464 lines
15 KiB
TeX
464 lines
15 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
|
|
|
|
\usepackage{mathpazo,libertine}
|
|
\usepackage[normalem]{ulem}
|
|
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
|
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
|
\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
|
|
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
|
|
|
|
\usepackage{hyperref}
|
|
\hypersetup{
|
|
colorlinks=true,
|
|
linkcolor=blue,
|
|
filecolor=blue,
|
|
urlcolor=blue,
|
|
citecolor=blue
|
|
}
|
|
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\mr}{\multirow}
|
|
\newcommand{\SI}{\textcolor{blue}{supporting information}}
|
|
|
|
\newcommand{\br}{\mathbf{r}}
|
|
|
|
% energies
|
|
\newcommand{\EHF}{E_\text{HF}}
|
|
\newcommand{\Ec}{E_\text{c}}
|
|
\newcommand{\EPT}{E_\text{PT2}}
|
|
\newcommand{\EFCI}{E_\text{FCI}}
|
|
\newcommand{\EsCI}{E_\text{sCI}}
|
|
\newcommand{\EDMC}{E_\text{DMC}}
|
|
\newcommand{\EexFCI}{E_\text{exFCI}}
|
|
\newcommand{\EexDMC}{E_\text{exDMC}}
|
|
\newcommand{\Ead}{\Delta E_\text{ad}}
|
|
\newcommand{\Eabs}{\Delta E_\text{abs}}
|
|
|
|
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
|
|
\newcommand{\ra}{\rightarrow}
|
|
|
|
% units
|
|
\newcommand{\IneV}[1]{#1 eV}
|
|
\newcommand{\InAU}[1]{#1 a.u.}
|
|
\newcommand{\InAA}[1]{#1 \AA}
|
|
|
|
\newcommand{\pis}{\pi^\star}
|
|
\newcommand{\si}{\sigma}
|
|
\newcommand{\sis}{\sigma^\star}
|
|
|
|
|
|
|
|
\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}
|
|
%%% %%% %%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\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}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
|
|
\bibliography{Ex-srDFT}
|
|
|
|
\end{document}
|