add sentence in appendix about spatial degeneracy

Revised_Manuscript/srDFT_SC.tex

@@ -837,7 +837,7 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
  \ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}}, 
 \label{PsiAB}
 \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 explicitly 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 $S_z$, \ie, 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 (\eg, coming from atomic $p$ states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work.
+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 explicitly 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 $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, \alert{for the systems treated in this work}, the lack of size consistency which could arise from spatial degeneracies (coming from atomic $p$ states) can also be avoided by selecting the \alert{same state} in the supersystem and in the isolated fragment. \alert{For example, for the F$_2$ molecule, the CASSCF wave function dissociates into the atomic configuration $\text{p}_\text{x}^2 \text{p}_\text{y}^2 \text{p}_\text{z}^1$ for each fragment, and we thus choose the same configuration for the calculation of the isolated atom. The same argument applies to the N$_2$ and O$_2$ molecules. For other systems, it may not be always possible to do so.}
 
 \subsection{Intensivity of the on-top pair density and the local range-separation function}