up to RSDFT
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@ -82,8 +82,8 @@
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\modX}{\mathcal{X}}
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\newcommand{\modY}{\mathcal{Y}}
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\newcommand{\modY}{Y}
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\newcommand{\modZ}{Z}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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@ -184,16 +184,13 @@ The present basis set correction relies on the RS-DFT formalism to capture the m
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Here, we only provide the main working equations.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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%=================================================================
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%\subsection{Correcting the basis set error of a general WFT model}
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%=================================================================
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Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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\approx \E{\modX}{\Bas}
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+ \bE{}{\Bas}[\n{\modY}{\Bas}],
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\approx \E{\modY}{\Bas}
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+ \bE{}{\Bas}[\n{\modZ}{\Bas}],
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\end{equation}
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where
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\begin{equation}
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@ -206,21 +203,19 @@ is the basis-dependent complementary density functional, $\hT$ is the kinetic op
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
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Both wave functions yield the same target density $\n{}{}$.
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%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
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Importantly, in the limit of a complete basis set (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
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\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
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\end{equation}
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where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
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In the case $\modX = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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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.
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where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
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In the case $\modY = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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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 $\modY$ and $\modZ$ for the \titou{FCI} energy and density within $\Bas$, respectively.
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Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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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.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
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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 Coulomb interaction at $r_{12} = 0$.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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@ -229,6 +224,9 @@ The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the
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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.
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator as
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\begin{equation}
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\label{eq:def_weebasis}
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@ -243,15 +241,15 @@ where
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\begin{equation}
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\label{eq:n2basis}
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\n{2}{}(\br{1},\br{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{equation}
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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,
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and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin pair density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital,
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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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= \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 $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
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and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
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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 do not have any basis set correction.
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\PFL{I don't agree with this. There must be a correction for one-electron system.
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However, it does not come from the e-e cusp but from the e-n cusp.}
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@ -266,37 +264,39 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\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},
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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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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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An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons
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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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and which is necessarily \textit{finite} for an incomplete basis set as long as the on-top two-body density is non vanishing.
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ is non vanishing.
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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})
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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}) = r_{12}^{-1}\
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\end{equation}
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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.
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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.
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%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.
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%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.
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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.
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To do so, we choose a range-separation \textit{function}
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\alert{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.}
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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{})
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\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) ,
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\end{equation}
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such that the long-range interaction
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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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\end{equation}
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\PFL{This expression looks like a cheap spherical average.}
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\PFL{This expression looks like a cheap spherical average.
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What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
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coincides with the effective interaction $\W{\Bas}{}(\br{})$ at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
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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}
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Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals defined to approximate $\bE{}{\Bas}[\n{}{}]$.
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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}
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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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@ -356,7 +356,7 @@ Therefore, the PBE complementary functional reads
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
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\end{equation}
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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}.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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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 general subset of molecular orbitals.
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We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
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@ -388,11 +388,11 @@ and the corresponding valence range separation function
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\end{equation}
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It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
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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 $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
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Defining $\n{\modZ}{\Val}$ as the valence one-electron density obtained with the model $\modZ$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modZ}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
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Regarding now the main computational source of the present approach, it consists in the evaluation
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of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
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All through this paper, we use two-body density matrix of a single Slater determinant (typically HF)
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All through this paper, we use pair density matrix of a single Slater determinant (typically HF)
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for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the evaluation
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at each quadrature grid point of
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\begin{equation}
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@ -505,10 +505,10 @@ In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCV
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In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
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This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08,Gro-JCP-09,FelPet-JCP-09,NemTowNee-JCP-10,FelPetHil-JCP-11,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
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%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
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As a method $\modX$ we employ either CCSD(T) or exFCI.
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As a method $\modY$ we employ either CCSD(T) or exFCI.
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Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
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We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy.
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In the case of the CCSD(T) calculations, we have $\modZ = \HF$ as we use the restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy.
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\titou{For exFCI, we use the density of a converged variational wave function.}
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For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculations, and built with the natural orbitals of the converged variational wave function for the exFCI calculations.
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The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
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