Sec IIA

Manuscript/srDFT_SC.tex

@@ -50,16 +50,6 @@
 \newcommand{\CBS}{\text{CBS}}                                                                                             
 
 	
-%operators 
-\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
-\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}
-
-%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
-%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}}
-
-% 
-
-	
 % energies
 \newcommand{\Ec}{E_\text{c}}
 \newcommand{\EPT}{E_\text{PT2}}
@@ -146,7 +136,7 @@
 
 
 % effective interaction
-\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
+\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
 \newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
 \newcommand{\ntwo}[0]{n_{2}}
 \newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
@@ -347,7 +337,7 @@ Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the p
 \section{Theory}
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
-As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section  \ref{sec:final_def_func}. 
+As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the \titou{density functional complementary to a basis set $\Bas$}. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Sec.~\ref{sec:final_def_func}. 
 
 \subsection{Basic formal equations}
 \label{sec:basic}
@@ -360,43 +350,45 @@ The exact ground-state energy $E_0$ of a $N$-electron system can in principle be
 where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83}
 \begin{equation}
  \label{eq:levy_func}
- F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
+ F[\den] = \min_{\Psi \rightarrow \den} \mel{\Psi}{\kinop +\weeop}{\Psi},
 \end{equation}
-where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas \approx E_0$.
+\PFL{$\kinop$ and $\weeop$ are not defined.}
+where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an \titou{acceptable} approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.
 
-In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
+In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
 \begin{equation}
  \label{eq:def_levy_bas}
- F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, 
+ F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, 
 \end{equation}
-where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as
+where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and 
 \begin{equation}
  \begin{aligned}
- \efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}  
+ \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi}  
-                  - \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. 
+                  - \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} 
  \end{aligned}
 \end{equation}
-Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$
+is the \titou{density functional complementary to the basis set $\Bas$.}
-\begin{eqnarray}
+Introducing the decomposition in Eq.~\eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy} yields 
+\begin{multline}
  \label{eq:E0basminPsiB}
- E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} 
+ E_0^\Bas = \min_{\Psi^{\Bas}} \bigg\{ \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} 
-\nonumber\\
+	\\
-&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
+	+ \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
-\end{eqnarray}
+\end{multline}
-where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set.
+where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects that are not included in the basis set.
 
-As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
+As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy [see Eqs.~(12)--(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}]
 \begin{equation}
  \label{eq:e0approx}
  E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
 \end{equation}
-where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy.
+where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two key requirements for this purpose are i) size consistency, and ii) spin-multiplet degeneracy.
 
 \subsection{Definition of an effective interaction within $\Bas$}
 \label{sec:wee}
 As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point. 
 
-As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
+As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
 \begin{equation}
  \label{eq:wbasis}
  \wbasis =
@@ -423,10 +415,10 @@ With such a definition, one can show that $\wbasis$ satisfies
 \nonumber\\
 \frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. 
 \end{eqnarray}
-As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, \ie,
+As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, \ie,
 \begin{equation}
  \label{eq:cbs_wbasis}
- \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis. 
+ \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis. 
 \end{equation}
 The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.
 
@@ -438,7 +430,7 @@ As the effective interaction within a basis set, $\wbasis$, is non divergent, it
  \label{eq:weelr}
  w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, 
 \end{equation}
-where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondence between these two interactions by using the local range-separation parameter $\murpsi$
+where $\mu$ is the range-separation parameter. As originally proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondence between these two interactions by using the local range-separation parameter $\murpsi$
 \begin{equation}
  \label{eq:def_mur}
  \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, 
@@ -450,7 +442,7 @@ such that the interactions coincide at the electron-electron coalescence point f
 Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis}) 
 \begin{equation}
  \label{eq:cbs_mu}
-  \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, 
+  \lim_{\Bas \to \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, 
 \end{equation}
 which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. 
 
@@ -490,7 +482,7 @@ It is noteworthy that, with the present definition, $\wbasisval$ still tends to
 \subsubsection{Generic form of the approximate functionals}
 \label{sec:functional_form}
 
-As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the 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
+As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the 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
 \begin{equation}
  \begin{aligned}
  \label{eq:def_ecmdpbebasis}
@@ -511,14 +503,14 @@ where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlati
 
 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
 \begin{equation}
-   \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),   
+   \lim_{\mu \to 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),   
 \end{equation}
 which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as
 \begin{equation}
  \label{eq:lim_mularge}
     \ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3}  \frac{\ntwo}{n},
 \end{equation}
-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}.
+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}.
 
 Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes 
 \begin{equation}
@@ -555,7 +547,7 @@ Another important requirement is spin-multiplet degeneracy, \ie, the independenc
 
 \subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
 
-A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}). 
+A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}]. 
 
 It has been proposed to replace in functionals the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01})
 \begin{equation}
@@ -643,7 +635,7 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
         \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps}
         \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
         \caption{
-        Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the 1) cc-pVDZ, 2) cc-pVTZ, and 3) cc-pVQZ basis sets. The MRCI+Q energies and the estimated exact energies are from Ref. \onlinecite{h10_prx}.
+        Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the 1) cc-pVDZ, 2) cc-pVTZ, and 3) cc-pVQZ basis sets. The MRCI+Q energies and the estimated exact energies are from Ref.~\onlinecite{h10_prx}.
         \label{fig:H10}}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -652,7 +644,7 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
 
 The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a H$_{10}$ chain with equally distant atoms and the C$_2$, N$_2$, O$_2$, and F$_2$ molecules. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$. 
 
-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). 
+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). 
 
 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$_2$. 
 
@@ -701,8 +693,8 @@ F$_2$ & aug-cc-pVDZ          &   49.6  [12.6]          &    54.8  [7.4]      &
 \end{ruledtabular}
 \begin{flushleft}
 \vspace{-0.2cm}
-$^a$ From Ref. \onlinecite{h10_prx}.  \\
+$^a$ From Ref.~\onlinecite{h10_prx}.  \\
-$^b$ From the extrapolated valence-only non-relativistic calculations of Ref. \onlinecite{BytLaiRuedenJCP05}.
+$^b$ From the extrapolated valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.
 \end{flushleft}
 
 \label{tab:extensiv_closed}
@@ -713,7 +705,7 @@ $^b$ From the extrapolated valence-only non-relativistic calculations of Ref. \o
 \subsection{Dissociation of the H$_{10}$ chain with equally distant atoms}
 
 
-The study of the H$_{10}$ chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for a detailed study of this problem). 
+The study of the H$_{10}$ chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem). 
 
 We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.  
 
@@ -730,7 +722,7 @@ Regarding in more details the performance of the different types of approximate
         \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps}
         \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps}
         \caption{
-        Potential energy curves of the C$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
+        Potential energy curves of the C$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
         \label{fig:C2}}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -742,7 +734,7 @@ Regarding in more details the performance of the different types of approximate
         \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps}
         \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
         \caption{
-        Potential energy curves of the N$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
+        Potential energy curves of the N$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
         \label{fig:N2}}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -754,7 +746,7 @@ Regarding in more details the performance of the different types of approximate
 %        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps}
 %        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps}
         \caption{
-        Potential energy curves of the O$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
+        Potential energy curves of the O$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
         \label{fig:O2}}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -766,7 +758,7 @@ Regarding in more details the performance of the different types of approximate
         \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.eps}
         \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
         \caption{
-        Potential energy curves of the F$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
+        Potential energy curves of the F$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
         \label{fig:F2}}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -786,7 +778,7 @@ In the present paper we have extended the recently proposed DFT-based basis-set
 
 The DFT-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$; ii) the fitting of this effective interaction with the long-range interaction used in RS-DFT, iii) the use of a complementary correlation functional of RS-DFT. In the present paper, we investigated points i) and iii) in the context of strong correlation and focused on potential energy curves  and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling a) spin-multiplet degeneracy, and b) size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. 
 
-The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin polarization without loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref. \onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of the on-top pair density. 
+The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin polarization without loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of the on-top pair density. 
 
 Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-zeta quality basis sets chemically accurate atomization energies $D_0$ are obtained for all systems but C$_2$, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction.