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Manuscript/sfBSE.bib
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@ -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.},

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Manuscript/sfBSE.tex
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@ -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}