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\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\' e de Toulouse, CNRS, UPS, France}
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\begin { document}
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\title { Excitation Energies Near The Complete Basis Set Limit}
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\author { Emmanuel Giner}
\affiliation { \LCT }
\author { Anthony Scemama}
\affiliation { \LCPQ }
\author { Julien Toulouse}
\affiliation { \LCT }
\author { Pierre-Fran\c { c} ois Loos}
\email [Corresponding author: ] { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
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\begin { abstract}
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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.
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\end { abstract}
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\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}
%=======================
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%%% TABLE 1 %%%
\begin { squeezetable}
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\begin { table*}
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\caption {
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Total energies $ E $ (in hartree) and adiabatic transition energies $ \Ead $ (in eV) of excited states of methylene for various methods and basis sets.}
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\begin { ruledtabular} { }
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\begin { tabular} { llddddddd}
& & \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)} \\
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\hline
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exFCI & AVDZ & -39.04846(1)
& -39.03225(1) & 0.441
& -38.99203(1) & 1.536
& -38.95076(1) & 2.659 \\
& AVTZ & -39.08064(3)
& -39.06565(2) & 0.408
& -39.02833(1) & 1.423
& -38.98709(1) & 2.546 \\
& AVQZ & -39.08854(1)
& -39.07402(2) & 0.395
& -39.03711(1) & 1.399
& -38.99607(1) & 2.516 \\
& AV5Z & -39.09079(1)
& -39.07647(1) & 0.390
& -39.03964(3) & 1.392
& -38.99867(1) & 2.507 \\
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exFCI+LDA & AVDZ & -39.07450(1)
& -39.06213(1) & 0.337
& -39.02233(1) & 1.420
& -38.97946(1) & 2.586 \\
& AVTZ & -39.09099(3)
& -39.07779(2) & 0.359
& -39.04051(1) & 1.374
& -38.99859(1) & 2.514 \\
& AVQZ & -39.09319(1)
& -39.07959(2) & 0.370
& -39.04267(1) & 1.375
& -39.00135(1) & 2.499 \\
exFCI+PBE & AVDZ & -39.07282(1)
& -39.06150(1) & 0.308
& -39.02181(1) & 1.388
& -38.97873(1) & 2.560 \\
& AVTZ & -39.08948(3)
& -39.07639(2) & 0.356
& -39.03911(1) & 1.371
& -38.99724(1) & 2.510 \\
& AVQZ & -39.09247(1)
& -39.07885(2) & 0.371
& -39.04193(1) & 1.375
& -39.00066(1) & 2.498 \\
exFCI+PBEot & AVDZ & -39.06924(1)
& -39.05651(1) & 0.347
& -39.01777(1) & 1.401
& -38.97698(1) & 2.511 \\
& AVTZ & -39.08805(3)
& -39.07430(2) & 0.374
& -39.03742(1) & 1.378
& -38.99652(1) & 2.491 \\
& AVQZ & -39.09189(1)
& -39.07795(2) & 0.379
& -39.04124(1) & 1.378
& -39.00044(1) & 2.489 \\
SHCI & AVQZ & -39.08849(1)
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& -39.07404(1) & 0.393
& -39.03711(1) & 1.398
& -38.99603(1) & 2.516 \\
CR-EOMCC (2,3)D& AVQZ & -39.08817
& -39.07303 & 0.412
& -39.03450 & 1.460
& -38.99457 & 2.547 \\
FCI & TZ2P & -39.066738
& -39.048984 & 0.483
& -39.010059 & 1.542
& -38.968471 & 2.674 \\
DMC & &
& & 0.406
& & 1.416
& & 2.524 \\
Exp. & &
& & 0.400
& & 1.411
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\end { tabular}
\end { ruledtabular}
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\end { table*}
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\end { squeezetable}
%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%
\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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\bibliography { Ex-srDFT}
\end { document}