changes in appendix

Manuscript/srDFT_SC.tex

@@ -924,7 +924,7 @@ with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\
 
 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.
 
-\section{Efficient computation of the basis-set correction for a CASSCF wave function}
+\section{Computation cost of the basis-set correction for a CASSCF wave function}
 \label{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 parameter $\mur$ 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.
@@ -954,30 +954,26 @@ The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ o
 
 \subsection{Computation of the local range-separation parameter}
 
-At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function. 
-
-One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as 
+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 
 \begin{equation}
  \label{eq:f_good}
- f_{\wf{}{}}(\bfr{},\bfr{})  =  \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
+ f(\bfr{},\bfr{})  =  \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}),
 \end{equation}
 where 
 \begin{equation}
- \mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
+ V^{rs}(\bfr{}) = \sum_{pq \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}),
 \end{equation}
 and 
 \begin{equation}
- \mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
+ \Gamma_{rs}(\bfr{}) = \sum_{pq \in \Bas} \Gam{rs}{pq} \phi_p(\br{}) \phi_q(\br{}) .
 \end{equation}
-\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$. 
+For a general multideterminant wave function, the computational cost of $f(\bfr{},\bfr{})$ thus scales as $O(N_\text{grid}N_{\Bas}^4)$.
 
-In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong 
-to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$. 
-As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as 
+In the case of a CASSCF wave function, $\Gam{rs}{pq}$ vanishes if one index $p,q,r,s$ does not belong to the set of inactive or active occupied orbitals $\mathcal{I}\cup \mathcal{A}$. Therefore, at a given grid point, the number of non-zero elements $\Gamma_{rs}(\bfr{})$ is only at most $(N_{\mathcal{I}}+N_{\mathcal{A}})^2$, which is usually much smaller than $N_{\Bas}^2$. As a consequence, one can also restrict the sum in the calculation of $f(\bfr{},\bfr{})$
 \begin{equation}
- f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}). 
+ f(\bfr{},\bfr{}) = \sum_{rs \in \mathcal{I}\cup\mathcal{A}} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}). 
 \end{equation}
-Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$. 
+The overall computational cost is dominated by that of $V^{rs}(\bfr{})$, which scales as $O(N_\text{grid}(N_{\mathcal{I}}+N_{\mathcal{A}})^2 N_{\Bas}^2)$, which is much smaller than $O(N_\text{grid}N_{\Bas}^4)$.
 
 \bibliography{srDFT_SC}