------------------------------------
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ 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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@ -515,14 +518,6 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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%\end{equation}
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%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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The main computational source of the present approach is the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of
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\begin{equation}
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\label{eq:fcoal}
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f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
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\end{equation}
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which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtaine the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}, but this step has in general to be performed before a correlated WFT calculations.
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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 reach a linear scaling method.
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%=================================================================
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%\subsection{Valence effective interaction}
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@ -531,29 +526,27 @@ As most WFT calculations are performed within the frozen-core (FC) approximation
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We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
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%According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$.
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Accounting solely for the valence electrons, we define
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
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\\
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= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
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\end{multline}
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and the valence part of the effective interaction is
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\begin{subequations}
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\begin{gather}
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We therefore define the valence-only effective interaction
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\begin{equation}
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\label{eq:Wval}
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\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}),
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\end{equation}
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with
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\begin{subequations}
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\begin{gather}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
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= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
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\\
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\label{eq:muval}
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\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}),
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\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
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= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2},
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\end{gather}
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\end{subequations}
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where
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and the corresponding valence range separation function $\rsmu{\wf{}{\Bas}}{\Val}(\br{})$
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\begin{equation}
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\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
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= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}
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\label{eq:muval}
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\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}).
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\end{equation}
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is the two body density associated to the valence electrons.
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%\begin{equation}
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% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
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%\end{equation}
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@ -583,7 +576,16 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
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%\end{equation}
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Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
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To conclude this theory session, it is important to notice that in the limit of a complete basis set, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the choice of functional}.
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Regarding now the main computational source of the present approach, it consists in the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of
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\begin{equation}
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\label{eq:fcoal}
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f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
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\end{equation}
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which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation.
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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 spped up the calculations.
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To conclude this theory session, it is important to notice that in the limit of a complete basis set, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part.
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Thefore in the limit of a complete basis set one recovers the correct limit of the WFT model whatever approximations are made in the DFT part, just like in equation \eqref{eq:limitfunc}.
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%%%%%%%%%%%%%%%%%%%%%%%%
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