changes in Appendix B

Manuscript/srDFT_SC.tex

@@ -570,7 +570,7 @@ An alternative way to eliminate the $S_z$ dependence is to simply set $\zeta=0$,
 
 \subsubsection{Size consistency}
 
-Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket{\Psi_{\ce{A}}^{\basis}} \otimes \ket{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{sizeconsistency} for a detailed proof and discussion of the latter statement. 
+Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket{\Psi_{\ce{A}}^{\basis}} \otimes \ket{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{app:sizeconsistency} for a detailed proof and discussion of the latter statement. 
 In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.
 
 
@@ -713,7 +713,7 @@ Regarding the complementary functional, we first perform full-valence CASSCF cal
 
 Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}. 
 
-Regarding the computational cost of the present approach, it should be stressed (see Appendix~\ref{computational} for additional details) that the basis-set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. 
+Regarding the computational cost of the present approach, it should be stressed (see Appendix~\ref{app:computational} for additional details) that the basis-set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. 
 %We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -823,7 +823,7 @@ Also, it is shown that the basis-set correction gives substantial differential c
 \appendix
 
 \section{Size consistency of the basis-set correction}
-\label{sizeconsistency}
+\label{app:sizeconsistency}
 
 \subsection{Sufficient condition for size consistency}
 
@@ -921,8 +921,8 @@ with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\
 
 We can therefore conclude that, 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{Computation cost of the basis-set correction for a CASSCF wave function}
+\section{Computational cost of the basis-set correction for a CASSCF wave function}
-\label{computational}
+\label{app:computational}
 
 The computational cost of the basis-set correction is determined by the calculation of the on-top pair density $n_{2}(\br{})$ and the local range-separation function $\mur$ on the real-space grid. For a general multideterminant wave function, the computational cost is of order $O(N_\text{grid}N_{\Bas}^4)$ where $N_\text{grid}$ is the number of grid points and $N_{\Bas}$ is the number of basis functions.\cite{LooPraSceTouGin-JCPL-19} For a CASSCF wave function, a significant reduction of the scaling of the computational cost can be achieved.
 
@@ -951,7 +951,7 @@ The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ o
 
 \subsection{Computation of the local range-separation function}
 
-In addition to the on-top pair density, the computation of $\mur$ needs the computation of $f(\bfr{},\bfr{})$ [Eq.~\eqref{eq:def_f}] at each grid point. It can be factorized as 
+In addition to the on-top pair density, the computation of $\mur$ needs the computation of $f(\bfr{},\bfr{})$ [see Eq.~\eqref{eq:def_f}] at each grid point. It can be factorized as 
 \begin{equation}
  \label{eq:f_good}
  f(\bfr{},\bfr{})  =  \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}),