From 816334d694a0050d3ab09ef44412aeba08dda1d9 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 6 Dec 2020 22:54:28 +0100 Subject: [PATCH] more work on the intro --- Manuscript/sfBSE.bib | 14 +++++++++++++- Manuscript/sfBSE.tex | 23 +++++++++++++++-------- 2 files changed, 28 insertions(+), 9 deletions(-) diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib index 88481e4..ed2ef35 100644 --- a/Manuscript/sfBSE.bib +++ b/Manuscript/sfBSE.bib @@ -1,13 +1,25 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-12-06 15:06:52 +0100 +%% Created for Pierre-Francois Loos at 2020-12-06 22:23:42 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Liu_2020, + author = {C. Liu and J. Kloppenburg and Y. Yao and X. Ren and H. Appel and Y. Kanai and V. Blum}, + date-added = {2020-12-06 22:23:41 +0100}, + date-modified = {2020-12-06 22:23:41 +0100}, + doi = {10.1063/1.5123290}, + journal = {J. Chem. Phys.}, + pages = {044105}, + title = {All-electron ab initio Bethe-Salpeter equation approach to neutral excitations in molecules with numeric atom-centered orbitals}, + volume = {152}, + year = {2020}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5123290}} + @article{Krylov_2008, abstract = { The equation-of-motion coupled-cluster (EOM-CC) approach is a versatile electronic-structure tool that allows one to describe a variety of multiconfigurational wave functions within single-reference formalism. This review provides a guide to established EOM methods illustrated by examples that demonstrate the types of target states currently accessible by EOM. It focuses on applications of EOM-CC to electronically excited and open-shell species. The examples emphasize EOM's advantages for selected situations often perceived as multireference cases [e.g., interacting states of different nature, Jahn-Teller (JT) and pseudo-JT states, dense manifolds of ionized states, diradicals, and triradicals]. I also discuss limitations and caveats and offer practical solutions to some problematic situations. The review also touches on some formal aspects of the theory and important current developments. }, author = {Krylov, Anna I.}, diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 856487f..0db0ba3 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -44,19 +44,26 @@ Accurately predicting ground- and excited-state energies (hence excitation energ An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a,Casanova_2020} The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new excited-state methodologies, their main goal being to get the most accurate excitation energies (and properties) at the lowest possible computational cost in the most general context. \cite{Loos_2020a} -Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996} the Bethe-Salpeter equation (BSE) formalism within the static approximation \cite{Salpeter_1951,Strinati_1988} is plagued by the lack of double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019} +Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} and popularized in condensed-matter physics, \cite{Sham_1966,Strinati_1984,Delerue_2000} one of the new emerging method in the computational chemistry landscape is the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999,Blase_2018,Blase_2020} from many-body perturbation theory (MBPT) \cite{Onida_2002,Martin_2016} which, based on an underlying $GW$ calculation to compute accurate charged excitations, \cite{Hedin_1965,Golze_2019} is able to provide accurate optical (\ie, neutral) excitations for molecular systems at a rather modest computational cost.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020} + +Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit. +Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996} the static BSE formalism is plagued by the lack of double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019} +Indeed both adiabatic TD-DFT and static BSE can only access single excitation with respect to the reference determinant. One way to access double excitations is via the spin-flip formalism established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002} with earlier attempts by Bethe, \cite{Bethe_1931} as well as Shibuya and McKoy. \cite{Shibuya_1970} -Nowadays, spin-flip techniques are widely available for many types of methods such as equation-of-motion coupled cluster (EOM-CC), \cite{Krylov_2001a} configuration interaction (CI), \cite{Krylov_2001b} TD-DFT, \cite{Shao_2003,Wang_2004,Li_2011a,Bernard_2012,Zhang_2015} and others \cite{Krylov_2002,Levchenko_2004,Manohar_2008,Casanova_2008,Casanova_2009a,Casanova_2009b,Mayhall_2014a,Mayhall_2014b,Bell_2013,Dutta_2013,Mayhall_2014c,Lefrancois_2015,Mato_2018} with successful applications in bond breaking processes, \cite{Golubeva_2007} radical chemistry, \cite{Slipchenko_2002,Wang_2005,Slipchenko_2003,Rinkevicius_2010,Ibeji_2015,Hossain_2017,Orms_2018,Luxon_2018} and the photochemistry of conical intersections \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} to mention a few. +The idea behind the spin-flip formalism is rather simple: instead of starting the calculation from the singlet ground state, one can start from the lowest triplet state. +In such a way, one can access the singlet ground state and the doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these excitation energies providing an estimate of the double excitation where one promotes two electrons from the singlet ground state. +One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way. +Moreover, spin-flip methods are usually hampered by spin-contamination. + +Nowadays, spin-flip techniques are widely available for many types of methods such as equation-of-motion coupled cluster (EOM-CC), \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} configuration interaction (CI), \cite{Krylov_2001b,Krylov_2002,Mato_2018,Casanova_2008,Casanova_2009b} TD-DFT, \cite{Shao_2003,Wang_2004,Li_2011a,Bernard_2012,Zhang_2015} algebraic-diagrammatic construction (ACD),\cite{Lefrancois_2015} and others \cite{Mayhall_2014a,Mayhall_2014b,Bell_2013,Mayhall_2014c} with successful applications in bond breaking processes, \cite{Golubeva_2007} radical chemistry, \cite{Slipchenko_2002,Wang_2005,Slipchenko_2003,Rinkevicius_2010,Ibeji_2015,Hossain_2017,Orms_2018,Luxon_2018} and the photochemistry of conical intersections \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} to mention a few. We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a more detailed review of spin-flip methods. -The idea behind the spin-flip formalism is quite simple: instead of starting the calculation from the singlet ground state, one can start from the lowest triplet state. -With a similar idea. -One of the main issue of spin-flip methods is the spin contamination. Here we apply the spin-flip formalism to the BSE formalism in order to access, in particular, double excitations. -We also go beyond the static approximation \cite{Strinati_1982,Strinati_1984,Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019} -Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} +The BSE calculations will be based on the spin unrestricted version of $GW$ +To the best of our knowledge, the present study is the first to apply spin-flip formalism to the BSE method. +Moreover, we also go beyond the static approximation and takes into account dynamical effects via our recently developed perturbative method which builds on the seminal work of Strinati, \cite{Strinati_1982,Strinati_1984,Strinati_1988} Romaniello and collaborators, \cite{Romaniello_2009b,Sangalli_2011} and Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Lettmann_2019} Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript. @@ -337,7 +344,7 @@ Within the $GW$ approximation, the BSE kernel is \\ - \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) \end{multline} -where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980, Strinati_1982, Strinati_1984, Strinati_1988} +where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980,Strinati_1982,Strinati_1984,Strinati_1988} Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip BSE optical excitations are obtained by solving the usual Casida-like linear response (eigen)problem: \begin{equation} \label{eq:LR-BSE}