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 $S^\spf= S_{\up\dw}^\spf+ S_{\dw\up}^\spf= O_\up V_\dw+ O_\dw V_\up$.
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 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.
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}
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)
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 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
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
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 $\bB{}{}=\bO$.
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).
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}].
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
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
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
Like its TD-DFT cousin, 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}
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.
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
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:
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
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.
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_2020e,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:
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
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.
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.
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}.
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_2020,Loos_2020e}.
%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}.
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).}
