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BSE-PES.tex
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@ -105,6 +105,9 @@
\newcommand{\ERI}[2]{(#1|#2)} \newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\sERI}[2]{[#1|#2]} \newcommand{\sERI}[2]{[#1|#2]}
%% bold in Table
\newcommand{\bb}[1]{\textbf{#1}}
% excitation energies % excitation energies
\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
\newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}} \newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}}
@ -145,9 +148,10 @@
\newcommand{\InAA}[1]{#1 \AA} \newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol} \newcommand{\kcal}{kcal/mol}
\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France} \newcommand{\NEEL}{Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France} \newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
\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{\CEA}{ Univ. Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France }
\begin{document} \begin{document}
@ -156,6 +160,9 @@
\author{Xavier \surname{Blase}} \author{Xavier \surname{Blase}}
\email{xavier.blase@neel.cnrs.fr } \email{xavier.blase@neel.cnrs.fr }
\affiliation{\NEEL} \affiliation{\NEEL}
\author{Ivan \surname{Duchemin}}
\email{ivan.duchemin@cea.fr}
\affiliation{\CEA}
\author{Denis \surname{Jacquemin}} \author{Denis \surname{Jacquemin}}
\email{denis.jacquemin@univ-nantes.fr} \email{denis.jacquemin@univ-nantes.fr}
\affiliation{\CEISAM} \affiliation{\CEISAM}
@ -170,11 +177,11 @@
%\end{wrapfigure} %\end{wrapfigure}
The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalisms have shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems. The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalisms have shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems.
\sout{Although no clear consensus has been reached for the definition of the BSE ground-state energy,} the BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). \sout{Although no clear consensus has been reached for the definition of the BSE ground-state energy,} the BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
\sout{Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules. Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near the equilibrium distance.
Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground-state PES.} \sout{Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground-state PES.}
Thanks to comparisons with \sout{both similar and} state-of-art computational approaches, we show that ACFDT-BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies, \xavier{ concerning the description of ground-state PES near equilibrium°. } Thanks to comparisons with \sout{both similar and} state-of-art computational approaches, we show that ACFDT-BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies, \xavier{equilibrium distances or stretching frequencies}.
However, we also observe, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak. However, we sometimes observe unphysical irregularities on the ground-state PES, in relation with the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak.
\xavier{[]184 words larger than 150]} \xavier{[Now below 150 words]}
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -184,10 +191,12 @@ However, we also observe, in some cases, unphysical irregularities on the ground
%\label{sec:intro} %\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida} the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a valuable alternative with early \textit{ab initio} calculations in condensed matter physics dated back to the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida} the many-body Green's function Bethe-Salpeter equation (BSE) formalism
In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative
\xavier{ [Removed Tiago2008 and Sai2008] } %%with early \textit{ab initio} calculations in condensed matter physics dated back to the end of the 90's.
and is now a serious candidate as a computationally inexpensive method that can effectively model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018} %% \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} In the past few years, BSE
that has gained in the past few years much momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017}
It now stands as a computationally inexpensive method that can effectively model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
One of the main advantages of BSE compared to TD-DFT is that it allows a faithful description of charge-transfer states. \cite{Lastra_2011,Blase_2011b,Baumeier_2012,Duchemin_2012,Cudazzo_2013,Ziaei_2016} One of the main advantages of BSE compared to TD-DFT is that it allows a faithful description of charge-transfer states. \cite{Lastra_2011,Blase_2011b,Baumeier_2012,Duchemin_2012,Cudazzo_2013,Ziaei_2016}
Moreover, when performed on top of a (partially) self-consistently {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\ie, on the xc functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018} Moreover, when performed on top of a (partially) self-consistently {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\ie, on the xc functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018}
However, similar to adiabatic TD-DFT, \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011} However, similar to adiabatic TD-DFT, \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011}
@ -431,7 +440,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.} Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. \xavier{As a guide to the eyes, reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold.} }
\label{tab:Req} \label{tab:Req}
\begin{ruledtabular} \begin{ruledtabular}
@ -441,8 +450,8 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain
Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\ Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
\hline \hline
CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.114 & 2.166 & 2.444 & 2.740 & 2.435 \\ CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.114 & 2.166 & 2.444 & 2.740 & 2.435 \\
& cc-pVTZ & 1.403 & 3.011 & 2.961 & 2.079 & 2.143 & 2.392 & 2.669 & 2.413 \\ & cc-pVTZ & 1.403 & \bb{3.011}& 2.961 & 2.079 & 2.143 & 2.392 & 2.669 & 2.413 \\
& cc-pVQZ & 1.402 & 3.019 & 2.963 & 2.075 & 2.136 & 2.390 & 2.663 & 2.403 \\ & cc-pVQZ & \bb{1.402}& 3.019 & 2.963 & 2.075 & 2.136 & 2.390 & 2.663 & 2.403 \\
CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.101 & 2.149 & 2.435 & 2.695 & 2.433 \\ CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.101 & 2.149 & 2.435 & 2.695 & 2.433 \\
& cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.064 & 2.126 & 2.382 & 2.629 & 2.409 \\ & cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.064 & 2.126 & 2.382 & 2.629 & 2.409 \\
& cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.059 & 2.118 & 2.118 & 2.621 & 2.398 \\ & cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.059 & 2.118 & 2.118 & 2.621 & 2.398 \\
@ -453,8 +462,8 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain
& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.095 & 2.144 & 2.383 & 2.636 & 2.405 \\ & cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.095 & 2.144 & 2.383 & 2.636 & 2.405 \\
& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.091 & 2.137 & 2.382 & 2.634 & 2.395 \\ & cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.091 & 2.137 & 2.382 & 2.634 & 2.395 \\
BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.107 & 2.153 & 2.407 & 2.700 & >2.440 \\ BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.107 & 2.153 & 2.407 & 2.700 & >2.440 \\
& cc-pVTZ & 1.404 & 3.023 & glitch & & & <2.420 & & <2.410 \\ & cc-pVTZ & 1.404 & \bb{3.023}& glitch & & & <2.420 & & <2.410 \\
& cc-pVQZ & 1.399 & & & & & & & \\ & cc-pVQZ & \bb{1.399}& & & & & & & \\
RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.083 & 2.144 & 2.403 & 2.691 & 2.436 \\ RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.083 & 2.144 & 2.403 & 2.691 & 2.436 \\
& cc-pVTZ & 1.388 & 3.013 & glitch & & & <2.420 & & <2.410 \\ & cc-pVTZ & 1.388 & 3.013 & glitch & & & <2.420 & & <2.410 \\
& cc-pVQZ & 1.382 & & & & & & & \\ & cc-pVQZ & 1.382 & & & & & & & \\