Major change of notations
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@ -189,7 +189,7 @@ 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 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 \alert{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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@ -204,19 +204,19 @@ where
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- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
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\end{equation}
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is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and a complete basis, respectively.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
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Both wave functions yield the same target density $\n{}{}$.
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%\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.}
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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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Importantly, in the limit of a complete basis set (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}{\infty} \approx E,
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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 $\E{\modX}{\infty}$ 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, 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 $\Bas$, respectively.
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where $\E{\modX}{}$ 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 $\E{\FCI}{} = \E{}{}$.
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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 \titou{FCI} energy and density within $\Bas$, respectively.
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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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@ -309,18 +309,16 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
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%\end{equation}
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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, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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for the lack of electronic cups in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the electron-electron (e-e)
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coalescence points.
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Therefore, the physical role of $\bE{}{\Bas}[\n{}{}]$ is to account for a universal condition of exact wave functions.
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Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions.
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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 approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth non divergent two-electron interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{}{}(\br{})$ which automatically adapts to quantify the incompleteness of a basis set $\Bas$ for each point in ${\rm I\!R}^3$.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which represents the effect of the projection in an incomplete basis set $\Bas$ of the Coulomb operator.
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We use a definition for which $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ 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 interaction 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 short-range density functionals\cite{TouGorSav-TCA-05} with $\rsmu{}{}(\br{})$ as the range separation.
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The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$.
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The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
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In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
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In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} alongside $\rsmu{\Bas}{}(\br{})$ as range separation.
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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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%Second, 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 \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}).
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@ -338,12 +336,12 @@ In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as
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We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \left\{
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\W{\Bas}{}(\br{1},\br{2}) = \left\{
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\begin{array}{ll}
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\f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
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\f{\Bas}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
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\,\,\,\,+\infty & \mbox{otherwise.}
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\end{array}
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\right.
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@ -354,58 +352,58 @@ where $\n{2}{\wf{}{\Bas}}(\br{1},\br{2})$ is the opposite-spin two-body density
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\n{2}{\wf{}{\Bas}}(\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 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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$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\Bas}{}(\br{1},\br{2})$ is defined as
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\begin{multline}
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\br{1},\br{2})
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\f{\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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The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction.
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With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\Bas}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction.
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (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{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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where here $\hWee{}$ contains only the opposite-spins 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{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\iint r_{12}^{-1} \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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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 in the present context is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
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which intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ 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 in the present context is $\W{\Bas}{}(\br{})$ 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}}{}(\br{},{\br{}})
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\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}})
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\end{equation}
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and which is necessarily \textit{finite} at for an \textit{incomplete} basis set as long as the on-top two-body density is non vanishing.
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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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Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ 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}}{}(\br{1},\br{2}) = r_{12}^{-1}\
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\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
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\end{equation}
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for all points $(\br{1},\br{2})$ such that $\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
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An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can quantify 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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As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems.
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An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify 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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As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems.
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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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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 the so-called long-range interaction which are smooth bounded two-electron operators.
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To do so, we choose a range-separation \textit{function} $\rsmu{\wf{}{\Bas}}{}(\br{})$
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To do so, we choose a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$
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\begin{equation}
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\label{eq:mu_of_r}
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\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
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\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{})
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\end{equation}
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such that the long-range interaction $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2})$
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such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
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\begin{equation}
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\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{2}) r_{12}]}{ r_{12}} }
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\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
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\end{equation}
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coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all points in ${\rm I\!R}^3$
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coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$
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\begin{equation}
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\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{},\br{}) = \W{\wf{}{\Bas}}{}(\br{})\quad \forall \,\, \br{}.
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\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}).
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\end{equation}
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@ -417,7 +415,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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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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% \wbasiscoal{} = \w{}{\lr,\rsmu{\Bas}{}}(\bfrb{},\bfrb{})
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%\end{equation}
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%where the long-range-like interaction is defined as
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%\begin{equation}
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@ -426,7 +424,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
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%\begin{equation}
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% \label{eq:mu_of_r}
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% \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
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% \rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{})
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%\end{equation}
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%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
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%\begin{equation}
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@ -444,7 +442,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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%and therefore
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%\begin{equation}
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%\label{eq:lim_mur}
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% \lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
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% \lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty
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%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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%\end{equation}
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@ -455,7 +453,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
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%=================================================================
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%\label{sec:ecmd}
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Once defined the range-separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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Once defined the range-separation function $\rsmu{\Bas}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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@ -486,14 +484,14 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}].
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Indeed, provided that $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) \approx \W{\wf{}{\Bas}}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\wf{}{\Bas}}{}(\br{})}$ coincides with $\wf{}{\Bas}$.
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}(\br{})}$ coincides with $\wf{}{\Bas}$.
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%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
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%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
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%--------------------------------------------
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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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 $\rsmu{\wf{}{\Bas}}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
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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 $\rsmu{\Bas}{}(\br{})$. 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{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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@ -522,7 +520,7 @@ Therefore, the PBE complementary functional reads
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
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\end{equation}
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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%The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
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%Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
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|
@ -601,7 +599,7 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
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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 $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
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Regarding now the main computational source of the present approach, it consists in the evaluation
|
||||
of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
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of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
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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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|
@ -615,7 +613,7 @@ When the four-index transformation become prohibitive, by performing successive
|
|||
|
||||
To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any WFT model that provides an energy and a density, ii) it vanishes for one-electron systems,
|
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iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model.
|
||||
%, 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 approximated functional} used for the DFT part.
|
||||
%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\Bas}{}(\br{})$ 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 approximated functional} used for the DFT part.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results}
|
||||
|
|
|
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