From 89dfb9b91b64a9a90e8b464c9e29b1f2b8b39165 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 7 Jan 2020 17:57:39 +0100 Subject: [PATCH] Intro OK --- BSE-PES.tex | 34 ++++++++++++++++------------------ 1 file changed, 16 insertions(+), 18 deletions(-) diff --git a/BSE-PES.tex b/BSE-PES.tex index 9c9f716..a8bd1dc 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -207,34 +207,32 @@ An important point from a practical point of view is that the TD-DFT xc kernel i 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. -With a similar computational cost, the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a very valuable alternative to TD-DFT \xavier{with early \textit{ab initio} calculations in condensed matter physics appearing at the end of the 90s. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} } -In the past few years, BSE has gained momentum \xavier{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 \xavier{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} +With a similar computational cost, the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a very valuable alternative to TD-DFT 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} 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,Rangel16,Kaplan_2018,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 TD-DFT, the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011} -\xavier{A significant limitation of the Bethe-Salpeter formalism as compared to TD-DFT lies in the lack of analytic forces in the excited state, preventing efficient applications to the study of photoluminescence, photocatalysis, etc. While calculations of the $GW$ quasiparticle energies ionic gradients is becoming very popular, -\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the Bethe-Salpeter gradients in the excited state is available. \cite{Beigi_2003} In this study devoted to small molecules (CO and NH$_3$), only the Bethe-Salpeter excitation energy gradients were calculated, while taking for the ground-state energy the DFT (LDA) forces. } +A significant limitation of the BSE formalism as compared to TD-DFT lies in the lack of analytic forces for the excited states, preventing efficient applications to the study of photoluminescence, photocatalysis, etc. While calculations of the {\GW} quasiparticle energies ionic gradients is becoming very 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. -Contrary to TD-DFT, there is no clear definition of the ground- and excited-state energies within BSE, which makes the development of analytical nuclear gradients a particularly tricky task. -Here, in analogy to the random-phase approximation (RPA) formalism, the ground-state BSE energy is calculated via the ``trace'' formula (see below). -The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy. -This definition of the energy has the advantage of treating at the same level of theory the ground state and the excited states. -Embracing this definition, the purpose of the present study is to investigate the quality of ground- and excited-state PES near equilibrium obtained within BSE -The location of the minima on the ground- and (singlet and triplet) excited-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. -There are very few studies about the ground-state BSE energy for atomic and molecular systems \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} and, to the best of our knowledge, the present study is the first to investigate the topology of the excited-state PES at the BSE level. +%Contrary to TD-DFT, there is no clear definition of the ground- and excited-state energies within BSE, which makes the development of analytical nuclear gradients a particularly tricky task. +%Here, in analogy to the random-phase approximation (RPA) formalism, the ground-state BSE energy is calculated via the ``trace'' formula (see below). +%The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy. +%This definition of the energy has the advantage of treating at the same level of theory the ground state and the excited states. +%Embracing this definition, the purpose of the present study is to investigate the quality of ground- and excited-state PES near equilibrium obtained within BSE +%The location of the minima on the ground- and (singlet and triplet) excited-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. +%There are very few studies about the ground-state BSE energy for atomic and molecular systems \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} and, to the best of our knowledge, the present study is the first to investigate the topology of the excited-state PES at the BSE level. -\xavier{ALTERNATIVE PARAGRAPH: Contrary to TD-DFT, the ground-state correlation energy calculated at the BSE level 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 formulation, was 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{Holzer_2018} Such a modified BSE polarization propagator was inspired by a previous study of the homogeneous interacting electron gaz.} \cite{Maggio_2016} +Contrary to TD-DFT, the ground-state correlation energy calculated at the BSE level 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, evaluated within the adiabatic connection formulation, was 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{Holzer_2018} Such a modified BSE polarization propagator was inspired by a previous study of the homogeneous interacting electron gas. \cite{Maggio_2016} -\xavier{Here, in analogy to the random-phase approximation (RPA) formalism, \cite{Furche_2008} the ground-state BSE energy is calculated via the ``trace'' formula (see below). The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy. +Here, in analogy to the random-phase approximation (RPA) formalism, \cite{Furche_2008} the ground-state BSE energy is calculated via the ``trace'' formula (see below). The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy. This definition of the energy has the advantage of treating at the same level of theory the ground state and the excited states. Embracing this definition, the purpose of the present study is to investigate the quality of ground- and excited-state PES near equilibrium obtained within the BSE approach. The location of the minima on the ground- and (singlet and triplet) excited-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. The paper is organized as follows. In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.