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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-05-13 21:13:34 +0200
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%% Created for Pierre-Francois Loos at 2020-02-05 14:17:56 +0100
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%% Saved with string encoding Unicode (UTF-8)
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48
BSE-PES.tex
48
BSE-PES.tex
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage{natbib}
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\usepackage[extra]{tipa}
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\bibliographystyle{achemso}
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\AtBeginDocument{\nocite{achemso-control}}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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@ -160,20 +166,20 @@
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\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Ivan \surname{Duchemin}}
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\email{ivan.duchemin@cea.fr}
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\affiliation{\CEA}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Anthony \surname{Scemama}}
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\email{scemama@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\email{scemama@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Ivan \surname{Duchemin}}
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\email{ivan.duchemin@cea.fr}
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\affiliation{\CEA}
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\author{Denis \surname{Jacquemin}}
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\email{denis.jacquemin@univ-nantes.fr}
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\affiliation{\CEISAM}
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\email{denis.jacquemin@univ-nantes.fr}
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\affiliation{\CEISAM}
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\author{Xavier \surname{Blase}}
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\email{xavier.blase@neel.cnrs.fr }
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\affiliation{\NEEL}
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\email{xavier.blase@neel.cnrs.fr }
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\affiliation{\NEEL}
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\begin{abstract}
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%\begin{wrapfigure}[12]{o}[-1.25cm]{0.4\linewidth}
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@ -335,7 +341,7 @@ In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme,
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& = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
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\end{align}
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\end{subequations}
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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 $W$.}
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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
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\begin{multline}
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\label{eq:W}
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\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
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@ -346,6 +352,7 @@ where the spectral weights at coupling strength $\IS$ read
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\begin{equation}
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\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
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\end{equation}
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In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
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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
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\begin{subequations}
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@ -515,7 +522,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
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%%% TABLE I %%%
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\begin{table*}
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\caption{
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Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
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The values in parenthesis have been obtained by fitting a Morse potential to the PES.
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}
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@ -542,7 +549,7 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
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& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
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& 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)}\\
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RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
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& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 &2.065(2.048) & 2.114 & (2.370) & (2.584) \\
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& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 & 2.055 & 2.114 & (2.370) & (2.584) \\
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& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
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RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
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& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
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@ -587,7 +594,8 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
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%\section{Conclusion}
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%\label{sec:conclusion}
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%%%%%%%%%%%%%%%%%%%%%%%%
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In this Letter, we have shown that calculating the BSE correlation energy within the ACFDT framework yield extremely accurate PES around equilibrium.
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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.
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To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yield extremely accurate PES around equilibrium.
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(Their accuracy near the dissociation limit remains an open question.)
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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).
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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.
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@ -596,11 +604,6 @@ We believe that this central issue must be resolved if one wants to expand the a
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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.
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We hope to be able to report on this in the near future.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting Information}
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%%%%%%%%%%%%%%%%%%%%%%%%
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See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
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@ -611,6 +614,11 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting Information}
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%%%%%%%%%%%%%%%%%%%%%%%%
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See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation.
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\bibliography{BSE-PES,BSE-PES-control}
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\end{document}
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