done with 1st draft of intro
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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2020-12-09 10:12:07 +0100
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%% Created for Pierre-Francois Loos at 2020-12-09 14:38:47 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@ -5469,22 +5469,6 @@
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volume = {72},
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year = {2005}}
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@article{Ball_2017,
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author = {Ball, Caleb J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
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date-added = {2020-01-01 21:36:51 +0100},
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date-modified = {2020-01-01 21:36:51 +0100},
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doi = {10.1039/C6CP06801D},
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file = {/Users/loos/Zotero/storage/7X5YE6WH/52.pdf},
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issn = {1463-9076, 1463-9084},
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journal = {Phys. Chem. Chem. Phys.},
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language = {en},
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number = {5},
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pages = {3987-3998},
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title = {Molecular Electronic Structure in One-Dimensional {{Coulomb}} Systems},
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volume = {19},
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year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1039/C6CP06801D}}
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@article{Bande_2006,
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author = {Bande, Annika and L{\"u}chow, Arne and Della Sala, Fabio and G{\"o}rling, Andreas},
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date-added = {2020-01-01 21:36:51 +0100},
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@ -44,17 +44,16 @@ Accurately predicting ground- and excited-state energies (hence excitation energ
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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}
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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}
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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}
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Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
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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}
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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.
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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}
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Indeed, both adiabatic TD-DFT and static BSE can only access (singlet and triplet) single excitations with respect to the reference determinant.
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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.
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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}
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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.
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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.
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We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a more detailed review of spin-flip methods.
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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}
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We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a detailed review of spin-flip methods.
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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}
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One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way.
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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.
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@ -62,13 +61,13 @@ This issue can be alleviated by increasing the excitation order at a significant
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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.
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Here we apply the spin-flip formalism to the BSE formalism in order to access, in particular, double excitations.
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The BSE calculations will be based on the spin unrestricted version of $GW$
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To the best of our knowledge, the present study is the first to apply spin-flip formalism to the BSE method.
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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}
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Here we apply the spin-flip technique to the BSE formalism in order to access, in particular, double excitations. \cite{Authier_2020}
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The present BSE calculations are based on the spin unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
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To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method.
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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}
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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.
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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}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -310,6 +309,7 @@ These are obtained via the diagonalization of an effective Fock matrix which inc
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%================================
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\section{Unrestricted Bethe-Salpeter equation formalism}
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\label{sec:UBSE}
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%================================
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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}
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Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
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@ -318,9 +318,9 @@ The purpose of the underlying $GW$ calculation is to provide quasiparticle energ
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%================================
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\subsection{Static approximation}
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\label{sec:BSE}
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%================================
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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
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\begin{multline}
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L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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@ -402,6 +402,7 @@ At the BSE level, these matrix elements are, of course, also present thanks to t
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%================================
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\subsection{Dynamical correction}
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\label{sec:dBSE}
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%================================
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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}
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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.
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@ -545,7 +546,7 @@ where
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= \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 )
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+ n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2
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\end{equation}
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is the expectation value of $\hS^2$ for the reference configuration, the first term correspoding to the exact value of $\expval{\hS^2}$, and
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is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval{\hS^2}$, and
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\begin{equation}
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\label{eq:OV}
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(p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br
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