From 7be8f6d0989aace3c99c0f1df5d12e382d97e624 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 27 Jan 2020 18:43:53 +0100 Subject: [PATCH] changes in intro --- BSE-PES.tex | 47 +++++++++++++++++++++++------------------------ 1 file changed, 23 insertions(+), 24 deletions(-) diff --git a/BSE-PES.tex b/BSE-PES.tex index f9c37d0..f12ff72 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -163,6 +163,9 @@ \author{Ivan \surname{Duchemin}} \email{ivan.duchemin@cea.fr} \affiliation{\CEA} +\author{Anthony \surname{Scemama}} +\email{scemama@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} \author{Denis \surname{Jacquemin}} \email{denis.jacquemin@univ-nantes.fr} \affiliation{\CEISAM} @@ -176,12 +179,10 @@ % \includegraphics[width=\linewidth]{TOC} %\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. -\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). -Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near the equilibrium distance. -\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{equilibrium distances or stretching frequencies}. -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{[Now below 150 words]} +The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). +Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium distance. +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, equilibrium distances or \titou{harmonic vibrational frequencies}. +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. \end{abstract} \maketitle @@ -192,10 +193,7 @@ However, we sometimes observe unphysical irregularities on the ground-state PES, %%%%%%%%%%%%%%%%%%%%%%%% 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,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} 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} 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} +\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 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} 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} @@ -207,16 +205,16 @@ While calculations of the $GW$ quasiparticle energies ionic gradients is becomin 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. 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} -As a matter of fact, in the largest recent available benchmark study \cite{Holzer_2018} of the total energies of the atoms H?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} +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} Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016} 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. 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} -Here, in analogy to 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. +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. 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. The location of the minima on the ground-state PES is of particular interest. This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism. -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. +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, equilibrium distances or \titou{harmonic vibrational 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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} %The paper is organized as follows. @@ -399,14 +397,13 @@ Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacin Several important comments are in order here. For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be labeled as weakly correlated. -However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, \xavier{ hampering in particular the calculation of atomisation energies}. -%% In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}. -Even for weakly correlated systems, triplet instabilities are much more common \xavier{but triplet excitations do not contribute to the correlation energy in the ACFDT formulation.} +However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, hampering in particular the calculation of atomization energies. \cite{Holzer_2018} +Even for weakly correlated systems, triplet instabilities are much more common, but triplet excitations do not contribute to the correlation energy in the ACFDT formulation. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011} -\xavier{ IS THIS NEEDED NOW ? However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Sawada_1957b, Rowe_1968} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach. -Indeed, although the plasmon and adiabatic connection formulations are equivalent for RPA, \cite{Sawada_1957b, Gell-Mann_1957, Fukuta_1964, Furche_2008} this is not the case at the BSE and RPAx levels. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011, Li_2020} -For RPAx, an alternative plasmon expression (equivalent to its ACFDT analog) can be found if exchange is also added to the interaction kernel [see Eq.~\eqref{eq:K}]. \cite{Angyan_2011} -However, to the best of our knowledge, such alternative plasmon expression does not exist for BSE. } +%\xavier{ IS THIS NEEDED NOW ? However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Sawada_1957b, Rowe_1968} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach. +%Indeed, although the plasmon and adiabatic connection formulations are equivalent for RPA, \cite{Sawada_1957b, Gell-Mann_1957, Fukuta_1964, Furche_2008} this is not the case at the BSE and RPAx levels. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011, Li_2020} +%For RPAx, an alternative plasmon expression (equivalent to its ACFDT analog) can be found if exchange is also added to the interaction kernel [see Eq.~\eqref{eq:K}]. \cite{Angyan_2011} +%However, to the best of our knowledge, such alternative plasmon expression does not exist for BSE. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{Computational details} @@ -417,7 +414,8 @@ Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculatio In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation. Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}. Finally, the infinitesimal $\eta$ has been set to zero for all calculations. -The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature. \xavier{ Comparison with the so-called plasmon (or Trace) formula~\cite{Furche_2008} at the RPA level confirmed the excellent convergency of such a $\lambda$-sampling scheme. } +The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature. +Comparison with the so-called plasmon (or trace) formula \cite{Furche_2008} at the RPA level has confirmed the excellent accuracy of the present quadrature scheme over $\IS$. For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} levels of theory using DALTON. \cite{dalton} All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018} @@ -440,7 +438,8 @@ 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. \xavier{As a guide to the eyes, reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold.} } +Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. +The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold for visual convenience.} \label{tab:Req} \begin{ruledtabular} @@ -595,8 +594,8 @@ See {\SI} for additional potential energy curves with other basis sets and withi %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} -PFL would like to thank Anthony Scemama for technical assistance. -\xavier{XB is indebted to Valerio Olevano for numerous discussions.} +PFL would like to thank Julien Toulouse for enlightening discussions about RPA. +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. 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''. }