changes in theory

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Julien Toulouse 2019-12-05 14:37:41 +01:00
parent f58e01e0e8
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2 changed files with 102 additions and 80 deletions

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@ -6,7 +6,7 @@
%Control: page (0) single %Control: page (0) single
%Control: year (1) truncated %Control: year (1) truncated
%Control: production of eprint (0) enabled %Control: production of eprint (0) enabled
\begin{thebibliography}{65}% \begin{thebibliography}{68}%
\makeatletter \makeatletter
\providecommand \@ifxundefined [1]{% \providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined} \@ifx{#1\undefined}
@ -531,15 +531,34 @@
{author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, \ and\ \bibinfo {author} {author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, \ and\ \bibinfo {author}
{\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {} {\bibinfo {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {} {\bibinfo
{journal} {arXiv:1910.12238}\ }\BibitemShut {NoStop}% {journal} {arXiv:1910.12238}\ }\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Levy}(1979)}]{Lev-PNAS-79}%
\BibitemOpen
\bibfield {journal} { }\bibfield {author} {\bibinfo {author} {\bibfnamefont
{M.}~\bibnamefont {Levy}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Proc. Natl. Acad. Sci. U.S.A.}\ }\textbf {\bibinfo {volume}
{76}},\ \bibinfo {pages} {6062} (\bibinfo {year} {1979})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Lieb}(1983)}]{Lie-IJQC-83}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont
{Lieb}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Int. J.
Quantum Chem.}\ }\textbf {\bibinfo {volume} {{24}}},\ \bibinfo {pages} {24}
(\bibinfo {year} {1983})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kato}(1957)}]{Kat-CPAM-57}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Kato}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Comm.
Pure Appl. Math.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages} {151}
(\bibinfo {year} {1957})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\ \bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\
\citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}% \citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}%
\BibitemOpen \BibitemOpen
\bibfield {journal} { }\bibfield {author} {\bibinfo {author} {\bibfnamefont \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{J.}~\bibnamefont {Toulouse}}, \bibinfo {author} {\bibfnamefont {Toulouse}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{P.}~\bibnamefont {Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont {Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{A.}~\bibnamefont {Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Theor.
{journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo {pages} {305}
{pages} {305} (\bibinfo {year} {2005})}\BibitemShut {NoStop}% (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Perdew}, \citenamefont {Burke},\ and\ \citenamefont \bibitem [{\citenamefont {Perdew}, \citenamefont {Burke},\ and\ \citenamefont
{Ernzerhof}(1996)}]{PerBurErn-PRL-96}% {Ernzerhof}(1996)}]{PerBurErn-PRL-96}%
\BibitemOpen \BibitemOpen

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@ -83,16 +83,16 @@
%pbeuegxiHF %pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} \newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}} \newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}}
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})} \newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})}
%pbeuegxiCAS %pbeuegxiCAS
\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} \newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} \newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} \newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS %pbeuegXiCAS
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}} \newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} \newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} \newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontxiCAS %pbeontxiCAS
\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta} \newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} \newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
@ -111,9 +111,9 @@
\newcommand{\argepbe}[0]{\den,\zeta,s} \newcommand{\argepbe}[0]{\den,\zeta,s}
\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}} \newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} \newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} \newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} \newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} \newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} \newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} \newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
@ -129,17 +129,17 @@
% effective interaction % effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} \newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})} \newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\ntwo}[0]{n^{(2)}} \newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}} \newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}} \newcommand{\ntwophi}[0]{n_2^{{\phi}}}
\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}} \newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\psibasis}}
\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}} \newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{CAS},\basis}}
\newcommand{\mur}[0]{\mu({\bf r})} \newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})} \newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} \newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})} \newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})} \newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})}
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} \newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
@ -149,9 +149,9 @@
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} \newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} \newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} \newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})} \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})} \newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
@ -239,6 +239,7 @@
\newcommand{\Cor}{\mathcal{C}} \newcommand{\Cor}{\mathcal{C}}
% operators % operators
\renewcommand{\d}{\text{d}}
\newcommand{\hT}{\Hat{T}} \newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} \newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}} \newcommand{\f}[2]{f_{#1}^{#2}}
@ -325,56 +326,56 @@ Then, in Section \ref{sec:results} we apply the method to the calculation of the
\section{Theory} \section{Theory}
\label{sec:theory} \label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
As the theoretical framework of the basis set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we briefly recall the main equations and concepts needed for this study in sections \ref{sec:basic}, \ref{sec:wee} and \ref{sec:mur}. 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}.
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 $\Bas$.
Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}.
Then, in section \ref{sec:functional} we expose the new approximated functionals complementary to a basis set $\Bas$ based on RSDFT functionals. The generic form of such functionals is exposed in section \ref{sec:functional_form}, their properties in the context of the basis set correction is discussed in \ref{sec:functional_prop}, and the requirements for strong correlation is discussed in section \ref{sec:requirements}. Then, the actual form of the functionals used in this work are introduced in section \ref{sec:final_def_func}.
\subsection{Basic formal equations} \subsection{Basic formal equations}
\label{sec:basic} \label{sec:basic}
The exact ground state energy $E_0$ of a $N-$electron system can be obtained by an elegant mathematical framework connecting WFT and DFT, that is the Levy-Lieb constrained search formalism which reads
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 $\denr$
\begin{equation} \begin{equation}
\label{eq:levy} \label{eq:levy}
E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\}, E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
\end{equation} \end{equation}
where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional 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} \begin{equation}
\label{eq:levy_func} \label{eq:levy_func}
F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}. F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
\end{equation} \end{equation}
The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density. 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 $\Bas$'', i.e. 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: $E_0^\Bas \approx E_0$.
Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$ and we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set. 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, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}
was to propose to split the minimization in the definition of $F[\denr]$ using $\wf{}{\Bas}$ which are wave functions developed in $\basis$
\begin{equation} \begin{equation}
\label{eq:def_levy_bas} \label{eq:def_levy_bas}
F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}, F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
\end{equation} \end{equation}
which leads to the following definition of $\efuncden{\denr}$ which is the density functional complementary to the basis set $\Bas$ 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
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\  \efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}  
&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. - \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
\end{aligned} \end{aligned}
\end{equation} \end{equation}
Therefore thanks to eq. \eqref{eq:def_levy_bas} one can properly connect the DFT formalism with the basis set error in WFT calculations. In other terms, the existence of $\efuncden{\denr}$ means that the correlation effects not taken into account in $\basis$ can be formulated as a density functional. Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$
\begin{eqnarray}
\label{eq:E0basminPsiB}
E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}}
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
\end{eqnarray}
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.
Assuming that the density $\denFCI$ associated to the ground state FCI wave function $\psifci$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state energy (see equations 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} \begin{equation}
\label{eq:e0approx} \label{eq:e0approx}
E_0 = \efci + \efuncbasisFCI E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
\end{equation} \end{equation}
where $\efci$ is the ground state FCI energy within $\Bas$. 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 errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set. 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.
The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it must provide size extensive energies, ii) it is invariant of the $S_z$ component of a given spin multiplicity.
\subsection{Definition of an effective interaction within $\Bas$} \subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee} \label{sec:wee}
As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, a cusp less 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 interaction at the electron coalescence point. 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 it was originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb 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. 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
More specifically, the effective interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
\begin{equation} \begin{equation}
\label{eq:wbasis} \label{eq:wbasis}
\wbasis = \wbasis =
@ -384,53 +385,55 @@ More specifically, the effective interaction associated to a given wave function
\infty, & \text{otherwise,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
\end{equation} \end{equation}
where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$ where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{\Bas}$
\begin{equation} \begin{equation}
\twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation} \end{equation}
$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals, and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\fbasis \fbasis
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$.
With such a definition, one can show that $\wbasis$ satisfies With such a definition, one can show that $\wbasis$ satisfies
\begin{equation} \begin{eqnarray}
\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
\end{equation} \nonumber\\
As it was shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set for any choice of wave function $\psibasis$, that is \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 $\psibasis$, i.e.
\begin{equation} \begin{equation}
\label{eq:cbs_wbasis} \label{eq:cbs_wbasis}
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}\quad \forall\,\psibasis. \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis.
\end{equation} \end{equation}
The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set. 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 range-separation parameter varying in real space} \subsection{Definition of a 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, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$ As the effective interaction within a basis set, $\wbasis$, is non divergent, it ressembles the long-range interaction used in RSDFT
\begin{equation} \begin{equation}
\label{eq:weelr} \label{eq:weelr}
w_{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}
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we use a range-separation parameter $\murpsi$ varying in real space where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$
\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 such that the interactions coincide at the electron-electron colescence point for each $\br{}$
\begin{equation} \begin{equation}
w_{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 properties at the CBS limit (see \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 \rightarrow \text{CBS}} \murpsi = \infty\quad \forall \,\psibasis, \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis,
\end{equation} \end{equation}
which is fundamental to guarantee the good behaviour of the theory at the CBS limit. which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen core density approximation} \subsubsection{Frozen-core approximation}
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}. As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}.
We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively)
and define the valence only range separation parameter and define the valence only range separation parameter
@ -538,17 +541,17 @@ Based on this reasoning, a similar approach has been used in the context of mult
In practice, these approaches introduce the effective spin polarisation In practice, these approaches introduce the effective spin polarisation
\begin{equation} \begin{equation}
\label{eq:def_effspin} \label{eq:def_effspin}
\tilde{\zeta}(n,\ntwo_{\psibasis}) = \tilde{\zeta}(n,\ntwo^{\psibasis}) =
% \begin{cases} % \begin{cases}
\sqrt{ n^2 - 4 \ntwo_{\psibasis} } \sqrt{ n^2 - 4 \ntwo^{\psibasis} }
% 0 & \text{otherwise.} % 0 & \text{otherwise.}
% \end{cases} % \end{cases}
\end{equation} \end{equation}
which uses the on-top pair density $\ntwo_{\psibasis}$ of a given wave function $\psibasis$. which uses the on-top pair density $\ntwo^{\psibasis}$ of a given wave function $\psibasis$.
The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation. The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo_{\psibasis}<0$ and also Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions. the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional. An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
@ -571,14 +574,14 @@ Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\ar
Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
\begin{equation} \begin{equation}
\label{eq:def_n2ueg} \label{eq:def_n2ueg}
\ntwo_{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big) \ntwo^{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
\end{equation} \end{equation}
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density. where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
\begin{equation} \begin{equation}
\label{eq:def_n2extrap} \label{eq:def_n2extrap}
\ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1} \ntwoextrap(\ntwo^{\psibasis},\mu,\br{}) = \ntwo^{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\end{equation} \end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT.
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot". When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot".