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Pierre-Francois Loos 2019-04-11 22:32:00 +02:00
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@ -94,14 +94,13 @@
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\updw}{\uparrow\downarrow}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
%\newcommand{\br}[1]{\mathbf{x}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
%\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\ra}{\rightarrow}
\newcommand{\De}{D_\text{e}}
@ -215,99 +214,9 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
\end{equation}
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = \E{}{}$.
In the case $\modX = \FCI$, 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 $\modX$ and $\modY$ for the \titou{FCI} energy and density within $\Bas$, respectively.
%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
%As any wave function model is necessary an approximation to the FCI model, one can write
%\begin{equation}
% \efci \approx \emodel
%\end{equation}
%and
%\begin{equation}
% \denfci \approx \denmodel
%\end{equation}
%and by defining the energy provided by the model $\model$ in the complete basis set
%\begin{equation}
% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, ,
%\end{equation}
%we can then write
%\begin{equation}
% \emodelcomplete \approx \emodel + \ecompmodel
%\end{equation}
%which verifies the correct limit since
%\begin{equation}
% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
%\end{equation}
%=================================================================
%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
%=================================================================
%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models.
%=================================================================
%\subsubsection{Basis set correction for the CCSD(T) energy}
%=================================================================
%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant.
%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have
%\begin{equation}
% \emodel = \ecc \,\, .
%\end{equation}
%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
%\begin{equation}
% \denmodel = \denhf \,\, .
%\end{equation}
%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory.
%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
%\begin{equation}
% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
%\end{equation}
%=================================================================
%\subsubsection{Correction of the CIPSI algorithm}
%=================================================================
%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
%\begin{align}
% E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
%\end{align}
%where $E_\text{v}$ is the variational energy
%\begin{align}
% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}} }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
%\end{align}
%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
%\begin{align}
% E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
%\end{align}
%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$.
%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$.
%
%In the context of the basis set correction, we use the following conventions
%\begin{equation}
% \emodel = \EexFCIbasis
%\end{equation}
%\begin{equation}
% \denmodelr = \dencipsir
%\end{equation}
%where the density $\dencipsir$ is defined as
%\begin{equation}
% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
%\end{equation}
%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$.
%
%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
%\begin{equation}
% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
%\end{equation}
Rigorously speaking, 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
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.
@ -318,32 +227,16 @@ Contrary to the conventional RS-DFT scheme which requires a range-separated \tex
The first step of the 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 $\Bas$.
The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} alongside $\rsmu{\Bas}{}(\br{})$ as 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}].
%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}).
In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
%=================================================================
%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
%=================================================================
%=================================================================
%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis set $\Bas$}
%\label{sec:weff}
%=================================================================
%=================================================================
%\subsection{Effective Coulomb operator}
%=================================================================
We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as
We define the effective operator as
\begin{equation}
\label{eq:def_weebasis}
\W{\Bas}{}(\br{1},\br{2}) =
\begin{cases}
\f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), & \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,}
\\
\infty, & \text{otherwise.}
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where
@ -352,108 +245,56 @@ where
\n{2}{}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}
\end{equation}
is the opposite-spin two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital, $\f{\Bas}{}(\br{1},\br{2})$ is defined as
and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin two-body density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital,
\begin{equation}
\label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2})
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\Bas}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction.
Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction.
\PFL{I don't agree with this. There must be a correction for one-electron system.
However, it does not come from the e-e cusp but from the e-n cusp.}
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (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}
\label{eq:int_eq_wee}
\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation}
where here $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
\begin{equation}
\iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation}
which 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.
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.
An important quantity to define in the present context is $\W{\Bas}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons
\begin{equation}
\label{eq:wcoal}
\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}})
\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
\end{equation}
and which is necessarily \textit{finite} at for an \textit{incomplete} basis set as long as the on-top two-body density is non vanishing.
and which is necessarily \textit{finite} for an incomplete basis set as long as the on-top two-body density is non vanishing.
Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
\end{equation}
for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify 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 shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems.
for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
%An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify 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 shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems.
%=================================================================
%\subsection{Range-separation function}
%=================================================================
As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.
To do so, we choose a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$
Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.
To do so, we choose a range-separation \textit{function}
\begin{equation}
\label{eq:mu_of_r}
\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{})
\end{equation}
such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
such that the long-range interaction
\begin{equation}
\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
\end{equation}
\PFL{This expression looks like a cheap spherical average.}
coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$
\begin{equation}
\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}).
\end{equation}
%More precisely, if we define the value of the interaction at coalescence as
%\begin{equation}
% \label{eq:def_wcoal}
% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
%\end{equation}
%where $(\br{},\Bar{\br{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
%we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
%\begin{equation}
% \wbasiscoal{} = \w{}{\lr,\rsmu{\Bas}{}}(\bfrb{},\bfrb{})
%\end{equation}
%where the long-range-like interaction is defined as
%\begin{equation}
% \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
%\end{equation}
%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
%\begin{equation}
% \label{eq:mu_of_r}
% \rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{})
%\end{equation}
%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
%\begin{equation}
% \label{eq:mu_of_r_val}
% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
%\end{equation}
%An important point to notice is that, in the limit of a complete basis set $\Bas$, as
%\begin{equation}
%\label{eq:lim_W}
% \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2})
%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
%\end{equation}
%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
%% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
%and therefore
%\begin{equation}
%\label{eq:lim_mur}
% \lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty
%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
%\end{equation}
%=================================================================
%\subsection{Complementary functional}
%=================================================================
%\label{sec:ecmd}
coincides with the effective interaction $\W{\Bas}{}(\br{})$ at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
Once defined the range-separation function $\rsmu{\Bas}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
@ -486,24 +327,17 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
\end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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}].
Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}(\br{})}$ coincides with $\wf{}{\Bas}$.
Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ 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}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
\begin{equation}
\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{},
\end{equation}
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range ECMD per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
%--------------------------------------------
%\subsubsection{New PBE functional}
%--------------------------------------------
The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} 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 ECMD functional inspired by the recent functional proposed by some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
\begin{subequations}
@ -515,8 +349,8 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
\end{gather}
\end{subequations}
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 of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \left(\n{}{}(\br{})\right)^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}$.
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 of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
Therefore, the PBE complementary functional reads
\begin{equation}
\label{eq:def_pbe_tot}
@ -524,27 +358,9 @@ Therefore, the PBE complementary functional reads
\end{equation}
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modY}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
%The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
%Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
%A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
%\begin{equation}
% \label{eq:approx_ecfuncbasis}
% \ecompmodel \approx \ecmuapproxmurmodel
%\end{equation}
%Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
%It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
%\begin{equation}
% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
%\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.
%=================================================================
%\subsection{Valence effective interaction}
%=================================================================
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.
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 $\wf{}{\Bas}$.
We therefore define the valence-only effective interaction
\begin{equation}
\W{\Bas}{\Val}(\br{1},\br{2}) =
@ -559,44 +375,19 @@ with
\begin{gather}
\label{eq:fbasisval}
\f{\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},
= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\\
\n{2}{\Val}(\br{1},\br{2})
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather}
\end{subequations}
and the corresponding valence range separation function $\rsmu{\Bas}{\Val}(\br{})$
and the corresponding valence range separation function
\begin{equation}
\label{eq:muval}
\rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}).
\end{equation}
%\begin{equation}
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
%\end{equation}
%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
%Defining the valence one-body spin density matrix as
%\begin{equation}
% \begin{aligned}
% \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
% & = 0 \qquad \text{in other cases}
% \end{aligned}
%\end{equation}
%then one can define the valence density as:
%\begin{equation}
% \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r})
%\end{equation}
%Therefore, we propose the following valence-only approximations for the complementary functional
%\begin{equation}
% \label{eq:def_lda_tot}
% \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
%\end{equation}
%\begin{equation}
% \label{eq:def_lda_tot}
% \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
%\end{equation}
Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the model $\modY$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modY}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
Regarding now the main computational source of the present approach, it consists in the evaluation
@ -606,15 +397,13 @@ for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the eva
at each quadrature grid point of
\begin{equation}
\label{eq:fcoal}
\f{\Bas}{\HF}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}
\f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{i \in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \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 speed up the calculations.
To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any WFT model that provides an energy and a density, ii) it vanishes for one-electron systems,
iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model.
%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\Bas}{}(\br{})$ 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.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}