work in progress

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Pierre-Francois Loos 2019-04-04 21:53:32 +02:00
parent de40e4c6cf
commit 1f3c11517e

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@ -25,9 +25,6 @@
\newcommand{\mr}{\multirow}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
% Titou's macros
\newcommand{\br}{\mathbf{r}}
% second quantized operators
\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
@ -74,13 +71,13 @@
\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\psibasis}[\denmodel]}
\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\psibasis}[\den]}
\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\psibasis}[\den]}
\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\psibasis}[\den]}
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)}
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\psibasis)\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)}
\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]}
\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
@ -97,30 +94,30 @@
% effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\psibasis)}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\psibasis)}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
\newcommand{\wbasis}[0]{W_{\psibasis}(\bfr{1},\bfr{2})}
\newcommand{\wbasisval}[0]{W_{\psibasis}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\fbasis}[0]{f_{\psibasis}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\psibasis}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\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{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ n^{(2)}_{\psibasis}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\psibasis}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\psibasis,\,\text{val}}(\rr{1}{2})}
\newcommand{\twodmrpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
\newcommand{\wbasiscoal}[1]{W_{\psibasis}({\bf r}_{#1})}
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\psibasis}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoalval}[1]{W_{\psibasis}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\psibasis}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
@ -146,7 +143,6 @@
% wave functions
\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
\newcommand{\psibasis}[0]{\Psi^{\Bas}}
\newcommand{\psimu}[0]{\Psi^{\mu}}
% operators
\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
@ -161,11 +157,40 @@
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\pis}{\pi^\star}
\newcommand{\si}{\sigma}
\newcommand{\sis}{\sigma^\star}
% methods
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\Nel}{N}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
\newcommand{\modX}{\text{X}}
\newcommand{\modY}{Y}
% basis sets
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}}
\newcommand{\Val}{\mathcal{V}}
\newcommand{\Cor}{\mathcal{C}}
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
\newcommand{\bx}[1]{\mathbf{x}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\dbx}[1]{d\bx{#1}}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
@ -203,7 +228,7 @@ For example, the coupled cluster (CC) family of methods offers a powerful WFT ap
By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually pricey.
One of the most fundamental drawback of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
This undesirable feature was put into light by Kutzelnigg more than thirty years ago, \cite{Kut-TCA-85}
who proposed, to palliate this, to introduce explicitly the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, TerKloKut-JCP-91, KloKut-JCP-91, KloRohKut-CPL-91, NogKut-JCP-94}
who proposed, to palliate this, to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, TerKloKut-JCP-91, KloKut-JCP-91, KloRohKut-CPL-91, NogKut-JCP-94}
The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12}
For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-zeta quality correlation energies with a triple-zeta basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals.
@ -229,35 +254,16 @@ Unless otherwise stated, atomic used are used.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
The basis set correction investigated here uses the RS-DFT formalism to capture the part of the short-range correlation effects missing from the description of the WFT in a finite basis set.
Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}.
The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set.
Here, we provide the main working equations.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about the formal derivation of the theory.
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\Nel}{N}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\modX}{\text{X}}
\newcommand{\modY}{Y}
% basis sets
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}}
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}{\Hat{W}_\text{ee}}
%=================================================================
%\subsection{Correcting the basis set error of a general WFT model}
%=================================================================
Let us assume we have both the density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$.
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a good approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a resonable approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
\begin{equation}
\label{eq:e0basis}
\E{}{}
@ -268,24 +274,23 @@ where
\begin{equation}
\label{eq:E_funcbasis}
\bE{}{\Bas}[\n{}{}]
= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee}{\wf{}{}}
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee}{\wf{}{\Bas}}
= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation}
is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee$ are the kinetic and interelectronic repulsion operators, respectively.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ is the wave function obtained from the $\Nel$-electron Hilbert space spanned by $\Bas$, and $\wf{}{}$ is a general $\Nel$-electron wave function being obtained in a complete basis.
is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
Both wave functions yield the same target density $\n{}{}$.
%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
An important aspect of such a theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
\begin{equation}
\label{eq:limitfunc}
\lim_{\Bas \rightarrow \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{\infty} \approx E,
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{} \approx E,
\end{equation}
where $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in complete basis set.
In the case of $\modX = \FCI$, we $\E{\FCI}{\infty} = E$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, at this stage, the only source of error lies in the potential approximate nature of the method $\modX$.
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the method $\modX$.
%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
%As any wave function model is necessary an approximation to the FCI model, one can write
@ -384,18 +389,18 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, at this sta
%=================================================================
However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br)$ varying in space (see Sec.~\ref{sec:weff}).
Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ (see Sec.~\ref{sec:ecmd}) with $\mu(\br)$.
First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br{})$ varying in space. %(see Sec.~\ref{sec:weff}).
Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
%=================================================================
\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
\label{sec:weff}
%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
%\label{sec:weff}
%=================================================================
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e. a discontinuous derivative) at the electron-electron coalescence points.
As the electron-electron cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent electron-electron interaction.
Therefore, the impact of the incompleteness of a finite basis set $\Bas$ can be viewed as a removal of the divergence of the coulomb interaction at $r_{12} = 0$.
The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which i) is finite at the electron-electron coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
The present paragraph briefly describes how to obtain an effective interaction $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
%----------------------------------------------------------------
%\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
@ -406,34 +411,34 @@ Consider the Coulomb operator projected in $\Bas$
\weeopbasis = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
\end{aligned}
\end{equation}
where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual coulomb two-electron integrals.
Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\Bas$.
After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
Consider now the expectation value of $\weeopbasis$ over a general wave function $\wf{}{\Bas}$ belonging to the $\Nel$-electron Hilbert space spanned by $\Bas$.
One can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
\begin{equation}
\label{eq:expectweeb}
\mel*{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis\,\,,
\label{eq:expectweeb}
\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}
\end{equation}
where
\begin{equation}
\label{eq:fbasis}
\begin{aligned}
\fbasis = \sum_{ijklmn\,\,\in\,\,\Bas} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}\,\,,
\end{aligned}
\end{equation}
\begin{multline}
\label{eq:fbasis}
\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
\\
= \sum_{ijklmn \in \Bas} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1} \SO{i}{1} \SO{j}{2},
\end{multline}
and
\begin{equation}
\gammamnpq{\psibasis} = \mel*{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis}\,\,,
\end{equation}
is the two-body density tensor of $\psibasis$ and $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \, \dr{} = \sum_{\sigma}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}$.
Then, consider the expectation value of the exact coulomb operator over $\psibasis$
\begin{equation}
\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}},
\end{equation}
is the two-body density tensor of $\wf{}{\Bas}$, $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$.
Then, consider the expectation value of the exact Coulomb operator over $\wf{}{\Bas}$
\begin{equation}
\label{eq:expectwee}
\mel*{\psibasis}{\weeop}{\psibasis} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
\mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
\end{equation}
where $\twodmrdiagpsi$ is the two-body density associated to $\psibasis$.
Because $\psibasis$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
where $\n{\wf{}{\Bas}}{(2)}$ is the two-body density associated with $\wf{}{\Bas}$.
Because $\wf{}{\Bas}$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
\begin{equation}
\mel*{\psibasis}{\weeopbasis}{\psibasis} = \mel*{\psibasis}{\weeop}{\psibasis},
\mel*{\wf{}{\Bas}}{\weeopbasis}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
\end{equation}
which can be rewritten as:
\begin{multline}
@ -445,23 +450,24 @@ which can be rewritten as:
where we introduced $\wbasis$
\begin{equation}
\label{eq:def_weebasis}
\wbasis = \frac{\fbasis}{\twodmrdiagpsi},
\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})},
\end{equation}
which is the effective interaction in the basis set $\Bas$.
As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\Bas$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\Bas$.
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\Bas$.
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ tends to the regular coulomb interaction $r_{12}^{-1}$ for all points $(\bx{1},\bx{2})$ and any choice of $\wf{}{\Bas}$ in the limit of a complete basis set.
%----------------------------------------------------------------
\subsubsection{Definition of a valence effective interaction}
%----------------------------------------------------------------
As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\Basval$.
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
\begin{equation}
\label{eq:expectweebval}
\mel*{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
\mel*{\wf{}{\Bas}}{\weeopbasisval}{\wf{}{\Bas}} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
\end{equation}
where $\weeopbasisval$ is the valence coulomb operator defined as
\begin{equation}
@ -474,7 +480,7 @@ Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defi
\begin{equation}
\label{eq:fbasisval}
\begin{aligned}
\fbasisval = \sum_{ij\,\,\in\,\,\Bas} \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
\fbasisval = \sum_{ij\,\,\in\,\,\Bas} \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\wf{}{\Bas}} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
\end{aligned}
\end{equation}
@ -485,7 +491,7 @@ Then, the effective interaction associated to the valence $\wbasisval$ is simply
\end{equation}
where $\twodmrdiagpsival$ is the two body density associated to the valence electrons:
\begin{equation}
\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\psibasis] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\wf{}{\Bas}] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
\end{equation}
It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
@ -495,7 +501,7 @@ To be able to approximate the complementary functional $\efuncbasis$ thanks to f
More precisely, if we define the value of the interaction at coalescence as
\begin{equation}
\label{eq:def_wcoal}
\wbasiscoal{} = W_{\psibasis}(\bfr{},\bar{{\bf x}}_{}).
\wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
\end{equation}
where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$,
we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
@ -589,7 +595,7 @@ It is important to notice that in the limit of a complete basis set, according t
\begin{equation}
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
\end{equation}
for whatever choice of density $\denmodel$, wave function $\psibasis$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
\subsubsection{LDA approximation for the complementary functional}
As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as
@ -648,13 +654,13 @@ We now introduce a valence-only approximation for the complementary functional w
Defining the valence one-body spin density matrix as
\begin{equation}
\begin{aligned}
\onedmval[\psibasis] & = \elemm{\psibasis}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\psibasis} \qquad \text{if }(i,j)\in \Basval \\
\onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
& = 0 \qquad \text{in other cases}
\end{aligned}
\end{equation}
then one can define the valence density as:
\begin{equation}
\denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\psibasis] \phi_i({\bf r}) \phi_j({\bf r})
\denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r})
\end{equation}
Therefore, we propose the following valence-only approximations for the complementary functional
\begin{equation}
@ -674,7 +680,7 @@ Therefore, we propose the following valence-only approximations for the compleme
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
%\subsubsection{CIPSI calculations and the basis-set correction}
%All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$.
%Regarding the wave function $\psibasis$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
%Regarding the wave function $\wf{}{\Bas}$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
\subsubsection{CCSD(T) calculations and the basis-set correction}
\begin{table*}