loos/srDFT_G2
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

working on the manuscript

Browse Source
This commit is contained in:
Emmanuel Giner 2019-04-09 00:14:26 +02:00
parent 466f2a61ff
commit fffb4a47fd
1 changed files with 44 additions and 42 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

86
Manuscript/G2-srDFT.tex
View File

@ -126,10 +126,10 @@
\affiliation{\LCT} \affiliation{\LCT}
\begin{abstract} \begin{abstract}
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 while keeping the correct limit when reaching the complete basis set (CBS).
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. 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.
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. 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.
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. 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.
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. 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.
\end{abstract} \end{abstract}
@ -195,7 +195,7 @@ where
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation} \end{equation}
is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator. is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and a complete basis, respectively.
Both wave functions yield the same target density $\n{}{}$. Both wave functions yield the same target density $\n{}{}$.
%\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.} %\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.}
@ -300,15 +300,14 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
%\end{equation} %\end{equation}
Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
Nevertheless, from the physical point of view $\bE{}{\Bas}[\n{}{}]$ plays a quite universal role as it aims at fixing the main 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.
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.
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 $r_{12} = 0$. 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 $r_{12} = 0$.
Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals. 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.
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. 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 ${\rm I\!R}^3$.
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. 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.
In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$. In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05} In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} with $\rsmu{}{}(\br{})$ as the range separation.
%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}]. %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}].
%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}) . %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}) .
%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}). %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}).
@ -326,7 +325,7 @@ In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density funct
%================================================================= %=================================================================
%\subsection{Effective Coulomb operator} %\subsection{Effective Coulomb operator}
%================================================================= %=================================================================
We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
@ -345,12 +344,12 @@ $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\w
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
\end{multline} \end{multline}
and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies (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{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2}, \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},
\end{equation} \end{equation}
where the $\hWee{}$ contains only the alpha-beta component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as where the $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
\begin{equation} \begin{equation}
\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}, \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},
\end{equation} \end{equation}
@ -368,7 +367,9 @@ Of course, there exists \textit{a priori} an infinite set of functions in ${\rm
\label{eq:lim_W} \label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}), \lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}),
\end{equation} \end{equation}
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. and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
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.
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.
%================================================================= %=================================================================
%\subsection{Range-separation function} %\subsection{Range-separation function}
@ -465,13 +466,15 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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. 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}].
This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. 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 $\wf{}{\Bas}$.
%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.
%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
%-------------------------------------------- %--------------------------------------------
%\subsubsection{Local density approximation} %\subsubsection{Local density approximation}
%-------------------------------------------- %--------------------------------------------
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 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
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}, \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
@ -515,14 +518,6 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad , % \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
%\end{equation} %\end{equation}
%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. %for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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
\begin{equation}
\label{eq:fcoal}
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}{}
\end{equation}
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.
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.
%================================================================= %=================================================================
%\subsection{Valence effective interaction} %\subsection{Valence effective interaction}
@ -531,29 +526,27 @@ As most WFT calculations are performed within the frozen-core (FC) approximation
We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$. We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
%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 $\wf{}{\Bas}$. %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 $\wf{}{\Bas}$.
Accounting solely for the valence electrons, we define We therefore define the valence-only effective interaction
\begin{multline} \begin{equation}
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
\\
= \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},
\end{multline}
and the valence part of the effective interaction is
\begin{subequations}
\begin{gather}
\label{eq:Wval} \label{eq:Wval}
\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}), \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}),
\end{equation}
with
\begin{subequations}
\begin{gather}
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
= \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},
\\ \\
\label{eq:muval} \n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}), = \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2},
\end{gather} \end{gather}
\end{subequations} \end{subequations}
where and the corresponding valence range separation function $\rsmu{\wf{}{\Bas}}{\Val}(\br{})$
\begin{equation} \begin{equation}
\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}) \label{eq:muval}
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2} \rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}).
\end{equation} \end{equation}
is the two body density associated to the valence electrons.
%\begin{equation} %\begin{equation}
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . % \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
%\end{equation} %\end{equation}
@ -583,7 +576,16 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
%\end{equation} %\end{equation}
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 $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$. 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 $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
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}. 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
\begin{equation}
\label{eq:fcoal}
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}{}
\end{equation}
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.
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.
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.
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}. 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}.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%