PTEROSOR/sfBSE
10
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

done with 1st draft of intro

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-12-09 14:45:52 +01:00
parent aac7bf87fb
commit a9ad96aa70
2 changed files with 17 additions and 32 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

18
Manuscript/sfBSE.bib
View File

@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-12-09 10:12:07 +0100 %% Created for Pierre-Francois Loos at 2020-12-09 14:38:47 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -5469,22 +5469,6 @@
volume = {72}, volume = {72},
year = {2005}} year = {2005}}
@article{Ball_2017,
author = {Ball, Caleb J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:51 +0100},
doi = {10.1039/C6CP06801D},
file = {/Users/loos/Zotero/storage/7X5YE6WH/52.pdf},
issn = {1463-9076, 1463-9084},
journal = {Phys. Chem. Chem. Phys.},
language = {en},
number = {5},
pages = {3987-3998},
title = {Molecular Electronic Structure in One-Dimensional {{Coulomb}} Systems},
volume = {19},
year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1039/C6CP06801D}}
@article{Bande_2006, @article{Bande_2006,
author = {Bande, Annika and L{\"u}chow, Arne and Della Sala, Fabio and G{\"o}rling, Andreas}, author = {Bande, Annika and L{\"u}chow, Arne and Della Sala, Fabio and G{\"o}rling, Andreas},
date-added = {2020-01-01 21:36:51 +0100}, date-added = {2020-01-01 21:36:51 +0100},

31
Manuscript/sfBSE.tex
View File

@ -44,17 +44,16 @@ 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_2020d,Casanova_2020} 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_2020d,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_2020d} 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_2020d}
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} 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,Holzer_2018a,Holzer_2018b,Loos_2020}
Most of BSE implementations rely on the so-called static approximation, \cite{Blase_2018,Bruneval_2016,Krause_2017,Liu_2020} which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
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} 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 (singlet and triplet) single excitations with respect to the reference determinant. Indeed, both adiabatic TD-DFT \cite{Levine_2006,Tozer_2000,Elliott_2011,Maitra_2012,Maitra_2016} and static BSE \cite{ReiningBook,Romaniello_2009b,Sangalli_2011,Loos_2020h,Authier_2020} can only access (singlet and triplet) single excitations with respect to the reference determinant usually taken as the closed-shell singlet ground state.
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} 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}
The idea behind spin-flip is rather simple: instead of considering the singlet ground state as reference, the reference is taken as the lowest triplet state. The idea behind spin-flip is rather simple: instead of considering the singlet ground state as reference, the reference is taken as the lowest triplet state.
In such a way, one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation. In such a way, one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation.
We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a more detailed review of spin-flip methods. We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a detailed review of spin-flip methods.
Note that a similar idea has been exploited by the group of Weito Yang to access double excitations in the context of particle-particle random-phase approximation. \cite{Peng_2013,Yang_2013b,Yang_2014a,Peng_2014,Zhang_2016,Sutton_2018} Note that a similar idea has been exploited by the group of Weito Yang to access double excitations in the context of the particle-particle random-phase approximation. \cite{Peng_2013,Yang_2013b,Yang_2014a,Peng_2014,Zhang_2016,Sutton_2018}
One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way. 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 (\ie, artificial mixing with configurations of different spin multiplicities) due to spin incompleteness of the configuration interaction expansion as well as the possible spin-contamination of the reference configuration. Moreover, spin-flip methods are usually hampered by spin-contamination (\ie, artificial mixing with configurations of different spin multiplicities) due to spin incompleteness of the configuration interaction expansion as well as the possible spin-contamination of the reference configuration.
@ -62,13 +61,13 @@ This issue can be alleviated by increasing the excitation order at a significant
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} the algebraic-diagrammatic construction (ADC) scheme,\cite{Lefrancois_2015,Lefrancois_2016} 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. 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} the algebraic-diagrammatic construction (ADC) scheme,\cite{Lefrancois_2015,Lefrancois_2016} 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.
Here we apply the spin-flip technique to the BSE formalism in order to access, in particular, double excitations. \cite{Authier_2020}
Here we apply the spin-flip formalism to the BSE formalism in order to access, in particular, double excitations. The present BSE calculations are based on the spin unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
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 the spin-flip formalism to the BSE method.
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 by taking into account dynamical effects (Sec.~\ref{sec:dBSE}) via an unrestricted generalization of our recently developed (renormalized) perturbative correction 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}
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} The computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}) is also discussed.
Computational details are reported in Sec.~\ref{sec:compdet} and our results for the beryllium atom \ce{Be}, the hydrogen molecule \ce{H2}, and cyclobutadiene \ce{C4H4} are discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript. Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -310,6 +309,7 @@ These are obtained via the diagonalization of an effective Fock matrix which inc
%================================ %================================
\section{Unrestricted Bethe-Salpeter equation formalism} \section{Unrestricted Bethe-Salpeter equation formalism}
\label{sec:UBSE}
%================================ %================================
Like its TD-DFT cousin, \cite{Runge_1984,Casida_1995,Petersilka_1996,Dreuw_2005} the BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \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} Like its TD-DFT cousin, \cite{Runge_1984,Casida_1995,Petersilka_1996,Dreuw_2005} the BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \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}
Using the BSE formalism, one can access the spin-conserved and spin-flip excitations. Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
@ -318,9 +318,9 @@ The purpose of the underlying $GW$ calculation is to provide quasiparticle energ
%================================ %================================
\subsection{Static approximation} \subsection{Static approximation}
\label{sec:BSE}
%================================ %================================
The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is
\begin{multline} \begin{multline}
L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
@ -402,6 +402,7 @@ At the BSE level, these matrix elements are, of course, also present thanks to t
%================================ %================================
\subsection{Dynamical correction} \subsection{Dynamical correction}
\label{sec:dBSE}
%================================ %================================
In order to go beyond the ubiquitous static approximation of BSE \cite{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} (which is somehow similar to the adiabatic approximation of TD-DFT \cite{Casida_2005,Huix-Rotllant_2011,Casida_2016,Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}), we have recently implemented, following Strinati's seminal work \cite{Strinati_1982,Strinati_1984,Strinati_1988} (see also the work of Romaniello \textit{et al.} \cite{Romaniello_2009b} and Sangalli \textit{et al.} \cite{Sangalli_2011}), a renormalized first-order perturbative correction in order to take into consideration the dynamical nature of the screened Coulomb potential $W$. \cite{Loos_2020h,Authier_2020} In order to go beyond the ubiquitous static approximation of BSE \cite{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} (which is somehow similar to the adiabatic approximation of TD-DFT \cite{Casida_2005,Huix-Rotllant_2011,Casida_2016,Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}), we have recently implemented, following Strinati's seminal work \cite{Strinati_1982,Strinati_1984,Strinati_1988} (see also the work of Romaniello \textit{et al.} \cite{Romaniello_2009b} and Sangalli \textit{et al.} \cite{Sangalli_2011}), a renormalized first-order perturbative correction in order to take into consideration the dynamical nature of the screened Coulomb potential $W$. \cite{Loos_2020h,Authier_2020}
This dynamical correction to the static BSE kernel (dubbed as dBSE in the following) does permit to recover additional relaxation effects coming from higher excitations. This dynamical correction to the static BSE kernel (dubbed as dBSE in the following) does permit to recover additional relaxation effects coming from higher excitations.
@ -545,7 +546,7 @@ where
= \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 ) = \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 )
+ n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2 + n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2
\end{equation} \end{equation}
is the expectation value of $\hS^2$ for the reference configuration, the first term correspoding to the exact value of $\expval{\hS^2}$, and is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval{\hS^2}$, and
\begin{equation} \begin{equation}
\label{eq:OV} \label{eq:OV}
(p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br (p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br