From 675cd395661c08d05e4a260022de4b684b13336f Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 7 Jan 2020 18:14:39 +0100 Subject: [PATCH] Intro OK --- BSE-PES.bib | 5 +++-- BSE-PES.tex | 14 ++++---------- 2 files changed, 7 insertions(+), 12 deletions(-) diff --git a/BSE-PES.bib b/BSE-PES.bib index 7f9f45f..c2b8436 100644 --- a/BSE-PES.bib +++ b/BSE-PES.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-01-07 16:02:18 +0100 +%% Created for Pierre-Francois Loos at 2020-01-07 18:13:25 +0100 %% Saved with string encoding Unicode (UTF-8) @@ -90,12 +90,13 @@ @article{Fukuta_1964, Author = {N. Fukuta and F. Iwamoto and K. Sawada}, Date-Added = {2020-01-04 20:13:22 +0100}, - Date-Modified = {2020-01-04 20:14:17 +0100}, + Date-Modified = {2020-01-07 18:13:24 +0100}, Doi = {10.1103/PhysRev.135.A932}, Journal = {Phys. Rev.}, Pages = {A932}, Title = {Linearized Many-Body Problem}, Volume = {135}, + Year = {1964}, Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.135.A932}} @article{Rowe_1968, diff --git a/BSE-PES.tex b/BSE-PES.tex index a8bd1dc..6ba01f1 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -216,17 +216,11 @@ However, similar to TD-DFT, the static version of BSE cannot describe multiple e 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, 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} +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 (AC-BSE), 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 AC-BSE correlation 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.