Theory almost done

Manuscript/G2-srDFT.bib

@@ -1,13 +1,24 @@
 %% This BibTeX bibliography file was created using BibDesk.
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-%% Created for Pierre-Francois Loos at 2019-04-11 14:20:29 +0200 
+%% Created for Pierre-Francois Loos at 2019-04-12 14:53:34 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Whi-JCP-73,
+	Author = {J. L. Whitten},
+	Date-Added = {2019-04-12 14:52:42 +0200},
+	Date-Modified = {2019-04-12 14:53:34 +0200},
+	Doi = {10.1063/1.1679012},
+	Journal = {J. Chem. Phys.},
+	Pages = {4496},
+	Title = {Coulombic potential energy integrals and approximations},
+	Volume = {58},
+	Year = {1973}}
+
 @article{BarLoo-JCP-17,
 	Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois},
 	Date-Added = {2019-04-11 14:20:15 +0200},

Manuscript/G2-srDFT.tex

@@ -420,23 +420,21 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
 %=================================================================
 %\subsection{Computational considerations}
 %=================================================================
-Regarding now the main computational source of the present approach, it consists in the evaluation 
+One of the most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
-of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
+This embarrassingly parallel step scales, in the general (multi-determinantal) case, as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant. 
-All through this paper, we use pair density matrix of a single Slater determinant (typically HF) 
+%\begin{equation}
-for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the evaluation 
+%	\label{eq:fcoal}
-at each quadrature grid point of 
+%	\f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ}  \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
-\begin{equation}
+%\end{equation}
-	\label{eq:fcoal}
+In our current implementation, the bottleneck is the four-index transformation to get the two-electron integrals in the molecular orbital basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. 
-	\f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ}  \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
+Nevertheless, this step usually has to be performed for most correlated WFT calculations. 
-\end{equation}
+Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) could be employed to significantly speed up this step.
-which scales as $\Nb^2\times \Ne^2 \times \Ng$ and is embarrassingly parallel. 
+%When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$  but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations.  
-Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation. 
-When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$  but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations.  
 
-To conclude this session, we point out that the present basis set correction has, independently of the DFT functional, the following properties: 
+To conclude this section, we point out that, independently of the DFT functional, the present basis set correction
-i) it can be applied to any WFT model that provides an energy and a density, 
+i) can be applied to any WFT model that provides an energy and a density, 
-ii) it does not correct one-electron systems, and
+ii) does not correct one-electron systems, and
-iii) it vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for the given WFT model. 
+iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}