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Theory almost done

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Manuscript/G2-srDFT.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% 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},

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Manuscript/G2-srDFT.tex
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@ -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
of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
All through this paper, we use pair density matrix of a single Slater determinant (typically HF)
for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the evaluation
at each quadrature grid point of
\begin{equation}
\label{eq:fcoal}
\f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
\end{equation}
which scales as $\Nb^2\times \Ne^2 \times \Ng$ and is embarrassingly parallel.
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.
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.
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.
%\begin{equation}
% \label{eq:fcoal}
% \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
%\end{equation}
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}.
Nevertheless, this step usually has to be performed for most correlated WFT calculations.
Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) could be employed to significantly speed up this step.
%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:
i) it can be applied to any WFT model that provides an energy and a density,
ii) it 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.
To conclude this section, we point out that, independently of the DFT functional, the present basis set correction
i) can be applied to any WFT model that provides an energy and a density,
ii) does not correct one-electron systems, and
iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}