Sec IID
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@ -447,7 +447,7 @@ Because of the very definition of $\wbasis$, one has the following property in t
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which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
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\subsubsection{Frozen-core approximation}
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the \titou{complementary basis functional} introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
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\begin{equation}
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\label{eq:def_mur_val}
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\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
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@ -475,32 +475,34 @@ is the valence-only effective interaction and
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One would note the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
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It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
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\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ }
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\subsection{Short-range correlation density functional approximations}
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\label{sec:functional}
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\subsubsection{Generic form of the approximate functionals}
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\subsubsection{Generic form}
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\label{sec:functional_form}
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As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the \titou{complementary basis} functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
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\begin{equation}
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\begin{aligned}
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As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the \titou{complementary basis functional} $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$.
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\trashPFL{Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form}
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\titou{Therefore, $\efuncden{\den}$ has the following generic form}
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\begin{multline}
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\label{eq:def_ecmdpbebasis}
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&\efuncdenpbe{\argebasis} = \;\;\;\;\;\;\;\; \\ &\int \d\br{} \,\denr \ecmd(\argrebasis),
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\end{aligned}
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\end{equation}
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where $\ecmd(\argecmd)$ is the correlation energy per particle taken as
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\efuncdenpbe{\argebasis} =
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\\
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\int \d\br{} \,\denr \ecmd(\argrebasis),
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\end{multline}
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where
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\begin{equation}
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\label{eq:def_ecmdpbe}
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\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe) \; \mu^3},
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\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\titou{\ntwo}) \; \mu^3},
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\end{equation}
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with
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is the correlation energy per particle, with
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\begin{equation}
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\label{eq:def_beta}
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\beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
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\beta(\argepbe,\titou{\ntwo}) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
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\end{equation}
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where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
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where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle. \cite{PerBurErn-PRL-96} Before introducing the different flavors of approximate functionals that we will use here (see Sec.~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
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The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional
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The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional, \ie,
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\begin{equation}
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\lim_{\mu \to 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),
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\end{equation}
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@ -509,7 +511,7 @@ which is relevant in the weak-correlation (or high-density) limit. In the large-
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\label{eq:lim_mularge}
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\ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n},
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\end{equation}
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which is the exact large-$\mu$ behavior of the exact ECMD correlation energy~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18}. Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq.~\eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers~\cite{CarTruGag-JPCA-17} and Pernal and coworkers\cite{GritMeePer-PRA-18}.
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which is the exact large-$\mu$ behavior of the exact ECMD correlation energy. \cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq.~\eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers \cite{CarTruGag-JPCA-17} and Pernal and coworkers.\cite{GritMeePer-PRA-18}
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Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes
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\begin{equation}
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@ -521,19 +523,19 @@ which is expected for systems with a vanishing on-top pair density, such as the
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\lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0.
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\end{equation}
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\subsubsection{Properties of approximate functionals}
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\subsubsection{Properties}
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\label{sec:functional_prop}
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Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary basis functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
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Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate \titou{complementary basis functional} $\efuncdenpbe{\argebasis}$ satisfies two important properties.
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First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, whatever the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$,
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First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$,
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\begin{equation}
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\label{eq:lim_ebasis}
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\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis,
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\end{equation}
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which guarantees an unaltered CBS limit.
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Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see Eq.~\eqref{eq:wbasis}) together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see Eq.~\eqref{eq:lim_n2}).
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Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ [see Eq.~\eqref{eq:wbasis}] together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
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\subsection{Requirements for the approximate functionals in the strong-correlation regime}
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\label{sec:requirements}
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@ -601,21 +603,21 @@ We define the following functionals:\\
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i) PBE-UEG-$\tilde{\zeta}$ which uses the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}
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\begin{equation}
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\label{eq:def_pbeueg}
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\label{eq:def_pbeueg_i}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeuegXi} = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi),
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\end{aligned}
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\end{equation}
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ii) PBE-ot-$\tilde{\zeta}$ which uses the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
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\begin{equation}
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\label{eq:def_pbeueg}
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\label{eq:def_pbeueg_ii}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi),
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\end{aligned}
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\end{equation}
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iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
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\begin{equation}
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\label{eq:def_pbeueg}
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\label{eq:def_pbeueg_iii}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
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\end{aligned}
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@ -646,7 +648,7 @@ The purpose of the present paper being the study of the basis-set correction in
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In the case of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules, approximations to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software~\cite{QP2}. The estimated exact potential energy curves are obtained from Ref.~\onlinecite{LieCle-JCP-74a}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be below 0.5 mH. In the case of the H$_{10}$ chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
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Regarding the complementary basis functional, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional (density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$) together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. These CASSCF wave functions correspond to the following active spaces: 10 electrons in 10 orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, 12 electrons in 8 orbitals for O$_2$, and 14 electrons in 8 orbitals for F$_2$.
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Regarding the \titou{complementary basis functional}, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional (density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$) together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. These CASSCF wave functions correspond to the following active spaces: 10 electrons in 10 orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, 12 electrons in 8 orbitals for O$_2$, and 14 electrons in 8 orbitals for F$_2$.
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Also, as the frozen core approximation is used in all our CIPSI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density-like quantities exclude any contribution from the core 1s orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
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@ -765,7 +767,7 @@ We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} t
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Just as in the case of \ce{H10}, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like \ce{O2}. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of \ce{C2} with the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the \ce{C2} molecule.
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Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary basis functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
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Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the \titou{complementary basis functionals} can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
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\section{Conclusion}
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\label{sec:conclusion}
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