From 5365979f32dae07a6956609b601d5d2e941cf055 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 26 May 2020 22:47:22 +0200 Subject: [PATCH] saving work --- BSEdyn.tex | 202 ++++++++++++++++++++++------------------------------- 1 file changed, 84 insertions(+), 118 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 1a48b80..890c98d 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -106,6 +106,7 @@ \newcommand{\XiBSE}[1]{\Xi_{#1}} \newcommand{\Po}[1]{P_{#1}} \newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}} \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} @@ -368,10 +369,10 @@ Dropping the (space/spin) variables, the Fourier components with respect to $t_1 with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation, \ie, \begin{equation} \label{eq:G-Lehman} - G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }, + G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \MO{p}(\bx_1) \MO{p}^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }, \end{equation} where $\mu$ is the chemical potential. -The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\phi_p$'s are their associated one-body (spin)orbitals. +The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\MO{p}$'s are their associated one-body (spin)orbitals. %where the $\eps_{p}$'s are proper addition/removal energies such that %\begin{equation} % e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}, @@ -379,12 +380,12 @@ The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper %$\hH$ being the exact many-body Hamiltonian. In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. %\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)} -Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Oms )$ onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields +Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Oms )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields \begin{multline} - \iint d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms) + \iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms) \\ = - \frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta } + \frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta } \qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms \tau_{34}) } ]. \end{multline} % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. @@ -395,7 +396,7 @@ For example, we have \begin{multline} \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s} \\ - = - \qty( e^{ -i \Omega_s \tau^{56} } ) \sum_{pq} \phi_p(\bx_6) \phi_q^*(\bx_5) + = - \qty( e^{ -i \Omega_s \tau^{56} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) \mel{N}{\ha_q^{\dagger} \ha_p}{N,s} \\ \times \qty[ \theta( \tau_{56} ) e^{- i ( \e{p} - \hOms ) \tau_{56} } + \theta( - \tau_{56} ) e^{ - i ( \e{q} + \hOms) \tau_{56} } ] @@ -419,12 +420,12 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal %Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). %Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$, -Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE): +Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE): \begin{equation} \label{eq:BSE-final} \begin{split} ( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s} - & + \sum_{jb} \qty[ (ia|jb) - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\ - & + \sum_{jb} \qty[ (ia|bj) - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s} + & + \sum_{jb} \qty[ \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\ + & + \sum_{jb} \qty[ \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s} = 0, \end{split} \end{equation} @@ -432,9 +433,9 @@ with $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb}^{s} = \mel{N Neglecting the term $Y_{jb}^{s}$ in the dBSE, which is much smaller than $X_{jb}^{s}$, leads to the well-known Tamm-Dancoff approximation (TDA). In Eq.~\eqref{eq:BSE-final}, \begin{equation} - (pq|rs) = \iint d\br d\br' \, \phi_p^*(\br) \phi_q(\br) v(\br -\br') \phi_r^*(\br') \phi_s(\br'), + \ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'), \end{equation} -are the bare two-electron integrals, $\phi_p(\br)$ is a spatial orbital, and +are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and \begin{multline} \widetilde{W}_{ij,ab}(\Oms) = \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) @@ -444,124 +445,26 @@ are the bare two-electron integrals, $\phi_p(\br)$ is a spatial orbital, and is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Oms - ( \eGW{q} - \eGW{p} )$ and \begin{equation} W_{pq,rs}({\omega}) - = \iint d\br d\br' \, \phi_p(\br) \phi_q^*(\br) W(\br ,\br'; \omega) \phi_r^*(\br') \phi_s(\br'). + = \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br'). \end{equation} \xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : } -\titou{T2: I propose to move what's below in the next section. What do you think Xavier?} -In the present study, we consider the exact spectral representation of $W(\omega)$ at the RPA level, which reads +In the present study, we consider the exact spectral representation of $W(\omega)$ at the random-phase approximation (RPA) level, which reads \begin{multline} +\label{eq:W} W_{ij,ab}(\omega) - = (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] + = \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m} \\ \times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ] \end{multline} where \begin{equation} \label{eq:sERI} - \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia} + \sERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia} \end{equation} are the spectral weights. -Because $\Om{m}{\RPA} > 0$, we have -\begin{multline} - \widetilde{W}_{ij,ab}( \Oms ) - = (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] - \\ - \times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ) -\end{multline} -\titou{Due to excitonic effects, the lowest BSE excitation energy, ${\Omega}_1$, stands lower than the lowest RPA excitation energy, $\Omega_m^{RPA}$, so that -e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. -Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g. -$$ -\abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} } < \frac{1}{ \Omega_m^{RPA}} -$$ -This leads to reduced electron-hole screening, namely larger electron-hole stabilising binding energy, as compared to the standard adiabatic BSE, leading to smaller (red-shifted) excitation energies. } - - -%In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016} -%\begin{multline} -%\label{eq:BSE} -% \LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2') -% \\ -% + \int d3 d4 d5 d6 \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2') -%\end{multline} -%as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential -%\begin{equation} -% \XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)}, -%\end{equation} -%which takes into account the self-consistent variation of the Hartree potential -%\begin{equation} -% \vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+), -%\end{equation} -%(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$. -%In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables. -%In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have -%\begin{equation} -% \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2), -%\end{equation} -%where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to -%\begin{equation} -% \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4), -%\end{equation} -%where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} -%Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. - -%================================ -\subsection{Perturbative dynamical correction} -%================================= - -For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988} -\begin{equation} -\label{eq:LR-dyn} - \begin{pmatrix} - \bA{}(\omega) & \bB{}(\omega) \\ - -\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\ - \end{pmatrix} - \cdot - \begin{pmatrix} - \bX{m}{}(\omega) \\ - \bY{m}{}(\omega) \\ - \end{pmatrix} - = - \omega - \begin{pmatrix} - \bX{m}{}(\omega) \\ - \bY{m}{}(\omega) \\ - \end{pmatrix}, -\end{equation} -where the dynamical matrices $\bA{}(\omega)$ and $\bB{}(\omega)$ are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}(\omega)$, and $\bY{m}{}(\omega)$ are (eigen)vectors of length $\Nocc \Nvir$. -In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. -Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $m$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016} - -The BSE matrix elements read -\begin{subequations} -\begin{align} - \label{eq:BSE-Adyn} - \A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \sigma \ERI{ia}{jb} - \W{ij,ab}{}(\omega), - \\ - \label{eq:BSE-Bdyn} - \B{ia,jb}{}(\omega) & = 2 \sigma \ERI{ia}{bj} - \W{ib,aj}{}(\omega), -\end{align} -\end{subequations} -where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively), -\begin{equation} - \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}' -\end{equation} -are the bare two-electron integrals in the molecular orbital basis $\lbrace \MO{p}(\br{}) \rbrace_{1 \le p \le \Norb}$, and the dynamically-screened Coulomb potential reads -\begin{multline} -\label{eq:W} - \W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m} - \\ - \times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}), -\end{multline} -where $\eta$ is a positive infinitesimal, and -\begin{equation} -\label{eq:sERI} - \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia} -\end{equation} -are the spectral weights. -In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem +In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem \begin{equation} \label{eq:LR-RPA} \begin{pmatrix} @@ -591,6 +494,69 @@ with \end{align} \end{subequations} where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$. +Because $\Om{m}{\RPA} > 0$, we have +\begin{multline} + \widetilde{W}_{ij,ab}( \Oms ) + = \ERI{ij}{ab} + 2 \sum_m^{OV} \sERI{ij}{m} \sERI{ab}{m} + \\ + \times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ) +\end{multline} +\titou{Due to excitonic effects, the lowest BSE excitation energy, ${\Omega}_1$, stands lower than the lowest RPA excitation energy, $\Omega_m^{RPA}$, so that +e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. +Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g. +$$ +\abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} } < \frac{1}{ \Omega_m^{RPA}} +$$ +This leads to reduced electron-hole screening, namely larger electron-hole stabilising binding energy, as compared to the standard adiabatic BSE, leading to smaller (red-shifted) excitation energies. } + +%================================ +\subsection{Perturbative dynamical correction} +%================================= + +For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988} +\begin{equation} +\label{eq:LR-dyn} + \begin{pmatrix} + \bA{}(\omega) & \bB{}(\omega) \\ + -\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{}(\omega) \\ + \bY{m}{}(\omega) \\ + \end{pmatrix} + = + \omega + \begin{pmatrix} + \bX{m}{}(\omega) \\ + \bY{m}{}(\omega) \\ + \end{pmatrix}, +\end{equation} +where the dynamical matrices $\bA{}(\omega)$ and $\bB{}(\omega)$ are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}(\omega)$, and $\bY{m}{}(\omega)$ are (eigen)vectors of length $\Nocc \Nvir$. +In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. +Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $m$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016} + +The BSE matrix elements read +\begin{subequations} +\begin{align} + \label{eq:BSE-Adyn} + \A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \sigma \ERI{ia}{jb} - \tW{ij,ab}{}(\omega), + \\ + \label{eq:BSE-Bdyn} + \B{ia,jb}{}(\omega) & = 2 \sigma \ERI{ia}{bj} - \tW{ib,aj}{}(\omega), +\end{align} +\end{subequations} +where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, and $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively). +%\begin{equation} +% \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}' +%\end{equation} +%are the bare two-electron integrals in the molecular orbital basis $\lbrace \MO{p}(\br{}) \rbrace_{1 \le p \le \Norb}$, and the dynamically-screened Coulomb potential reads +%\begin{multline} +%\label{eq:W} +% \W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m} +% \\ +% \times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}). +%\end{multline} Now, let us decompose, using basic perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that \begin{multline} @@ -625,16 +591,16 @@ and \begin{subequations} \begin{align} \label{eq:BSE-A1} - \A{ia,jb}{(1)}(\omega) & = - \W{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}}, + \A{ia,jb}{(1)}(\omega) & = - \tW{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}}, \\ \label{eq:BSE-B1} - \B{ia,jb}{(1)}(\omega) & = - \W{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}}, + \B{ia,jb}{(1)}(\omega) & = - \tW{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}}, \end{align} \end{subequations} where we have defined the static version of the screened Coulomb potential \begin{equation} \label{eq:Wstat} - \W{ij,ab}{\text{stat}} = \ERI{ij}{ab} - 4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}. + \W{ij,ab}{\text{stat}} = W_{ij,ab}(\omega = 0) = \ERI{ij}{ab} - 4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}. \end{equation} According to perturbation theory, the $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as \begin{subequations}