From 5d1afbdc56a3c727150f5744a8127e929eaec959 Mon Sep 17 00:00:00 2001
From: Julien Toulouse <julien.toulouse@upmc.fr>
Date: Mon, 20 Jan 2020 23:34:35 +0100
Subject: [PATCH] changes in SI

---
 Manuscript/SI/srDFT_SC-SI.tex | 24 ++++++------------------
 1 file changed, 6 insertions(+), 18 deletions(-)

diff --git a/Manuscript/SI/srDFT_SC-SI.tex b/Manuscript/SI/srDFT_SC-SI.tex
index a4d0d87..5c9ab54 100644
--- a/Manuscript/SI/srDFT_SC-SI.tex
+++ b/Manuscript/SI/srDFT_SC-SI.tex
@@ -286,16 +286,16 @@
 
 \section{Size consistency of the basis-set correction}
 
-\subsection{General considerations}
+\subsection{Sufficient condition for size consistency}
 
-The following paragraph gives a proof of the size consistency of the basis-set correction. The basis-set correction is expressed as an integral in real space
+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 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 $\Omega_\text{B}$
+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}
@@ -329,25 +329,13 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
 \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 well known that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. 
+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.
 
-For spatial degeneracies such as different p states, by selected the same member of the ensemble in the supersystem and the isolated fragements.
-
-applies to the systems treated in this work.
-
-Even though this does deal with the case of degeneracies, 
-
-To avoid difficulties arising from spin-multiplet degeneracy, we can use the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ depending only on $n(\br{})$ and $n_{2}(\br{})$ or a zero spin polarization $\zeta(\br{}) = 0$.
-
-
-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}
+\subsection{Intensivity 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}
  \label{eq:def_n2}
- n_{2,\wf{}{}}(\br{})  = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, 
+ n_{2{}}(\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 $B$