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Manuscript/G2-srDFT.tex
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@ -109,7 +109,13 @@
\newcommand{\dbr}[1]{d\br{#1}} \newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\ra}{\rightarrow} \newcommand{\ra}{\rightarrow}
\newcommand{\De}{D_\text{e}}
% frozen core
\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France}
@ -219,11 +225,11 @@ Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$,
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. 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.
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. 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.
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. 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.
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 $\mathbb{R}^3$. 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 $\mathbb{R}^3$.
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 $\Bas$. 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 $\Bas$.
In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$. 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} with $\rsmu{\Bas}{}(\br{})$ as range-separation function. In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
%================================================================= %=================================================================
%\subsection{Effective Coulomb operator} %\subsection{Effective Coulomb operator}
@ -231,9 +237,9 @@ In the final step, we employ short-range density functionals \cite{TouGorSav-TCA
We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18} We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
\begin{equation} \begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{\Bas}{}(\br{1},\br{2}) = \W{}{\Bas}(\br{1},\br{2}) =
\begin{cases} \begin{cases}
\f{\Bas}{}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,} \f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
\\ \\
\infty, & \text{otherwise,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
@ -247,70 +253,60 @@ where
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), 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),
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2}) \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}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (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} \begin{equation}
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
\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}, \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},
\end{equation} \end{equation}
where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
\begin{equation} \begin{equation}
\alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} = \alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},} \iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
\end{equation} \end{equation}
it 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.
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. 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.
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. 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.
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:lim_W} \label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \titou{r_{12}^{-1} } \lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = \titou{r_{12}^{-1} }
\end{equation} \end{equation}
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. for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
%================================================================= %=================================================================
%\subsection{Range-separation function} %\subsection{Range-separation function}
%================================================================= %=================================================================
A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{\Bas}{}(\br{},{\br{}})$, A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
%\begin{equation}
% \label{eq:wcoal}
% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
%\end{equation}
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. 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.
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. 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.
Although this choice is not unique, we choose here the range-separation function Although this choice is not unique, we choose here the range-separation function
\begin{equation} \begin{equation}
\label{eq:mu_of_r} \label{eq:mu_of_r}
\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{},\br{}), \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
\end{equation} \end{equation}
such that the long-range interaction of RS-DFT such that the long-range interaction of RS-DFT, \titou{$\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$},
%\begin{equation} %\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}} } % \w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
%\end{equation} %\end{equation}
\begin{equation} coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any point $\br{}$.
\w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
\end{equation}
coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}(\br{})}(0) = \W{\Bas}{}(\br{},\br{})$ at any point $\br{}$.
%================================================================= %=================================================================
%\subsection{Short-range correlation functionals} %\subsection{Short-range correlation functionals}
%================================================================= %=================================================================
Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$. Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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} 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}
%\begin{multline}
\begin{equation} \begin{equation}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} = \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
% \\
- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}}, - \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
%\end{multline}
\end{equation} \end{equation}
where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
\begin{equation} \begin{equation}
@ -318,55 +314,32 @@ where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$. with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%\begin{multline}
% \label{eq:ec_md_mu}
% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
% \\
% - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
%\end{multline}
%where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
%\begin{equation}
%\label{eq:argmin}
% \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
%\end{equation}
%with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%and
%\begin{equation}
%\label{eq:erf}
% \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
%\end{equation}
%is the long-range part of the Coulomb operator.
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
\begin{subequations}
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = 0, \lim_{\mu \to \infty} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
\\ &
\label{eq:small_mu_ecmd} % \label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}], \lim_{\mu \to 0} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
\end{align} \end{align}
\end{subequations}
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
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}]. 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}].
Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$. Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
%provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide. Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
%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.
Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{\Bas}{}(\br{})$.
The local-density approximation (LDA) of the ECMD complementary functional is defined as The local-density approximation (LDA) of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{}, \bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
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 $\mu$. 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 $\mu$.
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 $\mu$. 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 $\mu$.
In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
\\ \\
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{\Bas}{}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
\end{multline} \end{multline}
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 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
\begin{subequations} \begin{subequations}
@ -381,7 +354,7 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
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}. 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}.
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$. This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
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}. 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}.
%================================================================= %=================================================================
%\subsection{Frozen-core approximation} %\subsection{Frozen-core approximation}
@ -390,37 +363,33 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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. 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.
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 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
\begin{equation} \begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) = \WFC{}{\Bas}(\br{1},\br{2}) =
\begin{cases} \begin{cases}
\f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\Bas,\FC}(\br{1},\br{2}), & \text{if $\n{2}{\Bas,\FC}(\br{1},\br{2}) \ne 0$}, \fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
\\ \\
\infty, \! & \!\!\! \text{otherwise,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
\end{equation} \end{equation}
with with
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:fbasisval} \label{eq:fbasisval}
\f{\Bas}{\FC}(\br{1},\br{2}) \fFC{}{\Bas}(\br{1},\br{2})
= \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}, = \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},
\\ \\
\n{2}{\Bas,\FC}(\br{1},\br{2}) \nFC{2}{\Bas}(\br{1},\br{2})
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and the corresponding FC range-separation function and the corresponding FC range-separation function \titou{$\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$}.
\begin{equation} It is \titou{noteworthy} that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
\label{eq:muval}
\rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}).
\end{equation}
It is worth noticing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
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 $\bE{\PBE}{\Bas}[\n{\modZ}{\FC},\rsmu{\Bas}{\FC}]$. 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 $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
%================================================================= %=================================================================
%\subsection{Computational considerations} %\subsection{Computational considerations}
%================================================================= %=================================================================
The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{},\br{})$ at each quadrature grid point. The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
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. 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.
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 $\Bas$. 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. Hence, we will stick to this choice throughout the \titou{present} study.