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BSE-PES.tex
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BSE-PES.tex
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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\newcommand{\kcal}{kcal/mol}
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\newcommand{\NEEL}{Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
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\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
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\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
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\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\CEA}{ Univ. Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France }
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\newcommand{\CEA}{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France}
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\begin{document}
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\begin{document}
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\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
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\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
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\author{Xavier \surname{Blase}}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{xavier.blase@neel.cnrs.fr }
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\NEEL}
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\affiliation{\LCPQ}
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\author{Ivan \surname{Duchemin}}
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\author{Ivan \surname{Duchemin}}
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\email{ivan.duchemin@cea.fr}
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\email{ivan.duchemin@cea.fr}
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\affiliation{\CEA}
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\affiliation{\CEA}
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@ -171,9 +171,9 @@
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\author{Denis \surname{Jacquemin}}
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\author{Denis \surname{Jacquemin}}
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\email{denis.jacquemin@univ-nantes.fr}
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\email{denis.jacquemin@univ-nantes.fr}
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\affiliation{\CEISAM}
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\affiliation{\CEISAM}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\author{Xavier \surname{Blase}}
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\email{loos@irsamc.ups-tlse.fr}
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\email{xavier.blase@neel.cnrs.fr }
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\affiliation{\LCPQ}
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\affiliation{\NEEL}
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\begin{abstract}
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\begin{abstract}
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%\begin{wrapfigure}[12]{o}[-1.25cm]{0.4\linewidth}
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%\begin{wrapfigure}[12]{o}[-1.25cm]{0.4\linewidth}
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@ -182,7 +182,7 @@
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%\end{wrapfigure}
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%\end{wrapfigure}
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The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems.
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The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems.
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The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
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The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
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Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium distance.
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Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
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Thanks to comparisons with 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 and equilibrium bond distances.
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Thanks to comparisons with 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 and equilibrium bond distances.
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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.
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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.
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\end{abstract}
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\end{abstract}
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@ -195,28 +195,28 @@ However, we sometimes observe unphysical irregularities on the ground-state PES
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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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
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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
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\cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative that has gained momentum in the past few years for studying 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}
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\cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative that has gained momentum in the past few years for studying 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}
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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}
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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}
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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}
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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}
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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 exchange-correlation functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018}
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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 exchange-correlation functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018}
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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}
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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,Loos_2019}
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A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytic nuclear gradients (\ie, the first derivatives of the energy with respect to the nuclear displacements) for both the ground and excited states, \cite{Furche_2002} preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes \cite{Navizet_2011} associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
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A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients (\ie, the first derivatives of the energy with respect to the nuclear displacements) for both the ground and excited states, \cite{Furche_2002} preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes \cite{Navizet_2011} associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
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While calculations of the $GW$ quasiparticle energy ionic gradients is becoming popular,
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While calculations of the $GW$ quasiparticle energy ionic gradients is becoming popular,
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\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the KS-DFT (LDA) forces as its ground-state analog.
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\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state analog.
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Contrary to TD-DFT which relies on Kohn-Sham density-functional theory (KS-DFT) \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
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Contrary to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
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It then remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
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It then remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
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As a matter of fact, in the largest recent available benchmark study \cite{Holzer_2018} of the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation (ACFDT) framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
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As a matter of fact, in the largest recent available benchmark study \cite{Holzer_2018} of the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
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Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016}
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Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016}
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With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy.
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With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy.
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Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE correlation energy. \cite{Holzer_2018}
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Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability (see below) leads to two different versions of the BSE correlation energy. \cite{Holzer_2018}
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Here, in analogy to the random-phase approximation (RPA)-type formalisms \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} and similarly to Refs.~\onlinecite{Olsen_2014,Maggio_2016,Holzer_2018}, the ground-state BSE energy is calculated in the adiabatic connection framework.
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Here, in analogy to the random-phase approximation (RPA)-type formalisms \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} and similarly to Refs.~\onlinecite{Olsen_2014,Maggio_2016,Holzer_2018}, the ground-state BSE energy is calculated in the adiabatic connection framework.
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Embracing this definition, the purpose of the present study is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach for several diatomic molecules.
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Embracing this definition, the purpose of the present study is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach for several diatomic molecules.
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The location of the minima on the ground-state PES is of particular interest.
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The location of the minima on the ground-state PES is of particular interest.
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This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
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This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
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Thanks to comparison with both similar and state-of-art computational approaches, we show that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute energies and equilibrium distances.
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Thanks to comparisons with both similar and state-of-art computational approaches, we show that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute energies and equilibrium distances.
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However, we also observe that, 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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
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However, we also observe that, 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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
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%The paper is organized as follows.
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%The paper is organized as follows.
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@ -299,12 +299,12 @@ In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ is positive-definite
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where the excitation amplitudes are
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where the excitation amplitudes are
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\begin{subequations}
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\begin{subequations}
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\begin{align}
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\begin{align}
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\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS},
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\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{\IS},
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\\
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\\
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\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
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\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
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\end{align}
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\end{align}
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\end{subequations}
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\end{subequations}
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With Mulliken's notation of the bare two-electron integrals
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Introducing the so-called Mulliken notation for the bare two-electron integrals
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\begin{equation}
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\begin{equation}
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\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
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\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
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\end{equation}
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\end{equation}
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@ -324,7 +324,7 @@ the BSE matrix elements read
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\end{subequations}
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\end{subequations}
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where $\eGW{p}$ are the $GW$ quasiparticle energies.
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where $\eGW{p}$ are the $GW$ quasiparticle energies.
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%In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$.
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%In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$.
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In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is built within the direct RPA scheme:
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In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie,
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\begin{subequations}
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\begin{subequations}
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\label{eq:wrpa}
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\label{eq:wrpa}
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\begin{align}
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\begin{align}
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@ -335,7 +335,7 @@ In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is buil
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& = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
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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{align}
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\end{subequations}
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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 molecular orbitals 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 \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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\begin{multline}
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\begin{multline}
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\label{eq:W}
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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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\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
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@ -362,7 +362,7 @@ where $\eHF{p}$ are the HF orbital energies.
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The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening
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The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening
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%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
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%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
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so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
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so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
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In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
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In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock (HF) eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
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\begin{subequations}
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\begin{subequations}
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\begin{align}
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\begin{align}
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\label{eq:LR_RPAx-A}
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\label{eq:LR_RPAx-A}
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@ -437,7 +437,7 @@ Even for weakly correlated systems, triplet instabilities are much more common,
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%\section{Computational details}
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%\section{Computational details}
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%\label{sec:comp_details}
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%\label{sec:comp_details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
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All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) HF starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
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Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations.
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Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations.
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In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
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In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
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@ -520,7 +520,6 @@ The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold
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The values in parenthesis have been obtained by fitting a Morse potential to the PES.
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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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}
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\label{tab:Req}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{llcccccccc}
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\begin{tabular}{llcccccccc}
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& & \mc{8}{c}{Molecules} \\
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& & \mc{8}{c}{Molecules} \\
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@ -540,45 +539,17 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
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& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
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& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
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& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
|
& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
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BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
|
BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
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& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 &2.068(2.074) & 2.116 & (2.389) & (2.647) \\
|
& 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)}\\
|
& 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) \\
|
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 &2.065(2.048) & 2.114 & (2.370) & (2.584) \\
|
||||||
& cc-pVQZ & 1.382 &3.013(2.998) & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
|
& 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 \\
|
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 \\
|
& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
|
||||||
& cc-pVQZ & 1.394 & 3.011 &\gb{(2.943)} &\gb{(2.393)} &\gb{(2.041)} &\gb{(2.105)} &\gb{(2.367)} &\gb{(2.563)} \\
|
& cc-pVQZ & 1.394 & 3.011 &\gb{(2.943)} &\gb{(2.393)} &\gb{(2.041)} &\gb{(2.105)} &\gb{(2.367)} &\gb{(2.563)} \\
|
||||||
RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
|
RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
|
||||||
& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
|
& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
|
||||||
& cc-pVQZ & 1.386 & 2.994 &\gb{(2.946)} &\gb{(2.385)} &\gb{(2.042)} &\gb{(2.104)} &\gb{(2.365)} &\gb{(2.571)} \\
|
& cc-pVQZ & 1.386 & 2.994 &\gb{(2.946)} &\gb{(2.385)} &\gb{(2.042)} &\gb{(2.104)} &\gb{(2.365)} &\gb{(2.571)} \\
|
||||||
% FROZEN CORE VERSION
|
|
||||||
% Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
|
|
||||||
% \hline
|
|
||||||
% CC3 & cc-pVDZ & 1.438 & 3.052 & 3.014 & 2.115 & 2.167 & 2.447 & 2.741 & 2.438 \\
|
|
||||||
% & cc-pVTZ & 1.403 & 3.036 & 2.985 & 2.087 & 2.150 & 2.405 & 2.672 & 2.414 \\
|
|
||||||
% & cc-pVQZ & 1.402 & 3.037 & 2.985 & 2.080 & 2.142 & 2.398 & 2.667 & 2.413 \\
|
|
||||||
% 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-pVQZ & 1.402 & 3.020 & 2.953 & 2.059 & 2.118 & 2.380 & 2.621 & 2.398 \\
|
|
||||||
% CC2 & cc-pVDZ & 1.426 & & & & & & & \\
|
|
||||||
% & cc-pVTZ & 1.393 & & & & & & & \\
|
|
||||||
% & cc-pVQZ & 1.391 & & & & & & & \\
|
|
||||||
% MP2 & cc-pVDZ & 1.426 & 3.049 & 3.012 & 2.134 & 2.167 & 2.433 & 2.681 & 2.429 \\
|
|
||||||
% & cc-pVTZ & 1.393 & 3.026 & 2.990 & 2.104 & 2.151 & 2.395 & 2.640 & 2.407 \\
|
|
||||||
% & cc-pVQZ & 1.391 & 3.026 & 2.990 & 2.098 & 2.144 & 2.389 & 2.638 & 2.405 \\
|
|
||||||
% BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & & & & & & & \\
|
|
||||||
% & cc-pVTZ & 1.404 & & & & & & & \\
|
|
||||||
% & cc-pVQZ & 1.399 & & & & & & & \\
|
|
||||||
% RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & & & & & & & \\
|
|
||||||
% & cc-pVTZ & 1.388 & & & & & & & \\
|
|
||||||
% & cc-pVQZ & 1.382 & & & & & & & \\
|
|
||||||
% RPAx@HF & cc-pVDZ & 1.428 & & & & & & & \\
|
|
||||||
% & cc-pVTZ & 1.395 & & & & & & & \\
|
|
||||||
% & cc-pVQZ & 1.394 & & & & & & & \\
|
|
||||||
% RPA@HF & cc-pVDZ & 1.431 & & & & & & & \\
|
|
||||||
% & cc-pVTZ & 1.388 & & & & & & & \\
|
|
||||||
% & cc-pVQZ & 1.386 & & & & & & & \\
|
|
||||||
|
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\end{ruledtabular}
|
\end{ruledtabular}
|
||||||
\end{table*}
|
\end{table*}
|
||||||
@ -632,7 +603,8 @@ See {\SI} for additional potential energy curves with other basis sets and withi
|
|||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
\begin{acknowledgements}
|
\begin{acknowledgements}
|
||||||
PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and XB is indebted to Valerio Olevano for numerous discussions.
|
%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
|
||||||
|
XB is indebted to Valerio Olevano for numerous discussions.
|
||||||
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
||||||
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
|
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
|
||||||
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
||||||
|
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Reference in New Issue
Block a user