srDFT_Ex/Manuscript/Ex-srDFT.tex

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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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% energies
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% units
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\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}
\title{Excitation Energies Near The Complete Basis Set Limit}
\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}
\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}
\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 %%%
\begin{squeezetable}
\begin{table*}
\caption{
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{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)} \\
\hline
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 \\
& CBS & -39.09111
& -39.07682 & 0.389
& -39.04000 & 1.391
& -38.99904 & 2.505 \\
\\
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)
& -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
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%%
%%% TABLE 2 %%%
\begin{squeezetable}
\begin{table*}
\caption{
Vertical absorption energies $\Eabs$ (in eV) of excited states of water and ammonia for various methods and basis sets.}
\begin{ruledtabular}{}
\begin{tabular}{llddddddddddddd}
& & & \mc{12}{c}{Deviation with respect to TBE}
\\
\cline{4-15}
& & & \mc{3}{c}{exFCI}
& \mc{3}{c}{exFCI+PBEot}
& \mc{3}{c}{exFCI+PBE}
& \mc{3}{c}{exFCI+LDA}
\\
\cline{4-6} \cline{7-9} \cline{10-12} \cline{13-15}
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}
\\
\hline
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
& -0.03 & 0.00 &
& -0.07 & 0.02 &
& -0.07 & -0.01 &
\\
\\
Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi(\text{CT})$ & 7.86 & -0.04 & -0.02 & 0.02
& & &
& & &
& & &
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%%
%%% TABLE 3 %%%
\begin{squeezetable}
\begin{table*}
\caption{
Vertical absorption energies $\Eabs$ (in eV) of excited states of ethylene and formaldehyde for various methods and basis sets.}
\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
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
& &
& &
& &
\\
\\
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & 7.43 & -0.12 & -0.04
& -0.05 & -0.01
& -0.04 & -0.01
& -0.02 & 0.00
\\
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & 7.92 & 0.01 & 0.01
& 0.00 & 0.00
& 0.06 & 0.03
& 0.06 & 0.03
\\
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & 8.10 & -0.1 & -0.02
& -0.03 & 0.00
& -0.02 & 0.00
& 0.00 & 0.01
\\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & 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
\\
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & 3.58 & 0.00 & 0.00
& 0.09 & 0.05
& 0.11 & 0.06
& 0.07 & 0.04
\\
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 6.07 & 0.03 & 0.01
& 0.13 & 0.04
& 0.15 & 0.05
& 0.11 & 0.04
\\
& $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & 7.14 & -0.19 & -0.08
& 0.01 & 0.01
& 0.02 & 0.01
& -0.01 & 0.00
\\
& $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & 7.96 & -0.09 & -0.02
& 0.13 & 0.08
& 0.14 & 0.08
& 0.10 & 0.07
\\
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 8.15 & -0.14 & -0.05
& 0.07 & 0.05
& 0.07 & 0.04
& 0.04 & 0.04
\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%%
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\section{Conclusion}
\label{sec:ccl}
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%%%%%%%%%%%%%%%%%%%%%%%%
\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}