updated manuscript
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@ -80,7 +80,7 @@
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\modX}{\mathcal{X}}
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\newcommand{\modY}{\mathcal{Y}}
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@ -99,9 +99,9 @@
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\bx}[1]{\mathbf{x}_{#1}}
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%\newcommand{\br}[1]{\mathbf{x}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\dbx}[1]{d\bx{#1}}
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%\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\De}{D_\text{e}}
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@ -141,8 +141,8 @@ For example, CCSD(T)+LDA/cc-pVTZ and CCSD(T)+PBE/cc-pVTZ return mean absolute de
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Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
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Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.
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WFT is attractive as it exists a well-defined path for systematic improvement \manu{and powerful tools, such as perturbation theory, to guide the development of new attractive WFT models}.
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\trashMG{For example, t} The coupled cluster (CC) family of methods \trashMG{offers a powerful WFT approach} \manu{are a typical example of the WFT philosophy} for the description of weakly correlated systems and is well regarded as the gold standard of quantum chemistry.
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WFT is attractive as it exists a well-defined path for systematic improvement and powerful tools, such as perturbation theory, to guide the development of new attractive WFT models.
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The coupled cluster (CC) family of methods are a typical example of the WFT philosophy for the description of weakly correlated systems and is well regarded as the gold standard of quantum chemistry.
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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.
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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.
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This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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@ -180,7 +180,7 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
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%\subsection{Correcting the basis set error of a general WFT model}
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%=================================================================
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Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \titou{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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@ -203,11 +203,11 @@ Both wave functions yield the same target density $\n{}{}$.
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Importantly, 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
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{\infty} \approx E,
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\end{equation}
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where \trashMG{$\E{\modX}{}$} \manu{$\E{\modX}{\infty}$} is the energy associated with the method $\modX$ in the complete basis set.
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where $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in the complete basis set.
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In the case $\modX = \FCI$, we have as strict equality as $E_{\FCI}^\infty = E$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, \manu{the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$} \manu{for the FCI energy and density within $\Bas$}.
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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 methods $\modX$ and $\modY$ for the FCI energy and density within $\Bas$.
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%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 $\emodel$.
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%As any wave function model is necessary an approximation to the FCI model, one can write
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@ -299,15 +299,15 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, \manu{the o
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% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
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%\end{equation}
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\trashMG{However, in addition of being unknown,} \manu{Rigorously speaking, } the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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\manu{Nevertheless, from the physical point of view $\bE{}{\Bas}[\n{}{}]$ plays a quite universal role as it aims at fixing the main }
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\trashMG{One of the} consequences of the incompleteness of $\Bas$\manu{, which} is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Nevertheless, from the physical point of view $\bE{}{\Bas}[\n{}{}]$ plays a quite universal role as it aims at fixing the main
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consequence of the incompleteness of $\Bas$, which is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
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Therefore, as we shall do later on, it feels natural to \trashMG{evaluate} \manu{approximate} $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, the spatial inhomogeneity of $\Bas$ forces us to define a range-separated \textit{function} $\rsmu{}{}(\br{})$ as the value of $\rsmu{}{}$ must be known at any point in space.
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The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence point 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.
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In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$.
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The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which i) is finite at the e-e coalescence point 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.
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In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
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In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
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%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}].
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%First, 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 $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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@ -326,49 +326,49 @@ In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density funct
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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\trashMG{To compute} \manu{We define} the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ \trashMG{defined} as \manu{(see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})}
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We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{}(\bx{1},\bx{2})/\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})
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\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
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\end{equation}
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where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$
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where $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
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\begin{equation}
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\label{eq:n2basis}
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\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})
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\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2},
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\end{equation}
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$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), $\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is defined as
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$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\wf{}{\Bas}}{}(\br{1},\br{2})$ is defined as
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\begin{multline}
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
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\f{\wf{}{\Bas}}{}(\br{1},\br{2})
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\\
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
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\end{multline}
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and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
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With such a definition, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ verifies
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With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies
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\begin{equation}
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\label{eq:int_eq_wee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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which therefore can be rewritten as
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where the $\hWee{}$ contains only the alpha-beta component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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\iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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intuitively motivating $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ as a potential candidate for an effective interaction.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries.
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which intuitively motivates $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries.
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An important quantity to define is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
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\begin{equation}
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\label{eq:wcoal}
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\W{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\Bas}}{}(\bx{},\Bar{\bx{}})
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\W{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\Bas}}{}(\br{},{\br{}})
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\end{equation}
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and which is necessarily \textit{finite} at $r_{12} = 0$ for an \textit{incomplete} basis set.
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and which is necessarily \textit{finite} at for an \textit{incomplete} basis set.
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Of course, there exists \textit{a priori} an infinite set of functions satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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Of course, there exists \textit{a priori} an infinite set of functions in ${\rm I\!R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}\quad \forall \,\,(\bx{1},\bx{2}),
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\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}),
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\end{equation}
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which therefore guarantees a physically satisfying limit. An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
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which therefore guarantees a physically satisfying limit for all $\wf{}{\Bas}$. An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
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%=================================================================
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%\subsection{Range-separation function}
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@ -394,7 +394,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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% \label{eq:def_wcoal}
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% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
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%\end{equation}
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%where $(\bx{},\Bar{\bx{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
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%where $(\br{},\Bar{\br{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
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%we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
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%\begin{equation}
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% \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
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@ -416,7 +416,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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%An important point to notice is that, in the limit of a complete basis set $\Bas$, as
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%\begin{equation}
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%\label{eq:lim_W}
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% \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{2})
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% \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2})
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%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
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%\end{equation}
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%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
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@ -474,9 +474,9 @@ This makes them particularly well adapted to the present context where one aims
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\UEG}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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\end{equation}
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where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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%--------------------------------------------
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@ -515,10 +515,10 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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%\end{equation}
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%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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The computational bottleneck of the present approach is the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of
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The main computational source of the present approach is the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of
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\begin{equation}
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\label{eq:fcoal}
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f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{1} \SO{q}{2} \SO{i}{1} \SO{j}{2}
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f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
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\end{equation}
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which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtaine the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}, but this step has in general to be performed before a correlated WFT calculations.
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When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to reach a linear scaling method.
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@ -534,7 +534,7 @@ We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor
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Accounting solely for the valence electrons, we define
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
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\\
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= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
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\end{multline}
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@ -542,15 +542,15 @@ and the valence part of the effective interaction is
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\begin{subequations}
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\begin{gather}
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\label{eq:Wval}
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\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})/\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2}),
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\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}),
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\\
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\label{eq:muval}
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\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\bx{},\Bar{\bx{}}),
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\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}),
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\end{gather}
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\end{subequations}
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where
|
||||
\begin{equation}
|
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\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})
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\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
|
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= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}
|
||||
\end{equation}
|
||||
is the two body density associated to the valence electrons.
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@ -558,7 +558,7 @@ is the two body density associated to the valence electrons.
|
||||
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
|
||||
%\end{equation}
|
||||
%It is worth noting that, in Eq.~\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 $\Basval$.
|
||||
It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
|
||||
It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
|
||||
|
||||
%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
|
||||
%Defining the valence one-body spin density matrix as
|
||||
@ -583,6 +583,9 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
|
||||
%\end{equation}
|
||||
Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
|
||||
|
||||
To conclude this theory session, it is important to notice that in the limit of a complete basis set, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the choice of functional}.
|
||||
Thefore in the limit of a complete basis set one recovers the correct limit of the WFT model whatever approximations are made in the DFT part, just like in equation \eqref{eq:limitfunc}.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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Reference in New Issue
Block a user