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% second quantized operators
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\newcommand{\CBS}{\text{CBS}}                                                                                             


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% 

	
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\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{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}

\begin{abstract}
bla bla bla youpi tralala 
\end{abstract}

\maketitle

\section{Extensivity of the basis set correction}
\subsection{General considerations}
The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments. 
The present basis set correction being 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 quantities $\argrebasis$ are obtained from the same wave function $\Psi^{A+B}$. 
Such an integral can be rewritten as the sum of the contribution on $A$ and $B$
\begin{equation}
 \label{eq:def_ecmdpbebasis}
 \begin{aligned}
 & \efuncdenpbe{\argebasis} = \\ & \int_{ \br{} \in A}  \text{d}\br{} \,\denr   \ecmd(\argrebasis) \\ & + \int_{ \br{} \in B}  \text{d}\br{} \,\denr   \ecmd(\argrebasis),
 \end{aligned}
\end{equation}
Therefore, a sufficient condition to obtain size extensivity in the limit of dissociated fragments is that all arguments entering in the function $\ecmd(\argrebasis)$ are \textit{intensive}, which means that they \textit{locally} coincide in the system $A$ and in the sub system $A$ of the super system $A+B$. 
Regarding the density and its gradients, these are necessary intensive quantities. The remaining questions are therefore the local range-separation parameter $\murpsi$ and the on-top pair density. 


\subsection{Property of the on-top pair density}
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
\begin{equation}
 \ntwo_{\wf{}{}}(\br{})  = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, 
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. 
Assume now that the wave function $\wf{A+B}{}$  of the super system $A+B$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$ 
\begin{equation}
 \ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}. 
\end{equation}
Labelling the orbitals of fragment $A$ as $p_A,q_A,r_A,s_A$ and of fragment $B$ as $p_B,q_B,r_B,s_B$ and assuming that they don't overlap, one can split the two-body density operator as
\begin{equation}
 \begin{aligned}
 \hat{\Gamma}(\br{1},\br{2}) = \hat{\Gamma}_{AA}{}(\br{1},\br{2}) + \hat{\Gamma}_{BB}{}(\br{1},\br{2}) + \hat{\Gamma}_{AB}{}(\br{1},\br{2})
 \end{aligned}
\end{equation}
with 
\begin{equation}
 \begin{aligned}
 \hat{\Gamma}_{AA}(\br{1},\br{2}) = \sum_{p_A,q_A,r_A,s_A}&  \SO{r_A}{1} \SO{s_A}{2} \SO{p_A}{1} \SO{q_A}{2} \\ & \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}} , 
 \end{aligned}
\end{equation}
(and equivalently for $B$), 
%\begin{equation}
% \begin{aligned}
% \hat{\Gamma}_{BB} = \sum_{p_B,q_B,r_B,s_B} \aic{r_{B,\downarrow}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ai{p_{B,\downarrow}},  
% \end{aligned}
%\end{equation}
and 
\begin{equation}
 \begin{aligned}
 \hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} & \SO{r_A}{1} \SO{s_B}{2} \SO{p_A}{1} \SO{q_B}{2}  \\ & \left( \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} + \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \right) . 
 \end{aligned}
\end{equation}
Therefore, one can express the two-body density as 
\begin{equation}
 \twodmrdiagpsitot = \twodmrdiagpsiaa + \twodmrdiagpsibb + \twodmrdiagpsiab
\end{equation}
where $\twodmrdiagpsiaa$ and $\twodmrdiagpsibb$ are the two-body densities of the isolated fragments 
\begin{equation}
 \begin{aligned}
 & \twodmrdiagpsiaa = \bra{\wf{A}{}} \hat{\Gamma}_{AA}(\br{1},\br{2}) \ket{\wf{A}{}}
 \end{aligned}
\end{equation}
(and equivalently for $B$), 
and $\twodmrdiagpsiab$ is simply the product of the one body densities of the sub systems 
\begin{equation}
 \begin{aligned}
 & \twodmrdiagpsiab = n_{A}(\br{1}) n_B(\br{2}) + n_{B}(\br{1}) n_A(\br{2}), 
 \end{aligned}
\end{equation}
\begin{equation}
 \begin{aligned}
 & n_{A}(\br{}) = \sum_{p_A r_A} \SO{p_A}{}  \bra{\wf{A}{}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ket{\wf{A}{}} \SO{r_A}{} ,
 \end{aligned}
\end{equation}
(and equivalently for $B$).
As the densities of $A$ and $B$ are by definition non overlapping, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems 
\begin{equation}
 \begin{aligned}
 \ntwo_{\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
 \end{aligned}
\end{equation}
As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{  if }\br{} \in B$ (and equivalently for $\ntwo_{\wf{}{B/B}}(\br{}) $ on $A$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity. 
\subsection{Property of the local-range separation parameter}
The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator 
\begin{equation}
 \label{eq:def_f}
    f_{\wf{A+B}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
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. 
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. 
Therefore, one can rewrite eq. \eqref{eq:def_f} as
\begin{equation}
 \label{eq:def_fa+b}
    f_{\wf{A+B}{}}(\bfr{},\bfr{}) =  f_{\wf{AA}{}}(\bfr{},\bfr{})  +  f_{\wf{BB}{}}(\bfr{},\bfr{}),
\end{equation}
with 
\begin{equation}
\begin{aligned}
 \label{eq:def_faa}
    & f_{\wf{AA}{}}(\bfr{},\bfr{}) = \\ & \sum_{p_A q_A r_A s_A t_A u_A} \SO{p_A }{ } \SO{q_A}{ } \V{p_A q_A}{r_A s_A} \Gam{r_A s_A}{t_A u_A} \SO{t_A}{ } \SO{u_A}{ }, 
\end{aligned}
\end{equation}
(and equivalently for $B$). 
%\begin{equation}
% \begin{aligned}
% \label{eq:def_faa}
%   & f_{\wf{BB}{}}(\bfr{},\bfr{}) = \\ &\sum_{p_B q_B r_B s_B t_B u_B} \SO{p_B }{ } \SO{q_B}{ } \V{p_B q_B}{r_B s_B} \Gam{r_B s_B}{t_B u_B} \SO{t_B}{ } \SO{u_B}{ }. 
% \end{aligned}
%\end{equation}
As a consequence, the local range-separation parameter in the super system $A+B$ 
\begin{equation}
 \label{eq:def_mur}
 \murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{\ntwo_{\wf{A+B}{}}(\br{})}
\end{equation}
which, in the case of a multiplicative wave function is nothing but
\begin{equation}
 \label{eq:def_mur}
 \murpsi = \murpsia + \murpsib.  
\end{equation}
As $\murpsia = 0 \text{  if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent. 

\section{Computational considerations}

\bibliography{../srDFT_SC}

\end{document}