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Emmanuel Giner
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@ -139,8 +139,8 @@ 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.
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For example, the coupled cluster (CC) family of methods offers a powerful WFT approach 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 \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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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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@ -203,9 +203,9 @@ Importantly, in the limit of a complete basis set $\Bas$ (which we refer to as $
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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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\end{equation}
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where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
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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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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 $\modY$}.
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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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%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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@ -297,10 +297,11 @@ 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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However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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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.
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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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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 evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
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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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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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@ -323,61 +324,48 @@ 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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To compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined as
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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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\begin{equation}
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\label{eq:int_eq_wee}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\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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\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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\end{equation}
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where
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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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\begin{equation}
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\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{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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= \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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is the two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realize that
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
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\end{equation}
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which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
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\begin{equation}
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\label{eq:WeeB}
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\hWee{\Bas} = \frac{1}{2} \sum_{pqrs \in \Bas}\V{pq}{rs} \aic{r} \aic{s} \ai{q} \ai{p}
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\end{equation}
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over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals in $\Bas$ and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
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Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{subequations}
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\begin{align}
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\label{eq:expectweeb}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\\
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\label{eq:expectwee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{align}
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\end{subequations}
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where
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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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\begin{multline}
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{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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it comes naturally that
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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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\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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\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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\end{equation}
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which therefore 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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\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 and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set.
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Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}
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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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\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}.
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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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\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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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we associate the effective interaction to a long-range interaction characterized by a range-separation function $\rsmu{}{}(\br{})$.
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\manu{Working on that paragraph}
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As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses a smooth bounded two-electron interactions.
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In pracice, to be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$, we use the functionals developed in the field of RS-DFT thanks to we associate the effective interaction to a long-range interaction characterized by a range-separation function $\rsmu{}{}(\br{})$.
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Although this choice is not unique, the long-range interaction we have chosen is
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\begin{equation}
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\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{}{}(\br{2}) r_{12}]}{ r_{12}} }.
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