Similarly to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
Here, going beyond the static approximation, we compute molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
The present correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
The Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is to the $GW$ approximation \cite{Hedin_1965,Golze_2019} of many-body perturbation theory (MBPT) \cite{Martin_2016} what time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995} is to Kohn-Sham density-functional theory (KS-DFT), \cite{Hohenberg_1964,Kohn_1965} an affordable way of computing the neutral excitations of a given electronic system.
In recent years, it has shown to be useful for molecular systems with a large number of systematic benchmark studies appearing in the scientific literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).
Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
where $\EgFun= I^\Nel- A^\Nel$ is the the fundamental gap, \cite{Bredas_2014}$I^\Nel= E_0^{\Nel-1}- E_0^\Nel$ and $A^\Nel= E_0^{\Nel+1}- E_0^\Nel$ being the ionization potential and the electron affinity of the $\Nel$-electron system.
Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
One key consequence of this approximation is that double (and higher) excitations are completely absent from the BSE spectra.
Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007}
Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Nonetheless, it is worth mentioning the seminal work of Strinati, \cite{Strinati_1988} who \titou{bla bla bla.}
Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009}
With such as scheme, they have been able to compute the excited state of biological chromophores, showing that taking into account the electron-hole screening is important for an accurate description of the lowest $n \rightarrow\pi^*$ excitations. \cite{Ma_2009}
Zhang \text{et al.}\cite{Zhang_2013}, as well as Rebolini and Toulouse \cite{Rebolini_2016} (in a range-separated context) have separately studied the frequency-dependent second-order Bethe-Salpeter kernel showing a modest improvement over its static counterpart.
In the two latter studies, they also followed a perturbative approach within the TDA with a renormalization of the first-order perturbative correction.
It is important to note that, although these studies are clearly going beyond the static approximation of BSE, they are not able to recover double excitations as the perturbative treatment makes ultimately the BSE kernel static.
However, it does permit to recover additional relaxation effects coming from the higher excitations which would be present by ``unfolding'' the dynamical BSE kernel in order to recover a linear eigenvalue problem.
Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional excitations by solving the non-linear, frequency-dependent eigenvalue problem.
However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
In the present study, we extend the work of Rohlfing and coworkers \cite{Rohlfing_2000,Ma_2009} by proposing a renormalized first-order perturbative correction to the static neutral excitation energy.
Importantly, our correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
Moreover, we investigate quantitatively the effect of the TDA by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
i^2 G_2(1,2;1',2') &= \langle N | T {\hat\psi}(1) {\hat\psi}(2) {\hat\psi}^{\dagger}(2') {\hat\psi}^{\dagger}(1') | N \rangle
\end{align*}
where e.g. $1=(x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $ |N,s \rangle$ of the N-electron excited state with $| N \rangle= | N,0\rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing
The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+i \Oms t_2}$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by ${\tilde\chi}_s(x_2,x_{2'})$ on both side of the Bethe-Salpeter equation, we are left with the search of the $e^{-i \Oms t_1}$ Fourier component associated with the right-hand side of the BSE. For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, the $L_0(1,2;1',2')$ term cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionisation potential.
The $GW$ quasiparticle energies $\varepsilon_{n/m}^{GW}$ are good approximations to such removal/addition energies. Selecting (n,m)=(j,b) yields the largest components
$B_{jb}^{s}=\langle N | {\hat a}_b^{\dagger}{\hat a}_j | N,s \rangle$ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ weights leads to the Tamm Dancoff approximation (TDA). Working out the same expansion for $\langle N | T {\hat\psi}(5){\hat\psi}^{\dagger}(5) | N,s \rangle$ and $\langle N | T {\hat\psi}(x_1){\hat\psi}^{\dagger}(x_{1'}) | N,s \rangle$, and projecting onto $\phi_a^*(x_1)\phi_i(x_{1'})$,
one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (dBSE) :
\textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, 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.
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}
%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
%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, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
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}
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}
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
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
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$GW$.
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
In terms of computational consideration, because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum, we perform a complete diagonalization of the $\Nocc\Nvir\times\Nocc\Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3\Nvir^3}=\order{\Norb^6}$ computational cost.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%
The restricted HF formalism has been systematically employed in the present study.
All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent ev$GW$ calculations are employed as starting points to compute the BSE neutral excitations.
Within $GW$, all orbitals are corrected.
In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
For ev$GW$, the convergence creiterion has been set to $10^{-5}$.
Further details about our implementation of {\GOWO} and evGW can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
For comparison purposes, we employ the reference excitation energies and geometries from Ref.~\onlinecite{Loos_2018a} also computed the PES at the second-order M{\o}ller-Plesset perturbation theory (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b}
All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018b,Veril_2018}
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}
