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Pierre-Francois Loos 2019-12-12 15:23:04 +01:00
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Manuscript/srDFT_SC.tex
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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}.
\subsection{Basic formal equations}
\subsection{Basic equations}
\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 $\denr$
@ -384,7 +384,7 @@ As a simple non-self-consistent version of this approach, we can approximate the
\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.
\subsection{Definition of an effective interaction within $\Bas$}
\subsection{Effective interaction in a finite basis}
\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.}
@ -422,24 +422,24 @@ As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interacti
\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.
\subsection{Definition of a local range-separation parameter}
\subsection{Local range-separation parameter}
\label{sec:mur}
\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}
\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
\begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal,
\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}
w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}.
\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}
\label{eq:cbs_mu}
\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.
\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}
\label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
\end{equation}
where $\wbasisval$ is the valence-only effective interaction defined as
where
\begin{equation}
\label{eq:wbasis_val}
\wbasisval =
@ -462,19 +462,18 @@ where $\wbasisval$ is the valence-only effective interaction defined as
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where $\fbasisval$ and $\twodmrdiagpsival$ are defined as
\begin{equation}
\label{eq:fbasis_val}
\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},
\end{equation}
and
\begin{equation}
\label{eq:twordm_val}
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{equation}
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 noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
is the valence-only effective interaction and
\begin{gather}
\label{eq:fbasis_val}
\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},
\\
\label{eq:twordm_val}
\twodmrdiagpsival
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{gather}
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}.
It is also 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 $\efuncden{\den}$ }
\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}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results for the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ potential energy curves}
\section{Results}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -702,7 +701,7 @@ $^b$ From the extrapolated valence-only non-relativistic calculations of Ref.~\o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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).
@ -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.
\subsection{Dissociation of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules}
\subsection{Dissociation of diatomics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}