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Manuscript/Ex-srDFT.tex
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\newcommand{\mc}{\multicolumn} \newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext} \newcommand{\fnt}{\footnotetext}
\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\mr}{\multirow} \newcommand{\mr}{\multirow}
\newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\br}{\mathbf{r}}
% energies % energies
\newcommand{\EHF}{E_\text{HF}} \newcommand{\EHF}{E_\text{HF}}
\newcommand{\Ec}{E_\text{c}} \newcommand{\Ec}{E_\text{c}}
@ -33,8 +35,10 @@
\newcommand{\EDMC}{E_\text{DMC}} \newcommand{\EDMC}{E_\text{DMC}}
\newcommand{\EexFCI}{E_\text{exFCI}} \newcommand{\EexFCI}{E_\text{exFCI}}
\newcommand{\EexDMC}{E_\text{exDMC}} \newcommand{\EexDMC}{E_\text{exDMC}}
\newcommand{\Ead}{\Delta E_\text{ad}}
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} \newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
\newcommand{\ra}{\rightarrow}
% units % units
\newcommand{\IneV}[1]{#1 eV} \newcommand{\IneV}[1]{#1 eV}
@ -45,98 +49,169 @@
\newcommand{\si}{\sigma} \newcommand{\si}{\sigma}
\newcommand{\sis}{\sigma^\star} \newcommand{\sis}{\sigma^\star}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
\begin{document} \begin{document}
\title{Prout} \title{Excitation Energies Near The Complete Basis Set Limit}
%\author{Pierre-Fran\c{c}ois Loos} \author{Emmanuel Giner}
%\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} \affiliation{\LCT}
%\affiliation{\LCPQ} \author{Anthony Scemama}
%\author{Anthony Scemama} \affiliation{\LCPQ}
%\affiliation{\LCPQ} \author{Julien Toulouse}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract} \begin{abstract}
By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., J.~Chem.~Phys.~149, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
\end{abstract} \end{abstract}
%\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
This problem was already noticed thirty years ago by Kutzelnigg \cite{Kutzelnigg_1985} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg_1991, Termath_1991, Klopper_1991a, Klopper_1991b, Noga_1994}
This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson_1996, Persson_1997, May_2004, Tenno_2004b, Tew_2005, May_2005}
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno_2012a, Tenno_2012b, Hattig_2012, Kong_2012}
For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew_2007b}
In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{Giner_2018}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%
%=======================
\subsection{Water}
\label{sec:H2O}
%=======================
%=======================
\subsection{Formaldehyde}
\label{sec:CH2O}
%=======================
%=======================
\subsection{Methylene}
\label{sec:CH2}
%=======================
%%% TABLE 1 %%% %%% TABLE 1 %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table*}
\caption{ \caption{
Total energies (in hartree) and adiabatic transition energies (in eV) of excited states of methylene for various methods and basis sets.} Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.}
\begin{ruledtabular}{} \begin{ruledtabular}{}
\begin{tabular}{lccdd} \begin{tabular}{llddddddd}
Method & Basis set & State & \mcc{Total energy (a.u.)} & \mcc{Excitation energy (eV)} \\ & & \mc{1}{c}{$1\,^{3}B_1$}
& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$}
& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}B_1$}
& \mc{2}{c}{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\
\cline{3-3} \cline{4-5}
\cline{6-7} \cline{8-9}
Method & Basis set & \tabc{$E$ (a.u.)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} \\
\hline \hline
CIPSI & AVDZ & $1\,^{3}B_1$ & -39.04846(1) & \\ exFCI & AVDZ & -39.04846(1)
& & $1\,^{1}A_1$ & -39.03225(1) & 0.441 \\ & -39.03225(1) & 0.441
& & $1\,^{1}B_1$ & -38.99203(1) & 1.536 \\ & -38.99203(1) & 1.536
& & $2\,^{1}A_1$ & -38.95076(1) & 2.659 \\ & -38.95076(1) & 2.659 \\
CIPSI & AVTZ & $1\,^{3}B_1$ & -39.08064(3) & \\ & AVTZ & -39.08064(3)
& & $1\,^{1}A_1$ & -39.06565(2) & 0.408 \\ & -39.06565(2) & 0.408
& & $1\,^{1}B_1$ & -39.02833(1) & 1.423 \\ & -39.02833(1) & 1.423
& & $2\,^{1}A_1$ & -38.98709(1) & 2.546 \\ & -38.98709(1) & 2.546 \\
CIPSI & AVQZ & $1\,^{3}B_1$ & -39.08854(1) & \\ & AVQZ & -39.08854(1)
& & $1\,^{1}A_1$ & -39.07402(2) & 0.395 \\ & -39.07402(2) & 0.395
& & $1\,^{1}B_1$ & -39.03711(1) & 1.399 \\ & -39.03711(1) & 1.399
& & $2\,^{1}A_1$ & -38.99607(1) & 2.516 \\ & -38.99607(1) & 2.516 \\
CIPSI & AV5Z & $1\,^{3}B_1$ & -39.09079(1) & \\ & AV5Z & -39.09079(1)
& & $1\,^{1}A_1$ & -39.07647(1) & 0.390 \\ & -39.07647(1) & 0.390
& & $1\,^{1}B_1$ & -39.03964(3) & 1.392 \\ & -39.03964(3) & 1.392
& & $2\,^{1}A_1$ & -38.99867(1) & 2.507 \\ & -38.99867(1) & 2.507 \\
CIPSI+srLDA & AVDZ & $1\,^{3}B_1$ & & \\ exFCI+srLDA & AVDZ &
& & $1\,^{1}A_1$ & & 0.347 \\ & & 0.347
& & $1\,^{1}B_1$ & & 1.431 \\ & & 1.431
& & $2\,^{1}A_1$ & & 2.590 \\ & & 2.590 \\
CIPSI+srLDA & AVTZ & $1\,^{3}B_1$ & & \\ & AVTZ &
& & $1\,^{1}A_1$ & & 0.360 \\ & & 0.360
& & $1\,^{1}B_1$ & & 1.377 \\ & & 1.377
& & $2\,^{1}A_1$ & & 2.513 \\ & & 2.513 \\
CIPSI+srLDA & AVQZ & $1\,^{3}B_1$ & & \\ & AVQZ &
& & $1\,^{1}A_1$ & & 0.371 \\ & & 0.371
& & $1\,^{1}B_1$ & & 1.376 \\ & & 1.376
& & $2\,^{1}A_1$ & & 2.498 \\ & & 2.498 \\
CIPSI+srPBE & AVDZ & $1\,^{3}B_1$ & & \\ exFCI+srPBE & AVDZ &
& & $1\,^{1}A_1$ & & 0.358 \\ & & 0.358
& & $1\,^{1}B_1$ & & 1.420 \\ & & 1.420
& & $2\,^{1}A_1$ & & 2.529 \\ & & 2.529 \\
CIPSI+srPBE & AVTZ & $1\,^{3}B_1$ & & \\ & AVTZ &
& & $1\,^{1}A_1$ & & 0.373 \\ & & 0.373
& & $1\,^{1}B_1$ & & 1.383 \\ & & 1.383
& & $2\,^{1}A_1$ & & 2.496 \\ & & 2.496 \\
CIPSI+srPBE & AVQZ & $1\,^{3}B_1$ & & \\ & AVQZ &
& & $1\,^{1}A_1$ & & 0.380 \\ & & 0.380
& & $1\,^{1}B_1$ & & 1.381 \\ & & 1.381
& & $2\,^{1}A_1$ & & 2.492 \\ & & 2.492 \\
SHCI & AVQZ & $1\,^{3}B_1$ & -39.08849(1) & \\ HBCI & AVQZ & -39.08849(1)
& & $1\,^{1}A_1$ & -39.07404(1) & 0.393 \\ & -39.07404(1) & 0.393
& & $1\,^{1}B_1$ & -39.03711(1) & 1.398 \\ & -39.03711(1) & 1.398
& & $2\,^{1}A_1$ & -38.99603(1) & 2.516 \\ & -38.99603(1) & 2.516 \\
CR-EOMCC (2,3)D& AVQZ & $1\,^{3}B_1$ & -39.08817 & \\ CR-EOMCC (2,3)D& AVQZ & -39.08817
& & $1\,^{1}A_1$ & -39.07303 & 0.412 \\ & -39.07303 & 0.412
& & $1\,^{1}B_1$ & -39.03450 & 1.460 \\ & -39.03450 & 1.460
& & $2\,^{1}A_1$ & -38.99457 & 2.547 \\ & -38.99457 & 2.547 \\
FCI & TZ2P & $1\,^{3}B_1$ & -39.066738 & \\ FCI & TZ2P & -39.066738
& & $1\,^{1}A_1$ & -39.048984 & 0.483 \\ & -39.048984 & 0.483
& & $1\,^{1}B_1$ & -39.010059 & 1.542 \\ & -39.010059 & 1.542
& & $2\,^{1}A_1$ & -38.968471 & 2.674 \\ & -38.968471 & 2.674 \\
DMC & & $1\,^{3}B_1$ & & \\ DMC & &
& & $1\,^{1}A_1$ & & 0.406 \\ & & 0.406
& & $1\,^{1}B_1$ & & 1.416 \\ & & 1.416
& & $2\,^{1}A_1$ & & 2.524 \\ & & 2.524 \\
Exp. & & $1\,^{3}B_1$ & & \\ Exp. & &
& & $1\,^{1}A_1$ & & 0.400 \\ & & 0.400
& & $1\,^{1}B_1$ & & 1.411 \\ & & 1.411
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table*}
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for geometries and additional information (including total energies).
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
This work was performed using HPC resources from
i) GENCI-TGCC (Grant No. 2018-A0040801738),
ii) CALMIP (Toulouse) under allocations 2018-0510 and 2018-12158.
\end{acknowledgements}
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