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% energies
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\newcommand { \Ead } { \Delta E_ \text { ad} }
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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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\newcommand { \LCT } { Laboratoire de Chimie Th\' eorique, Universit\' e Pierre et Marie Curie, Sorbonne Universit\' e, CNRS, Paris, 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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& CBS & -39.09111
& -39.07682 & 0.389
& -39.04000 & 1.391
& -38.99904 & 2.505 \\
\\
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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 \\
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\\
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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 \\
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\\
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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 \\
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\\
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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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%%% TABLE 2 %%%
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\begin { squeezetable}
\begin { table*}
\caption {
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Vertical absorption energies $ \Eabs $ (in eV) of excited states of water and ammonia for various methods and basis sets.}
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\begin { ruledtabular} { }
\begin { tabular} { llddddddddddddd}
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& & & \mc { 12} { c} { Deviation with respect to TBE}
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\\
\cline { 4-15}
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& & & \mc { 3} { c} { exFCI}
& \mc { 3} { c} { exFCI+PBEot}
& \mc { 3} { c} { exFCI+PBE}
& \mc { 3} { c} { exFCI+LDA}
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\\
\cline { 4-6} \cline { 7-9} \cline { 10-12} \cline { 13-15}
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Molecule & Transition & \tabc { TBE} & \tabc { AVDZ} & \tabc { AVTZ} & \tabc { AVQZ}
& \tabc { AVDZ} & \tabc { AVTZ} & \tabc { AVQZ}
& \tabc { AVDZ} & \tabc { AVTZ} & \tabc { AVQZ}
& \tabc { AVDZ} & \tabc { AVTZ} & \tabc { AVQZ}
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\\
\hline
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Water & $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 1 } B _ 1 $ & 7.70 & -0.17 & -0.07 & -0.02
& 0.01 & 0.00 & 0.02
& -0.02 & -0.01 & 0.00
& -0.04 & -0.01 & 0.01
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 1 } A _ 2 $ & 9.47 & -0.15 & -0.06 & -0.01
& 0.03 & 0.01 & 0.03
& 0.00 & 0.00 & 0.02
& -0.03 & 0.00 & 0.00
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 2 \, ^ { 1 } A _ 1 $ & 9.97 & -0.03 & 0.02 & 0.06
& 0.13 & 0.08 & 0.09
& 0.10 & 0.07 & 0.08
& 0.09 & 0.07 & 0.03
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 3 } B _ 1 $ & 7.33 & -0.19 & -0.08 & -0.03
& 0.02 & 0.00 & 0.02
& 0.05 & 0.01 & 0.02
& 0.00 & 0.00 & 0.04
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 3 } A _ 2 $ & 9.30 & -0.16 & -0.06 & -0.01
& 0.04 & 0.02 & 0.04
& 0.07 & 0.03 & 0.04
& 0.03 & 0.03 & 0.04
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 3 } A _ 1 $ & 9.59 & -0.11 & -0.05 & -0.01
& 0.07 & 0.02 & 0.03
& 0.09 & 0.03 & 0.03
& 0.06 & 0.03 & 0.04
\\
\\
Hydrogen sulfide & $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 1 } A _ 2 $ & 6.10 & 0.19 & 0.08 & 0.05
& & &
& & &
& & &
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 1 } B _ 1 $ & 6.29 & -0.19 & -0.05 & 0.00
& & &
& & &
& & &
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 3 } A _ 2 $ & 5.74 & 0.16 & 0.07 & 0.05
& & &
& & &
& & &
\\
& $ 1 \, ^ { 1 } A _ 1 \ra 1 \, ^ { 3 } B _ 1 $ & 5.94 & -0.19 & -0.05 & -0.01
& & &
& & &
& & &
\\
\\
Ammonia & $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 1 } A _ { 2 } $ & 6.66 & -0.18 & -0.07 & -0.02
& -0.04 & -0.02 & 0.00
& -0.07 & -0.03 & 0.00
& -0.07 & -0.03 & 0.00
\\
& $ 1 \, ^ { 1 } A _ { 1 } \ra 2 \, ^ { 1 } A _ { 1 } $ & 8.65 & 1.03 & 0.68 & 0.49
& 1.17 & 0.73 & 0.75
& 1.13 & 0.72 & 0.74
& 1.13 & 0.71 & 0.78
\\
& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 3 } A _ { 2 } $ & 6.37 & -0.18 & -0.06 & -0.02
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& -0.03 & 0.00 &
& -0.07 & 0.02 &
& -0.07 & -0.01 &
\\
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\\
Hydrogen chloride& $ { } ^ 1 \Sigma \ra { } ^ 1 \Pi ( \text { CT } ) $ & 7.86 & -0.04 & -0.02 & 0.02
& & &
& & &
& & &
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\end { tabular}
\end { ruledtabular}
\end { table*}
\end { squeezetable}
%%% %%% %%%
%%% TABLE 3 %%%
\begin { squeezetable}
\begin { table*}
\caption {
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Vertical absorption energies $ \Eabs $ (in eV) of excited states of ethylene and formaldehyde for various methods and basis sets.}
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\begin { ruledtabular} { }
\begin { tabular} { llddddddddd}
& & & \mc { 8} { c} { Deviation with respect to TBE}
\\
\cline { 4-11}
& & & \mc { 2} { c} { exFCI}
& \mc { 2} { c} { exFCI+PBEot}
& \mc { 2} { c} { exFCI+PBE}
& \mc { 2} { c} { exFCI+LDA}
\\
\cline { 4-5} \cline { 6-7} \cline { 8-9} \cline { 10-11}
Molecule & Transition & \tabc { TBE} & \tabc { AVDZ} & \tabc { AVTZ}
& \tabc { AVDZ} & \tabc { AVTZ}
& \tabc { AVDZ} & \tabc { AVTZ}
& \tabc { AVDZ} & \tabc { AVTZ}
\\
\hline
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Acetylene & $ 1 \, ^ { 1 } \Sigma _ { g } ^ { + } \ra 1 \, ^ { 1 } \Sigma _ { u } ^ { - } $ & 7.10 & 0.10 & 0.00
& &
& &
& &
\\
& $ 1 \, ^ { 1 } \Sigma _ { g } ^ { + } \ra 1 \, ^ { 1 } \Delta _ { u } $ & 7.44 & 0.07 & 0.00
& &
& &
& &
\\
& $ 1 \, ^ { 1 } \Sigma _ { g } ^ { + } \ra 1 \, ^ { 3 } \Sigma _ { u } ^ { + } $ & 5.56 & -0.06 & -0.03
& &
& &
& &
\\
& $ 1 \, ^ { 1 } \Sigma _ { g } ^ { + } \ra 1 \, ^ { 3 } \Delta _ { u } $ & 6.40 & 0.06 & 0.00
& &
& &
& &
\\
& $ 1 \, ^ { 1 } \Sigma _ { g } ^ { + } \ra 1 \, ^ { 3 } \Sigma _ { u } ^ { - } $ & 7.09 & 0.05 & -0.01
& &
& &
& &
\\
\\
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Ethylene & $ 1 \, ^ { 1 } A _ { 1 g } \ra 1 \, ^ { 1 } B _ { 3 u } $ & 7.43 & -0.12 & -0.04
& -0.05 & -0.01
& -0.04 & -0.01
& -0.02 & 0.00
\\
& $ 1 \, ^ { 1 } A _ { 1 g } \ra 1 \, ^ { 1 } B _ { 1 u } $ & 7.92 & 0.01 & 0.01
& 0.00 & 0.00
& 0.06 & 0.03
& 0.06 & 0.03
\\
& $ 1 \, ^ { 1 } A _ { 1 g } \ra 1 \, ^ { 1 } B _ { 1 g } $ & 8.10 & -0.1 & -0.02
& -0.03 & 0.00
& -0.02 & 0.00
& 0.00 & 0.01
\\
& $ 1 \, ^ { 1 } A _ { 1 g } \ra 1 \, ^ { 3 } B _ { 1 u } $ & 4.54 & 0.01 & 0.00
& 0.07 & 0.03
& 0.10 & 0.04
& 0.08 & 0.04
\\
\\
Formaldehyde& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 1 } A _ { 2 } $ & 3.97 & 0.02 & 0.01
& 0.05 & 0.02
& 0.03 & 0.02
& 0.02 & 0.01
\\
& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 1 } B _ { 2 } $ & 7.30 & -0.19 & -0.07
& 0.00 & 0.00
& -0.02 & 0.00
& -0.04 & 0.00
\\
& $ 1 \, ^ { 1 } A _ { 1 } \ra 2 \, ^ { 1 } B _ { 2 } $ & 8.14 & -0.10 & -0.01
& 0.09 & 0.07
& 0.08 & 0.06
& 0.05 & 0.06
\\
& $ 1 \, ^ { 1 } A _ { 1 } \ra 2 \, ^ { 1 } A _ { 1 } $ & 8.27 & -0.15 & -0.04
& 0.03 & 0.04
& 0.02 & 0.03
& 0.00 & 0.03
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\\
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& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 3 } A _ { 2 } $ & 3.58 & 0.00 & 0.00
& 0.09 & 0.05
& 0.11 & 0.06
& 0.07 & 0.04
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\\
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& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 3 } A _ { 1 } $ & 6.07 & 0.03 & 0.01
& 0.13 & 0.04
& 0.15 & 0.05
& 0.11 & 0.04
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\\
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& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 3 } B _ { 2 } $ & 7.14 & -0.19 & -0.08
& 0.01 & 0.01
& 0.02 & 0.01
& -0.01 & 0.00
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\\
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& $ 1 \, ^ { 1 } A _ { 1 } \ra 2 \, ^ { 3 } B _ { 2 } $ & 7.96 & -0.09 & -0.02
& 0.13 & 0.08
& 0.14 & 0.08
& 0.10 & 0.07
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\\
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& $ 1 \, ^ { 1 } A _ { 1 } \ra 1 \, ^ { 3 } A _ { 1 } $ & 8.15 & -0.14 & -0.05
& 0.07 & 0.05
& 0.07 & 0.04
& 0.04 & 0.04
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\\
\end { tabular}
\end { ruledtabular}
\end { table*}
\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}
