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64
BSE-PES.bib
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@ -1,13 +1,60 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-01-25 14:38:41 +0100
%% Created for Pierre-Francois Loos at 2020-01-26 11:17:45 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Duchemin_2019,
Author = {Ivan Duchemin and Xavier Blase},
Date-Added = {2020-01-26 11:16:20 +0100},
Date-Modified = {2020-01-26 11:16:20 +0100},
Doi = {10.1063/1.5090605},
Journal = {J. Chem. Phys.},
Pages = {174120},
Title = {Separable Resolution-of-the-Identity with All-Electron Gaussian Bases: Application to Cubic-scaling RPA},
Volume = {150},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}
@article{Furche_2005,
Author = {Filipp Furche, and Troy Van Voorhis},
Date-Added = {2020-01-26 11:09:36 +0100},
Date-Modified = {2020-01-26 11:10:10 +0100},
Doi = {10.1063/1.1884112},
Journal = {J. Chem. Phys.},
Pages = {164106},
Title = {Fluctuation-dissipation theorem density- functional theory},
Volume = {122},
Year = {2005}}
@article{Toulouse_2009,
Author = {Julien Toulouse and Iann C. Gerber and Georg Jansen and Andreas Savin and Janos G. Angyan},
Date-Added = {2020-01-26 11:06:08 +0100},
Date-Modified = {2020-01-26 11:06:08 +0100},
Doi = {10.1103/PhysRevLett.102.096404},
Journal = {Phys. Rev. Lett.},
Pages = {096404},
Title = {Adiabatic-Connection Fluctuation-Dissipation Density-Functional Theory Based on Range Separation},
Volume = {102},
Year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.102.096404}}
@article{Toulouse_2010,
Author = {Julien Toulouse and Wuming Zhu and Janos G. Angyan and Andreas Savin},
Date-Added = {2020-01-26 11:00:55 +0100},
Date-Modified = {2020-01-26 11:07:07 +0100},
Doi = {10.1103/PhysRevA.82.032502},
Journal = {Phys. Rev. A},
Pages = {032502},
Title = {Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications},
Volume = {82},
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.82.032502}}
@article{Angyan_2011,
Author = {J. G. Angyan and R.-F. Liu and J. Toulouse and G. Jansen},
Date-Added = {2020-01-25 14:20:51 +0100},
@ -17,7 +64,8 @@
Pages = {3116--3130},
Title = {Correlation Energy Expressions from the Adiabatic-Connection Fluctuation Dissipation Theorem Approach},
Volume = {7},
Year = {2011}}
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1021/ct200501r}}
@article{Ghosh_2018,
Author = {Ghosh, Soumen and Verma, Pragya and Cramer, Christopher J. and Gagliardi, Laura and Truhlar, Donald G.},
@ -9410,15 +9458,13 @@
Year = {1995},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.74.1827}}
@article{Duchemin_2019,
@article{Duchemin_2020,
Author = {Ivan Duchemin and Xavier Blase},
Date-Added = {2019-10-23 10:00:45 +0200},
Date-Modified = {2019-10-23 10:01:39 +0200},
Doi = {10.1063/1.5090605},
Journal = {J. Chem. Phys.},
Pages = {174120},
Title = {Separable Resolution-of-the-Identity with All-Electron Gaussian Bases: Application to Cubic-scaling RPA},
Volume = {150},
Date-Modified = {2020-01-26 11:17:42 +0100},
Journal = {J. Chem. Theory Comput.},
Pages = {submitted},
Title = {Robust Analytic Continuation Approach to Many-Body GW Calculations},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}

26
BSE-PES.tex
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@ -169,10 +169,10 @@
% \includegraphics[width=\linewidth]{TOC}
%\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.
Although less popular, the BSE formalism can also be employed to compute ground-state energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
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).
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 potential energy surfaces.
Thanks to comparison 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.
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.
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}
@ -205,27 +205,27 @@ However, we also observe, in some cases, unphysical irregularities on the ground
%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}
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 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 \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}
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}
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}
A significant limitation of the BSE formalism, as compared to TD-DFT, \cite{Furche_2002} lies in the lack of analytic forces for both the ground and excited states, preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Navizet_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}
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 analogue.
\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.
Contrary to TD-DFT, the BSE ground-state correlation 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}
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, 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}
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}
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}
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 coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE 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}
Here, in analogy to the random-phase approximation (RPA)-type formalismes, \cite{Furche_2008, Angyan_2011, Holzer_2018} the ground-state BSE energy is calculated in the adiabatic-connection fluctuation-dissipation theorem 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.
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 present ACFDT-based approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods.
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{Loos_2018, Veril_2018}
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.
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.
%In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
@ -340,7 +340,7 @@ are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
\begin{subequations}
\begin{align}
\label{eq:LR_BSE}
\label{eq:LR_RPA}
\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{bj},
\\
\BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{jb},