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Pierre-Francois Loos 2019-05-28 15:02:19 +02:00
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Manuscript/Ex-srDFT.tex
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@ -153,7 +153,7 @@
\begin{document}
\title{Chemically-Accurate Excitation Energies With a Small Basis Set}
\title{Chemically-Accurate Excitation Energies With Small Basis Sets}
\author{Emmanuel Giner}
\affiliation{\LCT}
@ -189,6 +189,8 @@ This density-based correction relies on short-range correlation density function
RS-DFT combines rigorously density-functional theory (DFT) and wave function theory (WFT) via a decomposition of the electron-electron interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the convergence of these methods with respect to the size of the basis set is significantly improved. \cite{FraMusLupTou-JCP-15}
The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular electronically-excited systems which is a much tougher test for the reasons mentioned above.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -273,7 +275,7 @@ and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{
\f{}{\Bas}(\br{1},\br{2})
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals.
We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -314,11 +316,8 @@ This computationally-lighter functional will be refer to as PBE.
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
Here, we compute the ground and excited state energies, one-electron and on-top densities with a selected CI methods known as as CIPSI. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
exFCI stands for extrapolated FCI energies computed by increasing the number of determinants in the CI expansion.
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.
The one-electron density and on-top density is computed from a very large CIPSI expansion containing several million determinants.
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
@ -329,10 +328,10 @@ Frozen-core calculations are systematically performed and defined as such: a \ce
The FC density-based correction is used consistently with the FC approximation in WFT methods.
We refer the interested reader to Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for detailed explanations on how the previous equations have to be modified within the FC approximation.
The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO 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.
%The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
%In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
%In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO 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.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}