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Manuscript/BSE_JPCL.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-04 20:26:27 +0200
%% Created for Pierre-Francois Loos at 2020-06-04 22:08:23 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -959,21 +959,6 @@
Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevB.37.10159},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.37.10159}}
@article{Golze_2019rev,
Author = {Golze, Dorothea and Dvorak, Marc and Rinke, Patrick},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.3389/fchem.2019.00377},
Issn = {2296-2646},
Journal = {Front. Chem.},
Pages = {377},
Title = {The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy},
Url = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
Volume = {7},
Year = {2019},
Bdsk-Url-1 = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
Bdsk-Url-2 = {https://doi.org/10.3389/fchem.2019.00377}}
@article{Gui_2018,
Author = {Gui, Xin and Holzer, Christof and Klopper, Wim},
Date-Added = {2020-05-18 21:40:28 +0200},
@ -12349,18 +12334,6 @@
Volume = {49},
Year = {1977}}
@article{Schone_1998a,
Author = {Sch{\"o}ne, Wolf-Dieter and Eguiluz, Adolfo G.},
Date-Added = {2018-03-03 16:36:07 +0000},
Date-Modified = {2018-03-03 16:36:07 +0000},
File = {/Users/loos/Zotero/storage/J4XS8Z9Q/Schone_1998.pdf},
Journal = {Phys. Rev. Lett.},
Number = {8},
Pages = {1662},
Title = {Self-Consistent Calculations of Quasiparticle States in Metals and Semiconductors},
Volume = {81},
Year = {1998}}
@article{Pulay_1980,
Author = {Pulay, P{\'e}ter},
Date-Added = {2018-02-25 19:37:56 +0000},
@ -13252,6 +13225,7 @@
@article{Leng_2016,
Author = {Leng, Xia and Jin, Fan and Wei, Min and Ma, Yuchen},
Date-Modified = {2020-06-04 21:37:16 +0200},
Doi = {10.1002/wcms.1265},
Issn = {17590876},
Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.},
@ -13260,7 +13234,7 @@
Number = {5},
Pages = {532--550},
Shorttitle = {{{GW}} Method and {{Bethe}}-{{Salpeter}} Equation for Calculating Electronic Excitations},
Title = {{{GW}} Method and {{Bethe}}-{{Salpeter}} Equation for Calculating Electronic Excitations: {{GW}} Method and {{Bethe}}-{{Salpeter}} Equation},
Title = {{{GW}} Method and {{Bethe}}-{{Salpeter}} Equation for Calculating Electronic Excitations},
Volume = {6},
Year = {2016},
Bdsk-Url-1 = {https://dx.doi.org/10.1002/wcms.1265}}

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Manuscript/BSE_JPCL.tex
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@ -17,9 +17,7 @@
\usepackage{pifont}
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{braket}
\usepackage{multirow}
\usepackage{threeparttable}
\usepackage{xspace}
\usepackage{verbatim}
\usepackage[version=4]{mhchem} % Formula subscripts using \ce{}
@ -231,15 +229,16 @@ Simulations are so realistic that they predict the outcome of traditional experi
Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016} \xavier{[Xav: Good ref on theory for photocatalysis still needed.]}
The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Dreuw_2005} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
\\
%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%
The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques, \cite{Dreuw_2015} originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018} in quantum chemistry.
While the one-body density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015}
% originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018}
While the one-body density stands as the basic variable in density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
\begin{equation}
G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel},
\end{equation}
@ -258,7 +257,8 @@ A central property of the one-body Green's function is that its frequency-depend
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 $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
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 Green's function poles $\lbrace \varepsilon_s \rbrace$ 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, not only that associated with frontier orbitals.
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, not only that associated with frontier orbitals.
Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
\begin{equation}\label{eq:Gmotion}
@ -268,7 +268,7 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
where we introduce the usual composite index, \eg, $1 \equiv (\bx_1 t_1)$.
Here, $\delta$ is Dirac's delta function, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}],
dropping spin-variables for simplicity, one gets the familiar eigenvalue equation, \ie,
dropping spin variables for simplicity, one gets the familiar eigenvalue equation, \ie,
\begin{equation}
h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
\end{equation}
@ -284,7 +284,7 @@ The knowledge of $\Sigma$ allows to access the true addition/removal energies, n
While the equations reported above are formally exact, it remains to provide an expression for the xc self-energy operator $\Sigma$.
This is where Green's function practical theories differ.
Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Reining_2017,Golze_2019} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
The resulting equation, when compared with the equation for the time-evolution of $G$ [see Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
\begin{equation}\label{eq:Sig}
\Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
@ -308,9 +308,9 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
\caption{
Hedin's pentagon connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$ (gray arrows).
As input, one must provide KS (or HF) orbitals and their corresponding senergies.
As input, one must provide KS (or HF) orbitals and their corresponding energies.
Depending on the level of self-consistency of the $GW$ calculation, only the orbital energies or both the orbitals and their energies are corrected.
As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$ which can then be used to compute the BSE neutral excitations.
As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE neutral excitations.
\label{fig:pentagon}}
\end{figure}
%%% %%% %%%
@ -337,10 +337,10 @@ Because, one is usually interested only by the quasiparticle solution, in practi
Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique.
This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988}
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_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 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}.
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.
Further improvements may be obtained via self-consistency of the Hedin's equations (see Fig.~\ref{fig:pentagon}).
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,Shishkin_2007,Blase_2011,Faber_2011}
where the quasiparticle energies are updated at each iteration, and
@ -349,14 +349,14 @@ where one updates both the quasiparticle energies and the corresponding orbitals
Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006,Stan_2009,Rostgaard_2010, Caruso_2013,Caruso_2013a,Koval_2014,Wilhelm_2018,Loos_2018}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017,Loos_2018}
A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998,Holm_1999,Holm_2000,Garcia-Gonzalez_2001}
These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2013,Caruso_2013a,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999}
However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}
However, remaining a low-order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}
\\
%===================================
@ -374,7 +374,7 @@ The equation of motion for $G$ [see Eq.~\eqref{eq:Gmotion}] can be reformulated
\begin{equation}
G = G_0 + G_0 ( v_H + U + \Sigma ) G,
\end{equation}
that relates the full (interacting) Green's function, $G$, to its non-interacting version, $G_0$, with $v_H$ and $U$ the Hartree and external potential, respectively, and $\Sigma$ the xc self-energy.
that relates the full (interacting) Green's function, $G$, to its non-interacting version, $G_0$, where $v_H$ and $U$ are the Hartree and external potentials, respectively.
The derivative with respect to $U$ of this Dyson equation yields
\begin{multline}\label{eq:DysonL}
L(1,2;1',2') = L_0(1,2;1',2') +
@ -383,7 +383,7 @@ The derivative with respect to $U$ of this Dyson equation yields
\end{multline}
where $L_0(1,2;1',2') = G(1,2')G(2,1')$ is the non-interacting 4-point susceptibility and
\begin{equation}
i\Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) + i \pdv{\Sigma(3,4)}{G(6,5)}
i\,\Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) + i \pdv{\Sigma(3,4)}{G(6,5)}
\end{equation}
is the so-called BSE kernel.
This equation can be compared to its TD-DFT analog
@ -395,12 +395,14 @@ where
\Xi^{\DFT}(3,4) = v(3,4) + \pdv{V^{\XC}(3)}{\rho(4)}
\end{equation}
is the TD-DFT kernel.
Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{equation}\label{eq:BSEkernel}
i \Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
\end{equation}
where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. \cite{Hanke_1980,Strinati_1982,Strinati_1984}
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT:
Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that we label BSE@$GW$, leads to an approximate version of the BSE kernel
\begin{multline}\label{eq:BSEkernel}
i\,\Xi^{\BSE}(3,5;4,6)
\\
= v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
\end{multline}
where it is customary to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. \cite{Hanke_1980,Strinati_1982,Strinati_1984}
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied spatial orbitals and $(a,b)$ are unoccupied spatial orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT: \cite{Casida_1995}
\begin{equation} \label{eq:BSE-eigen}
\begin{pmatrix}
R & C
@ -445,16 +447,13 @@ $(ia|jb)$ bare Coulomb term defined as
\begin{equation}
(ai|bj) = \iint d\br d\br'
\phi_i(\br) \phi_a(\br) v(\br-\br')
\phi_j(\br') \phi_b(\br'),
\phi_j(\br') \phi_b(\br').
\end{equation}
Neglecting the coupling term $C$ between the resonant term $R$ and anti-resonant term $-R^*$ in Eq.~\eqref{eq:BSE-eigen}, leads to the well-known Tamm-Dancoff approximation.
As compared to TD-DFT,
\begin{itemize}
\item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues
\item the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
\end{itemize}
We emphasise that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations.
This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, emphasizing its pros and cons.
As compared to TD-DFT, i) the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues, and ii) the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
We emphasize that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations.
This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its pros and cons.
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%