size consistency
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@ -331,109 +331,59 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
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\end{equation}
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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.
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\subsection{Intensivity 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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\subsection{Intensivity of the on-top pair density and of the local range-separation parameter}
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The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as
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\begin{equation}
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\label{eq:def_n2}
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n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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\end{equation}
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with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. 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
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\begin{equation}
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\label{eq:def_n2}
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n_{2{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}),
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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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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 $B$
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where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
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\begin{equation}
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\ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}.
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\label{eq:def_n2}
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n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}.
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\end{equation}
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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
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This shows that the on-top pair density is a local intensive quantity.
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The local range-separation parameter is defined by
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\begin{equation}
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\begin{aligned}
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\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})
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\end{aligned}
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\label{eq:def_murAnnex}
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\mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})},
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\end{equation}
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with
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\begin{equation}
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\begin{aligned}
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\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}} ,
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\end{aligned}
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\end{equation}
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(and equivalently for $B$),
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%\begin{equation}
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% \begin{aligned}
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% \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}},
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% \end{aligned}
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%\end{equation}
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and
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\begin{equation}
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\begin{aligned}
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\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) .
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\end{aligned}
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\end{equation}
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Therefore, one can express the two-body density as
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\begin{equation}
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\twodmrdiagpsitot = \twodmrdiagpsiaa + \twodmrdiagpsibb + \twodmrdiagpsiab
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\end{equation}
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where $\twodmrdiagpsiaa$ and $\twodmrdiagpsibb$ are the two-body densities of the isolated fragments
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\begin{equation}
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\begin{aligned}
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& \twodmrdiagpsiaa = \bra{\wf{A}{}} \hat{\Gamma}_{AA}(\br{1},\br{2}) \ket{\wf{A}{}}
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\end{aligned}
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\end{equation}
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(and equivalently for $B$),
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and $\twodmrdiagpsiab$ is simply the product of the one body densities of the sub systems
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\begin{equation}
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\begin{aligned}
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& \twodmrdiagpsiab = n_{A}(\br{1}) n_B(\br{2}) + n_{B}(\br{1}) n_A(\br{2}),
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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& 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}{} ,
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\end{aligned}
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\end{equation}
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(and equivalently for $B$).
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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
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\begin{equation}
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\begin{aligned}
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n_{2,\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
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\end{aligned}
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\end{equation}
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As $n_{2,\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $n_{2,\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.
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\subsection{Property of the local-range separation parameter}
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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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where
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\begin{equation}
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\label{eq:def_f}
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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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f(\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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Therefore, one can rewrite eq. \eqref{eq:def_f} as
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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
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\begin{equation}
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\label{eq:def_fa+b}
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f_{\wf{A+B}{}}(\bfr{},\bfr{}) = f_{\wf{AA}{}}(\bfr{},\bfr{}) + f_{\wf{BB}{}}(\bfr{},\bfr{}),
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f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}),
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\end{equation}
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with
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\begin{equation}
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\begin{aligned}
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\label{eq:def_faa}
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& 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}{ },
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\end{aligned}
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\label{eq:def_fX}
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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}{ }.
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\end{equation}
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(and equivalently for $B$).
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%\begin{equation}
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% \begin{aligned}
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% \label{eq:def_faa}
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% & 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}{ }.
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% \end{aligned}
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%\end{equation}
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As a consequence, the local range-separation parameter in the super system $A+B$
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So, $f(\bfr{},\bfr{})$ is a local intensive quantity.
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As a consequence, the local range-separation parameter of the supersystem $\text{A}+\text{B}$ is
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\begin{equation}
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\label{eq:def_mur}
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\murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{n_{2,\wf{A+B}{}}(\br{})}
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\label{eq:def_murAB}
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\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{})},
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\end{equation}
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which, in the case of a multiplicative wave function is nothing but
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which gives
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\begin{equation}
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\label{eq:def_mur}
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\murpsi = \murpsia + \murpsib.
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\label{eq:def_murABsum}
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\mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}),
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\end{equation}
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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.
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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.
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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.
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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 $n_{2,\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
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