diff --git a/Manuscript/G2-srDFT.bib b/Manuscript/G2-srDFT.bib index f6580c4..81a4db5 100644 --- a/Manuscript/G2-srDFT.bib +++ b/Manuscript/G2-srDFT.bib @@ -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}, diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 6859815..c80b0a0 100644 --- a/Manuscript/G2-srDFT.tex +++ b/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 -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}