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still working on the notations

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Pierre-Francois Loos 2019-04-11 21:32:45 +02:00
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Manuscript/G2-srDFT.tex
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@ -602,11 +602,11 @@ Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the
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 two-body density matrix of a single Slater determinant (typically HF)
for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation
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_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
\f{\Bas}{\HF}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}
\end{equation}
which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly 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.
@ -719,7 +719,7 @@ This molecular set has been exhausively studied in the last 20 years (see, for e
As a method $\modX$ we employ either CCSD(T) or exFCI.
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the Restricted Open Shel Hartree-Fock (ROHF) one-electron density to compute the complementary energy, and for the CIPSI calculations we use the density of a converged variational wave function.
In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the Restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy, and for the CIPSI calculations we use the density of a converged variational wave function.
For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculations, and built with the natural orbitals of the converged variational wave function for the exFCI calculations.
The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}