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\begin{document}	

\title{Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism}

\author{Enzo \surname{Monino}}
	\affiliation{\LCPQ}	 
\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\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.
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.
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\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Due to the ubiquitous influence of processes involving electronic excited states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. 
Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community.
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_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.
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} 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}
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}
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. 
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 (\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.
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}
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}).
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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Unrestricted $GW$ formalism}
\label{sec:UGW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons, respectively) and $N$ one-electron basis functions.
The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming no linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
In a spin-flip (sf) excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
In the following, we assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
Moreover, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.

%================================
\subsection{The dynamical screening}
%================================
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook}
The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016a}
\begin{equation}
\label{eq:G}
	G^{\sig}(\br_1,\br_2;\omega) 
	= \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta}	
	+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta}	
\end{equation}
where $\eta$ is a positive infinitesimal.
As readily seen in Eq.~\eqref{eq:G}, the Green's function can be evaluated at different levels of theory depending on the choice of orbitals and energies, $\MO{p_\sig}$ and $\e{p_\sig}{}$. 
For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with KS orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$. 
Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below). \cite{Hedin_1965,Golze_2019}

Based on the spin-up and spin-down components of $G$ defined in Eq.~\eqref{eq:G}, one can easily compute the non-interacting polarizability (which is a sum over spins)
\begin{equation}
\label{eq:chi0}
	\chi_0(\br_1,\br_2;\omega) = - \frac{i}{2\pi} \sum_\sig \int G^{\sig}(\br_1,\br_2;\omega+\omega') G^{\sig}(\br_1,\br_2;\omega') d\omega'
\end{equation}
and subsequently the dielectric function
\begin{equation}
\label{eq:eps}
	\epsilon(\br_1,\br_2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
where $\delta(\br)$ is the Dirac delta function.
Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
\begin{equation}
\label{eq:W}
	W(\br_1,\br_2;\omega) =  \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates.

Within the $GW$ formalism, \cite{Hedin_1965,Onida_2002,Golze_2019} the dynamical screening is computed at the random-phase approximation (RPA) level by considering only the manifold of the spin-conserved neutral excitations.
In the orbital basis, the spectral representation of $W$ is
\begin{multline}
\label{eq:W_spectral}
	W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} 
	+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} 
	\\
	\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
\end{multline}
where the bare two-electron integrals are \cite{Gill_1994}
\begin{equation}
		\ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \int  \frac{\MO{p_\sig}(\br_1) \MO{q_\tau}(\br_1) \MO{r_\sigp}(\br_2) \MO{s_\taup}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation}
and the screened two-electron integrals (or spectral weights) are explicitly given by
\begin{equation}
\label{eq:sERI}
		\ERI{p_\sig q_\sig}{m} = \sum_{ia\sigp} \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{i_\sigp a_\sigp}
\end{equation}
In Eqs.~\eqref{eq:W_spectral} and \eqref{eq:sERI}, the spin-conserved RPA neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors, $\bX{m}{\spc,\RPA}$ and $\bY{m}{\spc,\RPA}$, are obtained by solving a linear response system of the form
\begin{equation}
\label{eq:LR-RPA}
	\begin{pmatrix}
		\bA{}{}	&	\bB{}{}	\\
		-\bB{}{}	&	-\bA{}{}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{}	\\
		\bY{m}{}	\\
	\end{pmatrix}
	=
	\Om{m}{}
	\begin{pmatrix}
		\bX{m}{}	\\
		\bY{m}{}	\\
	\end{pmatrix}
\end{equation}
where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold. 
The spin structure of these matrices, though, is general 
\begin{subequations}
\begin{align}
\label{eq:LR-RPA-AB-sc}
	\bA{}{\spc} & = \begin{pmatrix}
		\bA{}{\upup,\upup}	&	\bA{}{\upup,\dwdw}	\\
		\bA{}{\dwdw,\upup}	&	\bA{}{\dwdw,\dwdw}	\\
	\end{pmatrix}
	&
	\bB{}{\spc} & = \begin{pmatrix}
		\bB{}{\upup,\upup}	&	\bB{}{\upup,\dwdw}	\\
		\bB{}{\dwdw,\upup}	&	\bB{}{\dwdw,\dwdw}	\\
	\end{pmatrix}
\\
\label{eq:LR-RPA-AB-sf}
	\bA{}{\spf} & = \begin{pmatrix}
		\bA{}{\updw,\updw}	&	\bO	\\
		\bO					&	\bA{}{\dwup,\dwup}	\\
	\end{pmatrix}
	&
	\bB{}{\spf} & = \begin{pmatrix}
		\bO					&	\bB{}{\updw,\dwup}	\\
		\bB{}{\dwup,\updw}	&	\bO	\\
	\end{pmatrix}
\end{align}
\end{subequations}
In the absence of instabilities, the linear eigenvalue problem \eqref{eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $\pm \Om{m}{}$.
In such a case, $(\bA{}{}-\bB{}{})^{1/2}$ is positive definite, and Eq.~\eqref{eq:LR-RPA} can be recast as a Hermitian problem of half its original dimension 
\begin{equation}
\label{eq:small-LR}
	(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{} = \bOm{2} \cdot \bZ{}{}
\end{equation}
where the excitation amplitudes are
\begin{equation}
	\bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{}
\end{equation}
Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consists in setting $\bB{}{} = \bO$.
In such a case, Eq.~\eqref{eq:LR-RPA} reduces to a straightforward Hermitian problem of the form:
\begin{equation}
	\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}
\end{equation}
Note that, for spin-flip excitations, it is quite common to enforce the TDA especially when one considers a triplet reference as the first ``excited-state'' is usually the ground state of the closed-shell system (hence, corresponding to a negative excitation energy).

At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-A}
	\A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp}  \delta_{\tau \taup} (\e{a_\tau} - \e{i_\sig}) + \ERI{i_\sig a_\tau}{b_\sigp j_\taup}
	\\ 
	\label{eq:LR_RPA-B}
	\B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup}
\end{align}
\end{subequations}
from which we obtain the following expressions
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-Asc}
	\A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp} (\e{a_\sig} - \e{i_\sig}) + \ERI{i_\sig a_\sig}{b_\sigp j_\sigp}
	\\ 
	\label{eq:LR_RPA-Bsc}
	\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \ERI{i_\sig a_\sig}{j_\sigp b_\sigp}
\end{align}
\end{subequations}
for the spin-conserved excitations and 
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-Asf}
	\A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} & = \delta_{ij} \delta_{ab} (\e{a_\bsig} - \e{i_\sig})
	\\ 
	\label{eq:LR_RPA-Bsf}
	\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\RPA} & = 0
\end{align}
\end{subequations}
for the spin-flip excitations.

%================================
\subsection{The $GW$ self-energy}
%================================
Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy
\begin{equation}
\label{eq:Sig}
\begin{split}
	\Sig{}{\xc,\sig}(\br_1,\br_2;\omega) 
	& = \Sig{}{\x,\sig}(\br_1,\br_2) + \Sig{}{\co,\sig}(\br_1,\br_2;\omega) 
	\\
	& = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
\end{split}
\end{equation}
is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
\begin{subequations}
\begin{gather}
	\Sig{p_\sig q_\sig}{\x}
	= - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig}	 
	\\
\begin{split}
	\Sig{p_\sig q_\sig}{\co}(\omega) 
	& = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta}
	\\
	& + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta}			 
\end{split}
\end{gather}
\end{subequations}
where the self-energy has been split in its exchange (x) and correlation (c) contributions.
The Dyson equation linking the Green's function and the self-energy holds separately for each spin component 
\begin{equation}
\label{eq:Dyson_G}
\begin{split}
	\qty[ G^{\sig}(\br_1,\br_2;\omega) ]^{-1} 
	& = \qty[ G_{\KS}^{\sig}(\br_1,\br_2;\omega) ]^{-1} 
	\\
	& + \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) - v^{\xc}(\br_1) \delta(\br_1 - \br_2)
\end{split}
\end{equation}
where $v^{\xc}(\br)$ is the KS (local) exchange-correlation potential.
The target quantities here are the quasiparticle energies $\eGW{p_\sig}$, \ie, the poles of $G$ [see Eq.~\eqref{eq:G}], which correspond to well-defined addition/removal energies (unlike the KS orbital energies).
Because the exchange-correlation part of the self-energy is, itself, constructed with the Green's function [see Eq.~\eqref{eq:Sig}], the present process is, by nature, self-consistent.
The same comment applies to the dynamically-screened Coulomb potential $W$ entering the definition of $\Sig{}{\xc}$ [see Eq.~\eqref{eq:Sig}] which is also constructed from $G$ [see Eqs.~\eqref{eq:chi0}, \eqref{eq:eps}, and \eqref{eq:W}].

%================================
\subsection{Level of self-consistency}
%================================
This is where $GW$ schemes differ.
In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} a single iteration is performed, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation}
\label{eq:QP-eq}
	\omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc} 
\end{equation}
where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the KS level, and
\begin{equation}
	V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br 
\end{equation}
Because, from a practical point of view, one is usually interested by the so-called quasiparticle solution (or peak), the quasiparticle equation \eqref{eq:QP-eq} is often linearized around $\omega = \e{p_\sig}^{\KS}$, yielding
\begin{equation}
    \eGW{p_\sig}
    = \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ]
\end{equation}
where
\begin{equation}
\label{eq:Z_GW}
	Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1}
\end{equation}
is a renormalization factor (with $0 \le Z_{p_\sig} \le 1$) which also represents the spectral weight of the quasiparticle solution.
In addition to the principal quasiparticle peak which, in a well-behaved case, contains most of the spectral weight, the frequency-dependent quasiparticle equation \eqref{eq:QP-eq} generates a finite number of satellite resonances with smaller weights.

Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018}  several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level).

Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, both the one-electron energies and the orbitals are updated until convergence is reached.
These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as
\begin{equation}
	\Tilde{\Sigma}_{p_\sig q_\sig}^{\xc} = \frac{1}{2} \qty[ \Sig{p_\sig q_\sig}{\xc}(\e{p_\sig}{}) + \Sig{q_\sig p_\sig}{\xc}(\e{p_\sig}{}) ]
\end{equation}

%================================
\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} 
Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
In a nutshell, BSE builds on top of a $GW$ calculation by adding up excitonic effects (\ie, the electron-hole binding energy) to the $GW$ fundamental gap which is itself a corrected version of the KS gap. 
The purpose of the underlying $GW$ calculation is to provide quasiparticle energies and a dynamically-screened Coulomb potential that are used to build the BSE Hamiltonian from which the vertical excitations of the system are extracted.

%================================
\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
\begin{multline}
	L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) 
	= L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
	\\
	+ \int L_{0}^{\sig\sigp}(\br_1,\br_4;\br_1',\br_3;\omega) 
	\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)
	\\
	\times L^{\sig\sigp}(\br_6,\br_2;\br_5,\br_2';\omega)
	d\br_3 d\br_4 d\br_5 d\br_6
\end{multline}
where 
\begin{multline}
	L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) 
	\\
	= \frac{1}{2\pi} \int G^{\sig}(\br_1,\br_2';\omega+\omega') G^{\sig}(\br_1',\br_2;\omega') d\omega'
\end{multline}
is the non-interacting analog of the two-particle correlation function $L$.

Within the $GW$ approximation, the BSE kernel is
\begin{multline}
	i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega) 
	= \frac{\delta(\br_3 - \br_4) \delta(\br_5 - \br_6) }{\abs{\br_3-\br_6}} 
	\\
	- \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}
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}
	\begin{pmatrix}
		\bA{}{\BSE}		&	\bB{}{\BSE}	\\
		-\bB{}{\BSE}	&	-\bA{}{\BSE}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{\BSE}	\\
		\bY{m}{\BSE}	\\
	\end{pmatrix}
	=
	\Om{m}{\BSE}
	\begin{pmatrix}
		\bX{m}{\BSE}	\\
		\bY{m}{\BSE}	\\
	\end{pmatrix}
\end{equation}
Defining the elements of the static screening as $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, the general expressions of the BSE matrix elements are
\begin{subequations}
\begin{align}
	\label{eq:LR_BSE-A}
	\A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau}
	\\ 
	\label{eq:LR_BSE-B}
	\B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau}
\end{align}
\end{subequations}
from which we obtain the following expressions for the spin-conserved and spin-flip BSE excitations: 
\begin{subequations}
\begin{align}
	\label{eq:LR_BSE-Asc}
	\A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\sigp a_\sig}
	\\ 
	\label{eq:LR_BSE-Bsc}
	\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\sigp,j_\sigp a_\sig}
	\\
	\label{eq:LR_BSE-Asf}
	\A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\BSE} & = \A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} - W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}
	\\ 
	\label{eq:LR_BSE-Bsf}
	\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\BSE} & = - W^{\stat}_{i_\sig b_\sig,j_\bsig a_\bsig}
\end{align}
\end{subequations}

At this stage, it is of particular interest to discuss the form of the spin-flip matrix elements defined in Eqs.~\eqref{eq:LR_BSE-Asf} and \eqref{eq:LR_BSE-Bsf}.
As readily seen from Eq.~\eqref{eq:LR_RPA-Asf}, at the RPA level, the spin-flip excitations are given by the difference of one-electron energies, hence missing out on key exchange and correlation effects.
This is also the case at the TD-DFT level when one relies on (semi-)local functionals.
This explains why most of spin-flip TD-DFT calculations are performed with hybrid functionals containing a substantial amount of Hartree-Fock exchange as only the exact exchange integral of the form $\ERI{i_\sig j_\sig}{b_\bsig a_\bsig}$ survive spin-symmetry requirements.
At the BSE level, these matrix elements are, of course, also present thanks to the contribution of $W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}$ as evidenced in Eq.~\eqref{eq:W_spectral} but it also includes correlation effects.

%================================
\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}
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.

Our implementation follows closely the work of Rohlfing and co-workers \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} in which they computed the dynamical correction in the TDA and plasmon-pole approximation. 
However, our scheme goes beyond the plasmon-pole approximation as the spectral representation of the dynamically-screened Coulomb potential is computed exactly at the RPA level consistently with the underlying $GW$ calculation:
\begin{multline}
	\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} 
	+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} 
	\\
	\times \Bigg[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} 
	\\
	+ \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg]
\end{multline}
The dBSE non-linear response problem is
\begin{multline}
\label{eq:LR-dyn}
	\begin{pmatrix}
		\bA{}{\dBSE}(\Om{m}{\dBSE})		&	\bB{}{\dBSE}(\Om{m}{\dBSE})	
		\\
		-\bB{}{\dBSE}(-\Om{m}{\dBSE})	&	-\bA{}{\dBSE}(-\Om{m}{\dBSE})	
		\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{\dBSE}	\\
		\bY{m}{\dBSE}	\\
	\end{pmatrix}
	\\
	=
	\Om{m}{\dBSE}
	\begin{pmatrix}
		\bX{m}{\dBSE}	\\
		\bY{m}{\dBSE}	\\
	\end{pmatrix}
\end{multline}
where the dynamical matrices are generally defined as
\begin{subequations}
\begin{align}
	\label{eq:LR_dBSE-A}
	\A{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega) 
	\\ 
	\label{eq:LR_dBSE-B}
	\B{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) 
\end{align}
\end{subequations}
from which one can easily obtained the matrix elements for the spin-conserved and spin-flip manifolds similarly to Eqs.~\eqref{eq:LR_BSE-Asc}, \eqref{eq:LR_BSE-Bsc}, \eqref{eq:LR_BSE-Asf}, and \eqref{eq:LR_BSE-Bsf}. 
Following Rayleigh-Schr\"odinger perturbation theory, we then decompose the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (\ie, linear) reference and a first-order dynamic (\ie, non-linear) perturbation such that
\begin{multline}
\label{eq:LR-PT}
	\begin{pmatrix}
		\bA{}{\dBSE}(\omega)		&	\bB{}{\dBSE}(\omega)	\\
		-\bB{}{\dBSE}(-\omega)	&	-\bA{}{\dBSE}(-\omega)	\\
	\end{pmatrix}
	\\
	=
	\begin{pmatrix}
		\bA{}{(0)}	&	\bB{}{(0)}	
		\\
		-\bB{}{(0)}	&	-\bA{}{(0)}	
		\\
	\end{pmatrix}
	+
	\begin{pmatrix}
		\bA{}{(1)}(\omega)		&	\bB{}{(1)}(\omega)	\\
		-\bB{}{(1)}(-\omega)	&	-\bA{}{(1)}(-\omega)	\\
	\end{pmatrix}
\end{multline}
with
\begin{subequations}
\begin{align}
	\label{eq:BSE-A0}
	\A{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
	\\ 
	\label{eq:BSE-B0}
	\B{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
\end{align}
\end{subequations}
and 
\begin{subequations}
\begin{align}
	\label{eq:BSE-A1}
	\A{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau}
	\\ 
	\label{eq:BSE-B1}
	\B{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau}
\end{align}
\end{subequations}
The dBSE excitation energies are then obtained via
\begin{equation}
	\Om{m}{\dBSE} = \Om{m}{\BSE} + \zeta_{m} \Om{m}{(1)}
\end{equation}
where $\Om{m}{\BSE} \equiv \Om{m}{(0)}$ are the static (zeroth-order) BSE excitation energies obtained by solving Eq.~\eqref{eq:LR-BSE}, and  
\begin{equation}
\label{eq:Om1-TDA}
	\Om{m}{(1)} = \T{(\bX{m}{\BSE})} \cdot \bA{}{(1)}(\Om{m}{\BSE}) \cdot \bX{m}{\BSE}
\end{equation}
are first-order corrections (with $\bX{m}{\BSE} \equiv \bX{m}{(0)}$) obtained within the dynamical TDA (dTDA) with the renormalization factor
\begin{equation}
\label{eq:Z}
	\zeta_{m} = \qty[ 1 - \T{(\bX{m}{\BSE})} \cdot \left. \pdv{\bA{}{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{\BSE}} \cdot \bX{m}{\BSE} ]^{-1}
\end{equation}
which, unlike the $GW$ case [see Eq.~\eqref{eq:Z_GW}], is not restricted to be between $0$ and $1$.
In most cases, the value of $\zeta_{m}$ is close to unity which indicates that the perturbative expansion behaves nicely.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Oscillator strengths}
\label{sec:os}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the spin-conserved transitions, the $x$ component of the transition dipole moment is
\begin{equation}
	\mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig}
\end{equation}
where 
\begin{equation}
	(p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br
\end{equation}
are one-electron integrals in the orbital basis.
The total oscillator strength is given by
\begin{equation}
	f_{m}^{\spc} = \frac{2}{3} \Om{m}{\spc} \qty[ \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 ]
\end{equation}
For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix elements $(i_\sig|x|a_\bsig)$ vanish via integration over the spin coordinate.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin contamination}
\label{sec:spin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of the key issues of linear response formalism based on unrestricted references is spin contamination.
As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for a high-spin triplets, and ii) spin-contamination of the excited states due to spin incompleteness of the configuration interaction expansion.
The latter issue is an important source of spin contamination in the present context as BSE is limited to single excitations with respect to the reference configuration.
Specific schemes have been developed to palliate these shortcomings and we refer the interested reader to Ref.~\onlinecite{Casanova_2020} for a detailed discussion on this matter.

In order to monitor closely how contaminated are these states, we compute
\begin{equation}
	\expval{\hS^2}_m = \expval{\hS^2}_0 + \Delta \expval{\hS^2}_m
\end{equation}
where  
\begin{equation}
	\expval{\hS^2}_{0} 
	= \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 )
	+ n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2
\end{equation}
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}
\label{eq:OV}
	(p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br
\end{equation}
are overlap integrals between spin-up and spin-down orbitals.

For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the $\bX{m}{}$ and $\bY{m}{}$ vectors and the orbital overlaps defined in Eq.~\eqref{eq:OV}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the systems under investigation here, we consider either an open-shell doublet or triplet reference state.
We then adopt the unrestricted formalism throughout this work.
The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (unrestricted) UHF starting point.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a,Hybertsen_1986,vanSetten_2013} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
%Note that, for the present (small) molecular systems, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies and fundamental gap. 
%Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
%In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The dynamical correction is computed in the TDA throughout.
As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions. 
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
%It is important to mention that the small molecular systems considered here are particularly challenging for the BSE formalism, \cite{Hirose_2015,Loos_2018b} which is known to work best for larger systems where the amount of screening is more important. \cite{Jacquemin_2017b,Rangel_2017}

%For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted. 
%Various statistical quantities are reported in the following: the mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), and the maximum positive [Max($+$)] and maximum negative [Max($-$)] errors.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
	\caption{
	Spin-flip excitations (in  eV) of \ce{Be} obtained for various methods with the 6-31G basis.
	The $GW$ calculations are performed with a HF starting point.
	\label{tab:Be}}
	\begin{ruledtabular}
		\begin{tabular}{lcccccccccccc}
			State			&	TD-BLYP		&	TD-B3LYP	&	TD-BHHLYP		&	CIS		
							&	BSE@{\GOWO}	&	BSE@{\evGW}	&	dBSE@{\GOWO}	&	dBSE@{\evGW}	
							&	ADC(2)		&	ADC(2)-x	&	ADC(3)			&	FCI	\\
			\hline
			$^3P(1s^22s2p)$ &	3.210	&	3.332 &	2.874 &	2.111	&	2.399&	2.407&	2.363&	2.369 & 2.433	&	2.866 & 2.863	&	2.862	\\
			$^1P(1s^22s2p)$	&	3.210	&	4.275&	4.922&	6.036	&	6.191&	6.199 &	6.263 &	 6.273 & 6.255	&6.581	& 6.579	&	6.577	\\
			$^3P(1s^22p^2)$	&	6.691	&	6.864&	7.112 &	7.480	&	7.792 &	7.788 &	7.824 &	 7.820 & 7.745	&	7.664 & 7.658 &	7.669	\\
			$^1P(1s^22p^2)$	&	7.598	&	7.762&	 8.188 &	8.945	&9.373	&	9.388 &	9.424 & 9.441 & 9.047 & 8.612 & 8.618 &	8.624	\\
		\end{tabular}
	\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%

%%% FIG 1 %%%
\begin{figure*}
	\includegraphics[width=0.7\linewidth]{Be}
	\caption{
	Spin-flip excitation energies (in eV) of \ce{Be} obtained for various methods with the 6-31G basis.
	\label{fig:Be}}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions. 
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article and its supplementary material.

%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{sfBSE}
%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
