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\affiliation{\LCPQ}
\begin{abstract}
Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe-Salpeter equation (BSE) formalism in its static approximation is ``blind'' to double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes.
Here, we apply the spin-flip technique (which consists in considering the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism of many-body perturbation theory in order to access double excitations.
Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is ``blind'' to double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes.
Here, we apply the spin-flip \textit{ansatz} (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access double excitations.
The present scheme is based on a spin-unrestricted version of the $GW$ approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations.
Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment.
The performance of the present spin-flip BSE formalism is illustrated by computing the vertical excitation energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene.
The performance of the present spin-flip BSE formalism is illustrated by computing the vertical excitation energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.
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@ -48,25 +48,26 @@ 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_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}
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 and the dynamically-screened Coulomb potential, \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_2020e}
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 \cite{Onida_2002,Martin_2016} which, based on an underlying $GW$ calculation to compute accurate charged excitations and the dynamically-screened Coulomb potential, \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_2020e}
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.
Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996,UlrichBook} 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} or the ground state of open-shell molecules, \cite{Casida_2005,Huix-Rotllant_2011,Loos_2020f} and can be a real challenge to accurately predict even with state-of-the-art methods, \cite{Loos_2018a,Loos_2019,Loos_2020c,Loos_2020d,Veril_2020} like the approximate third-order coupled-cluster (CC3) method \cite{Christiansen_1995b,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996,UlrichBook} 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} or the ground state of open-shell molecules. \cite{Casida_2005,Huix-Rotllant_2011,Loos_2020f}
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.
Double excitations are even challenging for state-of-the-art methods, \cite{Loos_2018a,Loos_2019,Loos_2020c,Loos_2020d,Veril_2020} like the approximate third-order coupled-cluster (CC3) method \cite{Christiansen_1995b,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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 the spin-flip \textit{ansatz} 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.
We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for detailed reviews on 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 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.
Moreover, spin-flip methods are usually hampered by spin-contamination \cite{Casanova_2020} (\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. \cite{Krylov_2000b}
Moreover, spin-flip methods are usually hampered by spin contamination \cite{Casanova_2020} (\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. \cite{Krylov_2000b}
This issue can be alleviated by increasing the excitation order at a significant cost or by selectively complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
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 photochemistry in general \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} but not only.
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 present BSE calculations are based on the spin-unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
To the best of our knowledge, the present study is the first to apply the 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}
We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).