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BSE-PES-control.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-05-13 21:13:34 +0200
%% Created for Pierre-Francois Loos at 2020-02-05 14:17:56 +0100
%% Saved with string encoding Unicode (UTF-8)

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BSE-PES.tex
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
\usepackage{natbib}
\usepackage[extra]{tipa}
\bibliographystyle{achemso}
\AtBeginDocument{\nocite{achemso-control}}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
@ -160,20 +166,20 @@
\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Ivan \surname{Duchemin}}
\email{ivan.duchemin@cea.fr}
\affiliation{\CEA}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Anthony \surname{Scemama}}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Ivan \surname{Duchemin}}
\email{ivan.duchemin@cea.fr}
\affiliation{\CEA}
\author{Denis \surname{Jacquemin}}
\email{denis.jacquemin@univ-nantes.fr}
\affiliation{\CEISAM}
\email{denis.jacquemin@univ-nantes.fr}
\affiliation{\CEISAM}
\author{Xavier \surname{Blase}}
\email{xavier.blase@neel.cnrs.fr }
\affiliation{\NEEL}
\email{xavier.blase@neel.cnrs.fr }
\affiliation{\NEEL}
\begin{abstract}
%\begin{wrapfigure}[12]{o}[-1.25cm]{0.4\linewidth}
@ -335,7 +341,7 @@ In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme,
& = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
\end{align}
\end{subequations}
with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals
\begin{multline}
\label{eq:W}
\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
@ -346,6 +352,7 @@ where the spectral weights at coupling strength $\IS$ read
\begin{equation}
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
\end{equation}
In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
\begin{subequations}
@ -515,7 +522,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
%%% TABLE I %%%
\begin{table*}
\caption{
Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
The values in parenthesis have been obtained by fitting a Morse potential to the PES.
}
@ -542,7 +549,7 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{(2.408)} &\gb{(2.070)} &\gb{(2.130)} &\gb{(2.383)} &\rb{(2.640)}\\
RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 &2.065(2.048) & 2.114 & (2.370) & (2.584) \\
& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 & 2.055 & 2.114 & (2.370) & (2.584) \\
& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
@ -587,7 +594,8 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
%\section{Conclusion}
%\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
In this Letter, we have shown that calculating the BSE correlation energy within the ACFDT framework yield extremely accurate PES around equilibrium.
In this Letter, we hope to have illustrated that the ACFDT@BSE formalism is a promising methodology for the computation of accurate ground-state PES and their corresponding equilibrium structures.
To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yield extremely accurate PES around equilibrium.
(Their accuracy near the dissociation limit remains an open question.)
We have illustrated this for 8 diatomic molecules for which we have also computed reference ground-state energies using coupled cluster methods (CC2, CCSD, and CC3).
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
@ -596,11 +604,6 @@ We believe that this central issue must be resolved if one wants to expand the a
In the perspective of developing analytical nuclear gradients within the BSE@$GW$ formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
We hope to be able to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
@ -611,6 +614,11 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation.
\bibliography{BSE-PES,BSE-PES-control}
\end{document}