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Manuscript/BSE_JPCL.tex
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@ -201,7 +201,7 @@
\begin{abstract}
The many-body Green's function Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the ensemble of computational methods available to chemists in order to predict optical excitations in molecular systems.
In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable ionization energies and electron affinities, and the Bethe-Salpeter formalism, able to catch excitonic effects, has shown to provide accurate singlet excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable ionization energies and electron affinities, and the BSE formalism, able to catch excitonic effects, has shown to provide accurate singlet excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
With a similar computational cost as time-dependent density-functional theory (TD-DFT), the BSE formalism is then able to provide an accuracy on par with the most accurate global and range-separated hybrid functionals without the unsettling choice of the exchange-correlation functional, resolving further known issues (\textit{e.g.}, charge-transfer excitations) and offering a well-defined path to dynamical kernels.
In this \textit{Perspective} article, we provide a historical overview of the BSE formalism, with a particular focus on its condensed-matter roots.
We also propose a critical review of its strengths and weaknesses in different chemical situations.
@ -246,12 +246,13 @@ where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function.
The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
For $t > t'$, $G$ provides the amplitude of probability of finding, on top of the ground-state Fermi sea (\ie, higher in energy than the highest-occupied energy level, also known as Fermi level), an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of an electron hole (often simply called a hole) is monitored.
\xavier{ This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", more commonly labeled the electronic energy levels, as obtained e.g. in a direct or inverse photo-emission experiment where an electron is ejected or added to the N-electron system. In particular, and as compared to Kohn-Sham (KS) eigenvalues, the Green's function formalism offers in a first step a more rigorous path to the ionization potential, the electronic affinity and the experimental (photoemission) fundamental gap:
\begin{equation}
This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", \titou{more commonly labeled as electronic energy levels}, as obtained, \eg, in a direct or inverse photo-emission experiment where an electron is ejected or added to the $N$-electron system.
In particular, and as opposed to Kohn-Sham (KS) DFT, the Green's function formalism offers a more rigorous and systematically improvable path for the obtention of the ionization potential $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$, the electronic affinity $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$, and the experimental (photoemission) fundamental gap
\begin{equation}\label{eq:IPAEgap}
\EgFun = I^\Nel - A^\Nel,
\label{eq:IPAEgap}
\end{equation}
where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system. Since these energy levels obtained thanks to Green's function MBPT serve as an input to Bethe-Salpeter calculations, we start by discussing them in some details. }
of the $\Nel$-electron system, where $E_0^\Nel$ corresponds to its ground-state energy.
Since these energy levels are key input quantities for the subsequent BSE calculation, we start by discussing these in some details.
\\
%===================================
@ -262,7 +263,7 @@ A central property of the one-body Green's function is that its frequency-depend
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) },
\end{equation}
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{\Nel+1} - E_0^{\Nel}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{\Nel} - E_s^{\Nel-1}$ for $\varepsilon_s < \mu$.
Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{th} excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{th} excited state of the $\Nel$-electron system.
The $f_s$'s are the so-called Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinant many-body wave functions (see below).
Unlike Kohn-Sham (KS) eigenvalues, the poles of the Green's function $\lbrace \varepsilon_s \rbrace$ are proper addition/removal energies of the $\Nel$-electron system, leading to well-defined ionization potentials and electronic affinities.
Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission spectroscopy, not only that associated with frontier orbitals.
@ -329,8 +330,12 @@ Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V
\omega = \varepsilon_p^{\KS} +
\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\omega) - V^{\XC}}{\phi_p^{\KS}}.
\end{equation}
As a non-linear equation, the self-consistent quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with spectral weights $Z(\omega) = [ 1- \partial \Sigma^{\GW} / \partial {\omega} ]^{-1}$ taken at $\omega = \varepsilon_{p,s}^{\GW} }$. The existence of a well defined quasiparticle energy requires a solution with a large Z-factor, namely close to unity, a condition not always fulfilled for states far away from the gap.
\cite{Veril_2018}
As a non-linear equation, the self-consistent quasiparticle equation \eqref{eq:QP-eq} has various solutions associated with different spectral weights.
The existence of a well defined quasiparticle energy requires a solution with a large spectral weight, \ie, close to unity, a condition not always fulfilled for states far away from the fundamental gap. \cite{Veril_2018}
%As a non-linear equation, the self-consistent quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with different spectral weights $Z(\omega) = [ 1- \partial \Sigma^{\GW} / \partial {\omega} ]^{-1}$ evaluated at $\omega = \varepsilon_{p,s}^{\GW}$.
%The existence of a well defined quasiparticle energy requires a solution with a large spectral weight, namely close to unity, a condition not always fulfilled for states far away from the fundamental gap.
%\cite{Veril_2018}
%\begin{equation}
@ -352,7 +357,10 @@ This simple scheme was used in the early $GW$ studies of extended semiconductors
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the local-density approximation (LDA) KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV with a computational cost scaling quartically with the system size (see below). A compilation of data for $G_0W_0$ applied to extended inorganic semiconductors can be found in Ref.~\citenum{Shishkin_2007}.
Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature. Namely, the quasiparticle energies, and in particular the HOMO-LUMO gap, depends on the input Kohn-Sham eigenvalues. Tuning of the starting point functionals or self-consistency are two difference approaches to improve on this problem. We will comment on this below when addressing Bethe-Salpeter optical excitations.
Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature.
For example, the quasiparticle energies, and in particular the HOMO-LUMO gap, depends on the input KS eigenvalues.
Tuning the starting point functional or applying a self-consistent $GW$ scheme are two different approaches commonly employed to improve on this problem.
We will comment further on this particular point below when addressing the quality of the BSE optical excitations.
%Further improvements may be obtained via self-consistency of Hedin's equations (see Fig.~\ref{fig:pentagon}).
%There exists two main types of self-consistent $GW$ methods:
%i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986} where the quasiparticle energies are updated at each iteration, and
@ -456,13 +464,13 @@ where $m$ indexes the electronic excitations.
The $\lbrace \phi_{i/a} \rbrace$ are typically the input (KS) eigenstates used to build the $GW$ self-energy.
They are here taken to be real in the case of finite-size systems.
%(In the case of complex orbitals, we refer the reader to Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation.)
The resonant and coupling parts of the BSE Hamiltonian read
In the case of a closed-shell singlet ground state, the resonant and coupling parts of the BSE Hamiltonian read
\begin{gather}
R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab},
\\
C_{ai,bj} = \kappa (ia|bj) - W_{ib,aj},
\end{gather}
with $\kappa=2,0$ for singlets/triplets and
with $\kappa=2$ or $0$ if one targets singlet or triplet excited states (respectively), and
\begin{equation}\label{eq:Wmatel}
W_{ij,ab} = \iint d\br d\br'
\phi_i(\br) \phi_j(\br) W(\br,\br'; \omega=0)
@ -470,7 +478,7 @@ with $\kappa=2,0$ for singlets/triplets and
\end{equation}
where we notice that the two occupied (virtual) eigenstates are taken at the same position of space, in contrast with the $(ia|jb)$ bare Coulomb term defined as
\begin{equation}
(ai|bj) = \iint d\br d\br'
(ia|jb) = \iint d\br d\br'
\phi_i(\br) \phi_a(\br) v(\br-\br')
\phi_j(\br') \phi_b(\br').
\end{equation}
@ -497,10 +505,7 @@ An important conclusion drawn from these calculations was that the quality of th
\begin{equation}
\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
\end{equation}
with the experimental (photoemission) fundamental gap
of Eq.~\ref{eq:IPAEgap} (see Fig.~\ref{fig:gaps}).
with the experimental (photoemission) fundamental gap defined in Eq.~\eqref{eq:IPAEgap}.
Standard $G_0W_0$ calculations starting with KS eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input KS gap
\begin{equation}
@ -656,7 +661,7 @@ From a more practical point of view, dynamical effects have been found to affect
Thanks to first-order perturbation theory, Rohlfing and coworkers have developed an efficient way of taking into account the dynamical effects via a plasmon-pole approximation combined with TDA. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
Studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pi^*$ transitions, but for the lowest $n \ra \pi^*$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Zhang \textit{et al.}~have studied the frequency-dependent second-order BSE kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016}
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within TDA with a renormalization of the first-order perturbative correction.
\\