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@ -339,7 +339,7 @@ Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the p
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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}. 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} \subsection{Basic equations}
\label{sec:basic} \label{sec:basic}
The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$ The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$
@ -384,7 +384,7 @@ As a simple non-self-consistent version of this approach, we can approximate the
\end{equation} \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 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. 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 $\Bas$} \subsection{Effective interaction in a finite basis}
\label{sec:wee} \label{sec:wee}
As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of 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. \titou{In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction.} As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of 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. \titou{In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction.}
@ -422,24 +422,24 @@ As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interacti
\end{equation} \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. The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.
\subsection{Definition of a local range-separation parameter} \subsection{Local range-separation parameter}
\label{sec:mur} \label{sec:mur}
\subsubsection{General definition} \subsubsection{General definition}
As the effective interaction within a basis set, $\wbasis$, is non divergent, it resembles the long-range interaction used in RSDFT As the effective interaction within a finite basis, $\wbasis$ is bounded and resembles the long-range interaction used in RSDFT
\begin{equation} \begin{equation}
\label{eq:weelr} \label{eq:weelr}
w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}},
\end{equation} \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
\begin{equation} \begin{equation}
\label{eq:def_mur} \label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal,
\end{equation} \end{equation}
such that the interactions coincide at the electron-electron coalescence point for each $\br{}$ such that the two interactions coincide at the electron-electron coalescence point for each $\br{}$
\begin{equation} \begin{equation}
w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}. w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}.
\end{equation} \end{equation}
Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis}) Because of the very definition of $\wbasis$, one has the following property in the CBS limit [see Eq.~\eqref{eq:cbs_wbasis}]
\begin{equation} \begin{equation}
\label{eq:cbs_mu} \label{eq:cbs_mu}
\lim_{\Bas \to \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, \lim_{\Bas \to \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis,
@ -447,12 +447,12 @@ Because of the very definition of $\wbasis$, one has the following property in t
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen-core approximation} \subsubsection{Frozen-core approximation}
As all WFT calculations in this work are performed within the frozen-core approximation, we use the 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 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
\begin{equation} \begin{equation}
\label{eq:def_mur_val} \label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{}, \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
\end{equation} \end{equation}
where $\wbasisval$ is the valence-only effective interaction defined as where
\begin{equation} \begin{equation}
\label{eq:wbasis_val} \label{eq:wbasis_val}
\wbasisval = \wbasisval =
@ -462,19 +462,18 @@ where $\wbasisval$ is the valence-only effective interaction defined as
\infty, & \text{otherwise,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
\end{equation} \end{equation}
where $\fbasisval$ and $\twodmrdiagpsival$ are defined as is the valence-only effective interaction and
\begin{equation} \begin{gather}
\label{eq:fbasis_val} \label{eq:fbasis_val}
\fbasisval \fbasisval
= \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},
\end{equation} \\
and \label{eq:twordm_val}
\begin{equation} \twodmrdiagpsival
\label{eq:twordm_val} = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. \end{gather}
\end{equation} 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}.
Notice the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
It is noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ } \subsection{Generic form and properties of the approximations for $\efuncden{\den}$ }
\label{sec:functional} \label{sec:functional}
@ -623,7 +622,7 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
\end{equation} \end{equation}
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\section{Results for the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ potential energy curves} \section{Results}
\label{sec:results} \label{sec:results}
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@ -702,7 +701,7 @@ $^b$ From the extrapolated valence-only non-relativistic calculations of Ref.~\o
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\subsection{Dissociation of the H$_{10}$ chain with equally distant atoms} \subsection{H$_{10}$ chain}
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).
@ -713,7 +712,7 @@ More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an e
Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density. Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.
\subsection{Dissociation of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules} \subsection{Dissociation of diatomics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*} \begin{figure*}