pouet
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Emmanuel Giner
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@ -273,10 +273,11 @@
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\begin{document}
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\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}
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\title{A density-based basis-set correction for weak and strong correlation}
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\begin{abstract}
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bla bla bla youpi tralala
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\end{abstract}
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\maketitle
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@ -306,6 +307,7 @@ Regarding the density and its gradients, these are necessary intensive quantitie
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\subsection{Property of the on-top pair density}
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A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as
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\begin{equation}
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\label{eq:def_n2}
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\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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\end{equation}
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with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
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@ -371,7 +373,7 @@ As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for
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The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator
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\begin{equation}
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\label{eq:def_f}
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f_{\wf{A+B}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
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f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
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\end{equation}
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As the summations run over all orbitals in the basis set $\Bas$, the quantity $f_{\wf{}{\Bas}}(\bfr{},\bfr{})$ is orbital invariant and therefore can be expressed in terms of localized orbitals.
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In the limit of dissociated fragments, the coulomb interaction is vanishing between $A$ and $B$ and therefore any two-electron integral involving orbitals on both the system $A$ and $B$ vanishes.
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@ -408,11 +410,11 @@ As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $
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\section{Computational considerations}
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The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $\ntwo_{\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
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Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications and if one can afford a larger memory storage, one can substantially reduce the CPU time.
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Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
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\subsection{Computation of the on-top pair density for a CASSCF wave function}
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Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$.
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Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen.
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If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $r,s,t,u$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
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If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
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\begin{equation}
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\label{def_n2_good}
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\ntwo_{\wf{\Bas}{}}(\br{}) = \ntwo_{\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
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@ -420,7 +422,7 @@ If the active space is referred as the set of spatial orbitals $\mathcal{A}$ whi
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where
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\begin{equation}
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\label{def_n2_act}
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\ntwo_{\mathcal{A}}(\br{}) = \sum_{r,s,t,u \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{t_\uparrow}\ai{u_\downarrow}}{\wf{}{\Bas}} \phi_r (\br{}) \phi_s (\br{}) \phi_t (\br{}) \phi_u (\br{})
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\ntwo_{\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
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\end{equation}
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is the purely active part of the on-top pair density,
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\begin{equation}
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@ -428,12 +430,36 @@ is the purely active part of the on-top pair density,
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\end{equation}
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and
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\begin{equation}
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n_{\mathcal{A}}(\br{}) = \sum_{r,s\, \in \mathcal{A}} \phi_r (\br{}) \phi_s (\br{})
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\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\ai{s_\downarrow} + \aic{r_\uparrow}\ai{s_\uparrow}}{\wf{}{\Bas}}
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n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{}) \phi_u (\br{})
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\mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}}
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\end{equation}
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is the purely active one-body density.
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Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $n_{\mathcal{A}}^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$.
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Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
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\subsection{Computation of $\murpsibas$}
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At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function.
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One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as
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\begin{equation}
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\label{eq:f_good}
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f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
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\end{equation}
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where
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\begin{equation}
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\mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
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\end{equation}
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and
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\begin{equation}
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\mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
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\end{equation}
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\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$.
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In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong
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to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$.
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As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as
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\begin{equation}
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f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}).
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\end{equation}
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Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$.
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\bibliography{../srDFT_SC}
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\end{document}
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@ -273,7 +273,7 @@
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\begin{document}
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\title{A density-based basis-set correction for strong correlation}
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\title{A density-based basis-set correction for weak and strong correlation}
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\author{Emmanuel Giner}
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\email{emmanuel.giner@lct.jussieu.fr}
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@ -696,7 +696,7 @@ Regarding the computational cost of the present approach, it should be stressed
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\hline
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\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\
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& aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
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& aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
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& aug-cc-pVTZ & 60.1 [ ] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
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\hline
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& & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\
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\hline
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