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BSE-PES.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-01-26 13:32:50 +0100 %% Created for Pierre-Francois Loos at 2020-01-26 20:27:20 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Harding_2008,
Author = {M.E.Harding, J. Vazquez and B. Ruscic and A.K.Wilson and J. Gauss and J. F. Stanton},
Date-Added = {2020-01-26 20:25:15 +0100},
Date-Modified = {2020-01-26 20:27:17 +0100},
Doi = {10.1063/1.2835612},
Journal = {J. Chem. Phys.},
Pages = {114111},
Title = {High-Accuracy Extrapolated ab Initio Thermochemistry. III. Additional Improvements and Overview},
Volume = {128},
Year = {2008}}
@article{dalton, @article{dalton,
Author = {Aidas, Kestutis and Angeli, Celestino and Bak, Keld L. and Bakken, Vebj{\o}rn and Bast, Radovan and Boman, Linus and Christiansen, Ove and Cimiraglia, Renzo and Coriani, Sonia and Dahle, P{\aa}l and Dalskov, Erik K. and Ekstr{\"o}m, Ulf and Enevoldsen, Thomas and Eriksen, Janus J. and Ettenhuber, Patrick and Fern{\'a}ndez, Berta and Ferrighi, Lara and Fliegl, Heike and Frediani, Luca and Hald, Kasper and Halkier, Asger and H{\"a}ttig, Christof and Heiberg, Hanne and Helgaker, Trygve and Hennum, Alf Christian and Hettema, Hinne and Hjerten{\ae}s, Eirik and H{\o}st, Stinne and H{\o}yvik, Ida-Marie and Iozzi, Maria Francesca and Jans{\'\i}k, Branislav and Jensen, Hans J{\o}rgen Aa. and Jonsson, Dan and J{\o}rgensen, Poul and Kauczor, Joanna and Kirpekar, Sheela and Kj{\ae}rgaard, Thomas and Klopper, Wim and Knecht, Stefan and Kobayashi, Rika and Koch, Henrik and Kongsted, Jacob and Krapp, Andreas and Kristensen, Kasper and Ligabue, Andrea and Lutn{\ae}s, Ola B. and Melo, Juan I. and Mikkelsen, Kurt V. and Myhre, Rolf H. and Neiss, Christian and Nielsen, Christian B. and Norman, Patrick and Olsen, Jeppe and Olsen, J{\'o}gvan Magnus H. and Osted, Anders and Packer, Martin J. and Pawlowski, Filip and Pedersen, Thomas B. and Provasi, Patricio F. and Reine, Simen and Rinkevicius, Zilvinas and Ruden, Torgeir A. and Ruud, Kenneth and Rybkin, Vladimir V. and Sa{\l}ek, Pawel and Samson, Claire C. M. and de Mer{\'a}s, Alfredo S{\'a}nchez and Saue, Trond and Sauer, Stephan P. A. and Schimmelpfennig, Bernd and Sneskov, Kristian and Steindal, Arnfinn H. and Sylvester-Hvid, Kristian O. and Taylor, Peter R. and Teale, Andrew M. and Tellgren, Erik I. and Tew, David P. and Thorvaldsen, Andreas J. and Th{\o}gersen, Lea and Vahtras, Olav and Watson, Mark A. and Wilson, David J. D. and Ziolkowski, Marcin and {\AA}gren, Hans}, Author = {Aidas, Kestutis and Angeli, Celestino and Bak, Keld L. and Bakken, Vebj{\o}rn and Bast, Radovan and Boman, Linus and Christiansen, Ove and Cimiraglia, Renzo and Coriani, Sonia and Dahle, P{\aa}l and Dalskov, Erik K. and Ekstr{\"o}m, Ulf and Enevoldsen, Thomas and Eriksen, Janus J. and Ettenhuber, Patrick and Fern{\'a}ndez, Berta and Ferrighi, Lara and Fliegl, Heike and Frediani, Luca and Hald, Kasper and Halkier, Asger and H{\"a}ttig, Christof and Heiberg, Hanne and Helgaker, Trygve and Hennum, Alf Christian and Hettema, Hinne and Hjerten{\ae}s, Eirik and H{\o}st, Stinne and H{\o}yvik, Ida-Marie and Iozzi, Maria Francesca and Jans{\'\i}k, Branislav and Jensen, Hans J{\o}rgen Aa. and Jonsson, Dan and J{\o}rgensen, Poul and Kauczor, Joanna and Kirpekar, Sheela and Kj{\ae}rgaard, Thomas and Klopper, Wim and Knecht, Stefan and Kobayashi, Rika and Koch, Henrik and Kongsted, Jacob and Krapp, Andreas and Kristensen, Kasper and Ligabue, Andrea and Lutn{\ae}s, Ola B. and Melo, Juan I. and Mikkelsen, Kurt V. and Myhre, Rolf H. and Neiss, Christian and Nielsen, Christian B. and Norman, Patrick and Olsen, Jeppe and Olsen, J{\'o}gvan Magnus H. and Osted, Anders and Packer, Martin J. and Pawlowski, Filip and Pedersen, Thomas B. and Provasi, Patricio F. and Reine, Simen and Rinkevicius, Zilvinas and Ruden, Torgeir A. and Ruud, Kenneth and Rybkin, Vladimir V. and Sa{\l}ek, Pawel and Samson, Claire C. M. and de Mer{\'a}s, Alfredo S{\'a}nchez and Saue, Trond and Sauer, Stephan P. A. and Schimmelpfennig, Bernd and Sneskov, Kristian and Steindal, Arnfinn H. and Sylvester-Hvid, Kristian O. and Taylor, Peter R. and Teale, Andrew M. and Tellgren, Erik I. and Tew, David P. and Thorvaldsen, Andreas J. and Th{\o}gersen, Lea and Vahtras, Olav and Watson, Mark A. and Wilson, David J. D. and Ziolkowski, Marcin and {\AA}gren, Hans},
Date-Added = {2020-01-26 13:32:47 +0100}, Date-Added = {2020-01-26 13:32:47 +0100},

42
BSE-PES.tex
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@ -13,8 +13,8 @@
]{hyperref} ]{hyperref}
\urlstyle{same} \urlstyle{same}
\newcommand{\ie}{\textit{i.e}} \newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g}} \newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900} \definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
@ -171,10 +171,10 @@
% \includegraphics[width=\linewidth]{TOC} % \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure} %\end{wrapfigure}
The many-body Green's function Bethe-Salpeter equation (BSE) formalism performed on top of a $GW$ calculation has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies for medium and large molecular systems. The many-body Green's function Bethe-Salpeter equation (BSE) formalism performed on top of a $GW$ calculation has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies for medium and large molecular systems.
Although no consensus has been reached for the definition of the ground-state energy, the BSE formalism can also be employed to compute ground-state energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). 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. Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules.
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. 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 both similar and state-of-art computational approaches, we show that the present ACFDT-based approach is surprisingly accurate, and can even compete with coupled cluster methods. Thanks to comparisons with both similar and state-of-art computational approaches, we show that the present ACFDT-based approach is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies.
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 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.
\end{abstract} \end{abstract}
@ -206,24 +206,25 @@ However, we also observe, in some cases, unphysical irregularities on the ground
%One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} %One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011}
%Yet another problem is the choice of the xc functionals as the quality of excitation energies are substantially dependent on this choice. %Yet another problem is the choice of the xc functionals as the quality of excitation energies are substantially dependent on this choice.
With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Casida,Runge_1984} 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 appearing at 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{Casida,Runge_1984} 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}
In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Tiago_2008,Sai_2008,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} and is now a serious candidate as a computationally inexpensive method \cite{Gonzales_2012,Loos_2020a} that can effectively model excited states 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} In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Tiago_2008,Sai_2008,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} 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}
One of the main advantage 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}
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytic forces 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} 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}
While calculations of the {\GW} quasiparticle energies ionic gradients is becoming popular, While calculations of the {\GW} quasiparticle energies ionic gradients is becoming popular,
\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 ground-state analog. \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.
Contrary to TD-DFT which relies on density-functional theory (DFT) \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} 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.
As a matter of fact, in the largest recent available benchmark study of the 26 small molecules forming the HEAT test set, \cite{Holzer_2018} 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} 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}
Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016} 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. 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 coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE correlation energy. \cite{Holzer_2018} 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 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 of several diatomic molecules. 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. 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. 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.
@ -297,7 +298,7 @@ For a closed-shell system, to compute the singlet BSE excitation energies (withi
\bY{\IS}_m \\ \bY{\IS}_m \\
\end{pmatrix}, \end{pmatrix},
\end{equation} \end{equation}
where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interaction strength $\IS$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interaction strength $\IS$, $\T{}$ is the matrix transpose, and we assume real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively. The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively.
In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
@ -367,6 +368,7 @@ where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF en
\end{equation} \end{equation}
is the ground-state BSE correlation energy computed in the adiabatic connection framework, where is the ground-state BSE correlation energy computed in the adiabatic connection framework, where
\begin{equation} \begin{equation}
\label{eq:K}
\bK = \bK =
\begin{pmatrix} \begin{pmatrix}
\btA{\IS=1} & \bB{\IS=1} \\ \btA{\IS=1} & \bB{\IS=1} \\
@ -387,10 +389,10 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
\bO & \bI \\ \bO & \bI \\
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace and $\T{}$ the matrix transpose. is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer et al. \cite{Holzer_2018} Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer et al. \cite{Holzer_2018}
Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintain with respect to $\IS$. Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintain with respect to $\IS$.
Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} coming from the variation of the Green's function along the adiabatic connection should be added. Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added.
However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study.
Several important comments are in order here. Several important comments are in order here.
@ -398,8 +400,10 @@ For spin-restricted closed-shell molecular systems around their equilibrium geom
However, singlet instabilities may appear in the presence of strong correlation (\eg, when the bond is stretched). However, singlet instabilities may appear in the presence of strong correlation (\eg, when the bond is stretched).
In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}. In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}.
Even for weakly correlated systems, triplet instabilities are much more common. Even for weakly correlated systems, triplet instabilities are much more common.
However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach. 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 at the RPA level, the plasmon and adiabatic connection formulations are equivalent, \cite{Sawada_1957b, Fukuta_1964, Furche_2008} this is not the case at the BSE level. 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} %\section{Computational details}
@ -411,7 +415,7 @@ These will be labeled as BSE@{\GOWO}.
In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation. 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}. 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. 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 has been performed with a 21-point Gauss-Legendre quadrature. 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.
As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions. As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fairer comparison between methods. Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fairer comparison between methods.
However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}). However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}).