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Revised_Manuscript/srDFT_SC.tex

@@ -325,7 +325,7 @@ An alternative way to improve the convergence towards the complete-basis-set (CB
 
 Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} in which RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also originate from a Hamiltonian with a non-divergent long-range electron-electron interaction. Therefore, a mapping with RSDFT can be performed through the introduction of an effective non-divergent interaction representing the usual electron-electron Coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out, \cite{LooPraSceTouGin-JCPL-19} together with extensions of this approach to the calculations of excitation energies \cite{GinSceTouLoo-JCP-19} and ionization potentials. \cite{LooPraSceGinTou-JCTC-20} The goal of the present work is to further develop this approach for the description of strongly correlated systems. 
 
-The paper is organized as follows. In Sec.~\ref{sec:theory}, we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties which are highly desirable in the context of strong correlation: spin-multiplet degeneracy and size consistency. To do this, we introduce i) new functionals using different flavors of spin polarizations and on-top pair densities, and ii) an effective electron-electron interaction based on a multiconfigurational wave function. This generalizes the method used in previous works on weakly correlated systems\cite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JCPL-19} for which it was sufficient to use an effective interaction based on a single-determinant wave function and a functional depending only on the usual density and spin polarization. 
+The paper is organized as follows. In Sec.~\ref{sec:theory}, we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties which are highly desirable in the context of strong correlation: spin-multiplet degeneracy and size consistency. To do this, we introduce i) new functionals using different flavors of spin polarizations and on-top pair densities, and ii) an effective electron-electron interaction based on a multiconfigurational wave function. This generalizes the method used in previous works on weakly correlated systems\cite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JCPL-19} for which it was sufficient to use an effective interaction based on a single-determinant wave function and a functional depending only on the usual density, reduced density gradient and spin polarization. 
 Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit. Finally, we conclude in Sec.~\ref{sec:conclusion}. 
 
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@@ -478,7 +478,7 @@ It is also noteworthy that, with the present definition, $\wbasisval$ still tend
 \subsubsection{Generic approximate form}
 \label{sec:functional_form}
 
-As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary 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,Tou-THESIS-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to consider a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), 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 $\mur \equiv\murpsi$. 
+As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary 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,Tou-THESIS-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to consider a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), 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, $\efuncden{\den}$ has the following generic form
 \begin{multline}
  \label{eq:def_ecmdpbebasis}