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% second quantized operators
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\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
\newcommand{\ai}[1]{\hat{a}_{#1}}
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\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}

\newcommand{\CBS}{\text{CBS}}                                                                                             


%operators 
\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}

%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
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% 

	
% energies
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EsCI}{E_\text{sCI}}
\newcommand{\EDMC}{E_\text{DMC}}
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%pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
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%pbeuegxiCAS
\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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%pbeontxiCAS
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%pbeontXiCAS
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%pbeont0xiCAS
\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta}
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% numbers
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\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4}  }



% effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})}
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\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
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\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
 \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
 \newcommand{\twodmrpsi}[0]{ n^{2,\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
 \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
 \newcommand{\twodmrdiagpsitot}[0]{ n_{2,\wf{}{A+B}}(\rr{1}{2})}
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 \newcommand{\twodmrdiagpsibb}[0]{ n_{2,\wf{}{BB}}(\rr{1}{2})}
 \newcommand{\twodmrdiagpsibbd}[0]{ n_{2,\wf{}{BB}}(\rr{}{})}
 \newcommand{\twodmrdiagpsiab}[0]{ n_{2\wf{}{AB}}(\rr{1}{2})}
 \newcommand{\twodmrdiagpsival}[0]{ n_{2\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
 \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
 \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
 \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}



\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
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% MODEL 
\newcommand{\model}[0]{\mathcal{Y}}

% densities 
\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
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% wave functions
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% operators
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% units
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\newcommand{\InAA}[1]{#1 \AA}


% methods
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% basis sets 
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% operators
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% coordinates
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\newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}

\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}

\begin{document}	

\title{A density-based basis-set correction for weak and strong correlation}                                                                                           


\begin{abstract}

\end{abstract}

\maketitle


\section{Size consistency of the basis-set correction}

\subsection{Sufficient condition for size consistency}

The basis-set correction is expressed as an integral in real space
\begin{equation}
 \label{eq:def_ecmdpbebasis}
 \begin{aligned}
 & \efuncdenpbe{\argebasis} = \\ & \int  \text{d}\br{} \,\denr   \ecmd(\argrebasis),
 \end{aligned}
\end{equation}
where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$
\begin{equation}
 \label{eq:def_ecmdpbebasis}
 \begin{aligned}
 & \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}}  \text{d}\br{} \,\denr   \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}}  \text{d}\br{} \,\denr   \ecmd(\argrebasis).
 \end{aligned}
\end{equation}
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have
\begin{subequations}
\begin{equation}
n_\text{A+B}(\br{}) = n_\text{X}(\br{}),
\label{nAB}
\end{equation}
\begin{equation}
\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}),
\label{zAB}
\end{equation}
\begin{equation}
s_\text{A+B}(\br{}) = s_\text{X}(\br{}),
\label{sAB}
\end{equation}
\begin{equation}
n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}),
\label{n2AB}
\end{equation}
\begin{equation}
\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}),
\label{muAB}
\end{equation}
\end{subequations}
where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfil in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e.
\begin{equation}
 \ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}}, 
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicity consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, i.e. by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragement. This applies to the systems treated in this work.

\subsection{Intensivity of the on-top pair density and of the local range-separation parameter}

The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as
\begin{equation}
\label{eq:def_n2}
 n_{2{}}(\br{})  = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs}  \SO{r}{} \SO{s}{}, 
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then partitioned into orbitals localized on the fragment A and orbitals localized on B, i.e. $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
 \label{eq:def_n2}
 n_{2,\text{A}+\text{B}}(\br{})  = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}),
\end{equation}
where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
\begin{equation}
 \label{eq:def_n2}
 n_{2,\text{X}}(\br{})  = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs}  \SO{r}{} \SO{s}{}. 
\end{equation}
This shows that the on-top pair density is a local intensive quantity.

The local range-separation parameter is defined by 
\begin{equation}
 \label{eq:def_murAnnex}
 \mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})},
\end{equation}
where 
\begin{equation}
 \label{eq:def_f}
    f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both $A$ and $B$ vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
 \label{eq:def_fa+b}
    f_{\text{A}+\text{B}}(\bfr{},\bfr{}) =  f_{\text{A}}(\bfr{},\bfr{})  +  f_{\text{B}}(\bfr{},\bfr{}),
\end{equation}
with 
\begin{equation}
 \label{eq:def_fX}
    f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
So, $f(\bfr{},\bfr{})$ is a local intensive quantity.
As a consequence, the local range-separation parameter of the supersystem $\text{A}+\text{B}$ is
\begin{equation}
 \label{eq:def_murAB}
 \mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{})  +  f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})},
\end{equation}
which gives
\begin{equation}
 \label{eq:def_murABsum}
 \mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}),
\end{equation}
with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation parameter is thus a local intensive quantity.

In conclusion, if the wave function of the supersystem  $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent.

\section{Computational considerations}
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 $n_{2,\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively. 
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time. 
\subsection{Computation of the on-top pair density for a CASSCF wave function}
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 $\left(n_{\Bas}\right)^4$. 
Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen. 
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
\begin{equation}
 \label{def_n2_good}
 n_{2,\wf{\Bas}{}}(\br{})  = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
\end{equation}
where 
\begin{equation}
 \label{def_n2_act}
 n_{2,\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{})
\end{equation} 
is the purely active part of the on-top pair density, 
\begin{equation}
 n_{\mathcal{C}}(\br{}) = \sum_{i\, \in \mathcal{C}} \left(\phi_i (\br{}) \right)^2, 
\end{equation}
and 
\begin{equation}
 n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{})  \phi_u (\br{}) 
 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow}  +   \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}} 
\end{equation}
is the purely active one-body density. 
Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ 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. 
\subsection{Computation of $\murpsibas$}
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. 

One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as 
\begin{equation}
 \label{eq:f_good}
 f_{\wf{}{}}(\bfr{},\bfr{})  =  \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
\end{equation}
where 
\begin{equation}
 \mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
\end{equation}
and 
\begin{equation}
 \mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
\end{equation}
\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 $\left(n_{\Bas}\right)^4 n_G$. 

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 
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 $\left(n_{\Bas}\right)^2$. 
As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as 
\begin{equation}
 f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}). 
\end{equation}
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 $\left(n_{\Bas}\right)^4 n_G$. 
\bibliography{../srDFT_SC}

\end{document}
