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Manuscript/G2-srDFT.tex
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@ -117,7 +117,7 @@
\begin{document}
\title{A Density-Based Basis Set Correction For Wave Function Theory: Application to Coupled Cluster}
\title{A Basis Set Correction For Wave Function Theory Based on Density Functional Theory: Application to Coupled Cluster}
\author{Bath\'elemy Pradines}
\affiliation{\LCT}
@ -178,7 +178,8 @@ Therefore, a number of approximate RS-DFT schemes have been developed using eith
%Therefore, a number of approximate RS-DFT schemes have been developed using either single-reference WFT approaches (such as M{\o}ller-Plesset perturbation theory\cite{AngGerSavTou-PRA-05}, coupled cluster\cite{GolWerSto-PCCP-05}, random-phase approximations\cite{TouGerJanSavAng-PRL-09,JanHenScu-JCP-09}) or multi-reference WFT approaches (such as multi-reference CI\cite{LeiStoWerSav-CPL-97}, multiconfiguration self-consistent field\cite{FroTouJen-JCP-07}, multi-reference perturbation theory\cite{FroCimJen-PRA-10}, density-matrix renormalization group\cite{HedKneKieJenRei-JCP-15}, selected CI\cite{FerGinTou-JCP-18}).
The present work proposes an extension of a recently proposed basis set correction scheme based on RS-DFT \cite{GinPraFerAssSavTou-JCP-18} together with the first numerical tests on molecular systems.
\manu{Very recently, a step forward has been performed by some of the present authors thanks to a density-based basis set correction which merges WFT and RS-DFT\cite{GinPraFerAssSavTou-JCP-18}. }
The present work proposes an extension of \manu{this new theory to CCSD(T)} together with the first numerical tests on molecular systems.
Unless otherwise stated, atomic units are used.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -283,7 +284,6 @@ Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{G
\end{equation}
for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
%An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
%As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems.
%=================================================================
%\subsection{Range-separation function}
@ -423,17 +423,18 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
%\subsection{Computational considerations}
%=================================================================
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.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is \manu{strongly} reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant, \manu{which is the case used all through this work}.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is \manu{strongly} reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant.
\manu{As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a single Slater determinant wave function to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems, and therefore we use this framework all through this work. }
%\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}.
In our current implementation, the \manu{computational} bottleneck \manu{of the basis set correction} 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}) or atomic-orbital-based algorithms 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 section, we point out that \manu{because of the definitions \eqref{eq:def_weebasis},\eqref{eq:mu_of_r} and the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}}, independently of the DFT functional, the present basis set correction
To conclude this section, we point out that \manu{because of the definitions \eqref{eq:def_weebasis}, \eqref{eq:mu_of_r} and the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}}, 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.
@ -502,8 +503,8 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
\titou{In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.}
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2-1 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
In a second time, we compute the entire correlation energies of the G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}).
In a second time, we compute the \trashMG{entire} correlation energies of the \manu{entire} G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) \manu{ and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}.
%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
As a method $\modY$ 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}
@ -519,7 +520,7 @@ Frozen-core calculations are defined as such: an \ce{He} core is frozen from \ce
In the context of the basis set correction, the set of spinorbitals $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals.
The FC density-based correction is set consistently when the FC approximation was applied in WFT methods.
In order to estimate the complete basis set (CBS) limit for each model, \manu{following the work of Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}},
we employ the two-point extrapolation for the correlation energies \manu{in quadruple- and quintuple-$\zeta$ basis sets, which is refered to as $\CBS$, and we add to these the HF energies in the largest basis sets, \textit{i.e.} in quintuple-$\zeta$ quality basis sets, to estimate the CBS FCI energies.}
we employ the two-point extrapolation for the correlation energies \manu{for each model in quadruple- and quintuple-$\zeta$ basis sets, which is refered to as $\CBS$ correlation energies, and we add the HF energies in the largest basis sets (\textit{i.e.} quintuple-$\zeta$ quality basis sets) to the CBS correlation energies to estimate the CBS FCI energies.}
%\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations are converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values, and they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
@ -552,7 +553,7 @@ With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
\titou{Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.}
\manu{Encouraged by these results for weakly correlated ground states molecules, ongoing development based on the same strategy point towards the correction of the basis set error for strongly correlated systems, excited states and the treatment of the one-electron error in the basis set incompleteness. }
\manu{Encouraged by these results for weakly correlated ground states molecules, we are developing this theory towards the treatment of the basis set error for strongly correlated systems, excited states and the treatment of the one-electron error in the basis set incompleteness. }
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