change of notations
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@ -109,7 +109,13 @@
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\De}{D_\text{e}}
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% frozen core
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\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
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\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
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\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France}
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@ -219,11 +225,11 @@ Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$,
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for the lack of \titou{cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$} at the e-e coalescence points, a universal condition of exact wave functions.
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Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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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 an incomplete basis $\Bas$.
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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 function.
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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 an incomplete basis $\Bas$.
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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 function.
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%=================================================================
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%\subsection{Effective Coulomb operator}
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@ -231,9 +237,9 @@ In the final step, we employ short-range density functionals \cite{TouGorSav-TCA
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We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\Bas}{}(\br{1},\br{2}) =
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\W{}{\Bas}(\br{1},\br{2}) =
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\begin{cases}
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\f{\Bas}{}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
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\f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
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\\
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\infty, & \text{otherwise,}
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\end{cases}
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@ -247,70 +253,60 @@ where
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and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
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\begin{equation}
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\label{eq:fbasis}
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\f{\Bas}{}(\br{1},\br{2})
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\f{}{\Bas}(\br{1},\br{2})
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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 $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
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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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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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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
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Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
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\begin{equation}
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\alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
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\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
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\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
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\end{equation}
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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 basis set incompleteness error originating from the e-e cusp.
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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 basis set incompleteness error originating from the e-e cusp.
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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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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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Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \titou{r_{12}^{-1} }
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\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = \titou{r_{12}^{-1} }
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\end{equation}
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for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
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for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{\Bas}{}(\br{},{\br{}})$,
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%\begin{equation}
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% \label{eq:wcoal}
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% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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%\end{equation}
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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing.
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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 a non-divergent long-range interaction operator.
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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 a non-divergent long-range interaction operator.
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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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\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{},\br{}),
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
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\end{equation}
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such that the long-range interaction of RS-DFT
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such that the long-range interaction of RS-DFT, \titou{$\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$},
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%\begin{equation}
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% \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}} }
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% \w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
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%\end{equation}
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\begin{equation}
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\w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
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\end{equation}
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coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}(\br{})}(0) = \W{\Bas}{}(\br{},\br{})$ at any point $\br{}$.
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coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any point $\br{}$.
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%=================================================================
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%\subsection{Short-range correlation functionals}
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%=================================================================
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Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within 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 correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05}
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%\begin{multline}
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\begin{equation}
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\label{eq:ec_md_mu}
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\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
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= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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% \\
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- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
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%\end{multline}
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\end{equation}
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where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
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\begin{equation}
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@ -318,55 +314,32 @@ where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
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\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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%\begin{multline}
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% \label{eq:ec_md_mu}
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% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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% \\
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% - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
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%\end{multline}
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%where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
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%\begin{equation}
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%\label{eq:argmin}
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% \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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%\end{equation}
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%with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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%and
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%\begin{equation}
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%\label{eq:erf}
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% \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
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%\end{equation}
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%is the long-range part of the Coulomb operator.
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The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
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\begin{subequations}
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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\\
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\label{eq:small_mu_ecmd}
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&
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% \label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
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\end{align}
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\end{subequations}
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where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
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The choice of the 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 functional [Eq.~\eqref{eq:ec_md_mu}].
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The choice of \trashPFL{the} 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 functional [Eq.~\eqref{eq:ec_md_mu}].
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Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
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%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, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{\Bas}{}(\br{})$.
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Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
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The local-density approximation (LDA) 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}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{},
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\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
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\end{equation}
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where $\be{\text{c,md}}{\sr,\LDA}(\qty{\n{\sigma}{}},\rsmu{}{})$ is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
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The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] =
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\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
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\\
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{},
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
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\end{multline}
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inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional~\cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
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\begin{subequations}
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@ -381,7 +354,7 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
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The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(0,\qty{\n{\sigma}{}(\br{})})$, where $0$ refers to $r_{12}=0$ and $\n{2}{\UEG}(0,\qty{n_\sigma}) \approx 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is \titou{then evaluated as} $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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%=================================================================
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%\subsection{Frozen-core approximation}
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@ -390,37 +363,33 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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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 MOs.
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We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
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\begin{equation}
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\W{\Bas}{\FC}(\br{1},\br{2}) =
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\WFC{}{\Bas}(\br{1},\br{2}) =
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\begin{cases}
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\f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\Bas,\FC}(\br{1},\br{2}), & \text{if $\n{2}{\Bas,\FC}(\br{1},\br{2}) \ne 0$},
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\fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
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\\
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\infty, \! & \!\!\! \text{otherwise,}
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\infty, & \text{otherwise,}
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\end{cases}
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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{\Bas}{\FC}(\br{1},\br{2})
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\fFC{}{\Bas}(\br{1},\br{2})
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= \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\\
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\n{2}{\Bas,\FC}(\br{1},\br{2})
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\nFC{2}{\Bas}(\br{1},\br{2})
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= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{gather}
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\end{subequations}
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and the corresponding FC range-separation function
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\begin{equation}
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\label{eq:muval}
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\rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}).
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\end{equation}
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It is worth noticing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
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and the corresponding FC range-separation function \titou{$\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$}.
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It is \titou{noteworthy} that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
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Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\Bas}[\n{\modZ}{\FC},\rsmu{\Bas}{\FC}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\FC},\rsmu{\Bas}{\FC}]$.
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Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ \titou{in $\Bas$}, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
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%=================================================================
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%\subsection{Computational considerations}
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%=================================================================
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The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{},\br{})$ at each quadrature grid point.
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The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
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||||
Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a \titou{multi}-determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.
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||||
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
|
||||
Hence, we will stick to this choice throughout the \titou{present} study.
|
||||
|
||||
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