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Manuscript/BSE_JPCL.bib
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@ -1,14 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-08 22:10:57 +0200
%% Created for Denis Jacquemin at 2020-06-08 11:44:16 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Packer_1996,
Author = {Packer, M. K. and Dalskov, E. K. and Enevoldsen, T. and Jensen, H. J. and Oddershede, J.},
Date-Added = {2020-06-08 21:57:16 +0200},
Date-Modified = {2020-06-08 22:10:55 +0200},
Doi = {10.1063/1.472430},
Journal = {J. Chem. Phys.},
Pages = {5886--5900},
Title = {A New Implementation of the Second-Order Polarization Propagator Approximation (SOPPA): The Excitation Spectra of Benzene and Naphthalene},
Volume = {105},
Year = {1996}}
@article{Wu_2019, @article{Wu_2019,
Author = {Xin‐Ping Wu and Indrani Choudhuri and Donald G. Truhlar}, Author = {Xin‐Ping Wu and Indrani Choudhuri and Donald G. Truhlar},
Date-Added = {2020-06-05 20:35:01 +0200}, Date-Added = {2020-06-05 20:35:01 +0200},

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Manuscript/BSE_JPCL.tex
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@ -228,7 +228,7 @@ Future directions of developments and improvements are also discussed.
In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to Karplus, Levitt, and Warshel, the Royal Swedish Academy of Sciences concluded by stating \textit{``Today the computer is just as important a tool for chemists as the test tube. In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to Karplus, Levitt, and Warshel, the Royal Swedish Academy of Sciences concluded by stating \textit{``Today the computer is just as important a tool for chemists as the test tube.
Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003} Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003}
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. 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,Wu_2019} The study of optical excitations (also known as neutral 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. \cite{Kippelen_2009,Improta_2016,Wu_2019}
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 TD-DFT 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 TD-DFT issues. \cite{Blase_2018}
\\ \\
@ -236,21 +236,20 @@ The present \textit{Perspective} aims at describing the current status and upcom
\section{Theory} \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,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} \hl{je parlerais aussi de SOPPA ici ? A citer au moins une fois ?} 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 \cite{Dreuw_2015} or the polarization propagator approaches (like SOPPA\cite{Packer_1996}) in quantum chemistry.
% 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 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} \begin{equation}
G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel}, G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel},
\end{equation} \end{equation}
where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function. 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. 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, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of a hole is monitored. 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.
\\ \\
%=================================== %===================================
\subsection{Charged excitations} \subsection{Charged excitations}
%=================================== %===================================
A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies of the system A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies (\ie, the ionization potentials and electron affinities) of the system
\begin{equation}\label{eq:spectralG} \begin{equation}\label{eq:spectralG}
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) }, 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} \end{equation}
@ -274,7 +273,6 @@ dropping spin variables for simplicity, one gets the familiar eigenvalue equatio
\end{equation} \end{equation}
which formally resembles the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian. which formally resembles the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation. The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.
%% \titou{The spin variable has disappear. How do we deal with this?}
\\ \\
%=================================== %===================================
@ -289,21 +287,23 @@ The resulting equation, when compared with the equation for the time-evolution o
\begin{equation}\label{eq:Sig} \begin{equation}\label{eq:Sig}
\Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3), \Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
\end{equation} \end{equation}
where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$. \hl{vs ne vlz pas simplement dire que c'est des corrections de + grand ordre ?} where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is the so-called ``vertex" function.
The neglect of the vertex leads to the so-called $GW$ approximation of the self-energy %where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$.
The neglect of the vertex, \ie, $\Gamma(42,3) = \delta(23) \delta(24)$, leads to the so-called $GW$ approximation of the self-energy
\begin{equation}\label{eq:SigGW} \begin{equation}\label{eq:SigGW}
\Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}), \Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}),
\end{equation} \end{equation}
that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
\begin{gather} \begin{gather}
W(1,2) = v(1,2) + \int d34 \, v(1,2) \chi_0(3,4) W(4,2), \label{eq:defW} W(1,2) = v(1,2) + \int d34 \, v(1,2) \chi_0(3,4) W(4,2),
\label{eq:defW}
\\ \\
\chi_0(1,2) = -i \int d34 \, G(2,3) G(4,2), \chi_0(1,2) = -i \int d34 \, G(2,3) G(4,2),
\end{gather} \end{gather}
where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential. where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential.
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure}[h] \begin{figure}[ht]
\includegraphics[width=0.55\linewidth]{fig1/fig1} \includegraphics[width=0.55\linewidth]{fig1/fig1}
\caption{ \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} 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}
@ -363,13 +363,14 @@ However, remaining a low-order perturbative approach starting with single-determ
\subsection{Neutral excitations} \subsection{Neutral excitations}
%=================================== %===================================
While TD-DFT starts with the variation of the charge density $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility, or two-particle correlation function, that monitors the variation of the one-body Green's function $G(1,1')$ with respect to a non-local external perturbation $U(2,2')$: While TD-DFT starts with the variation of the charge density $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility, or two-particle correlation function, that monitors the variation of the one-body Green's function $G(1,1')$ with respect to a non-local external perturbation $U(2,2')$: \cite{Strinati_1988}
\begin{equation} \begin{equation}
\chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)} \chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
\quad \rightarrow \quad \quad \rightarrow \quad
L(1, 2;1',2' ) \stackrel{\BSE}{=} \pdv{G(1,1')}{U(2',2)}, L(1, 2;1',2' ) \stackrel{\BSE}{=} \pdv{G(1,1')}{U(2',2)}.
\end{equation} \end{equation}
where we follow the notations by Strinati.\cite{Strinati_1988} The formal relation $\chi(1,2) = -i L(1,2;1^+,2^+)$ with $\rho(1) = -iG(1,1^{+})$ offers a direct bridge between the TD-DFT and the BSE worlds. %where we follow the notations by Strinati.\cite{Strinati_1988}
The formal relation $\chi(1,2) = -i L(1,2;1^+,2^+)$ with $\rho(1) = -iG(1,1^{+})$ offers a direct bridge between the TD-DFT and BSE worlds.
The equation of motion for $G$ [see Eq.~\eqref{eq:Gmotion}] can be reformulated in the form of a Dyson equation The equation of motion for $G$ [see Eq.~\eqref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
\begin{equation} \begin{equation}
G = G_0 + G_0 ( v_H + U + \Sigma ) G, G = G_0 + G_0 ( v_H + U + \Sigma ) G,
@ -402,7 +403,8 @@ Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that w
= v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ), = v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
\end{multline} \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} 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}
At that stage, the BSE kernel is fully dynamical. 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} At that stage, the BSE kernel is fully dynamical, \ie, it explicitly depends on the frequency $\omega$.
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{equation} \label{eq:BSE-eigen}
\begin{pmatrix} \begin{pmatrix}
R & C R & C
@ -444,8 +446,7 @@ with $\kappa=2,0$ for singlets/triplets and
\phi_i(\br) \phi_j(\br) W(\br,\br'; \omega=0) \phi_i(\br) \phi_j(\br) W(\br,\br'; \omega=0)
\phi_a(\br') \phi_b(\br'), \phi_a(\br') \phi_b(\br'),
\end{equation} \end{equation}
where we notice that the two occupied (virtual) eigenstates are taken at the same position of space, in contrast with the 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
$(ia|jb)$ bare Coulomb term defined as
\begin{equation} \begin{equation}
(ai|bj) = \iint d\br d\br' (ai|bj) = \iint d\br d\br'
\phi_i(\br) \phi_a(\br) v(\br-\br') \phi_i(\br) \phi_a(\br) v(\br-\br')
@ -479,7 +480,7 @@ with the experimental (photoemission) fundamental gap
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 (see Fig.~\ref{fig:gaps}). 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 (see Fig.~\ref{fig:gaps}).
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure*}[h] \begin{figure*}[ht]
\includegraphics[width=0.7\linewidth]{gaps} \includegraphics[width=0.7\linewidth]{gaps}
\caption{ \caption{
Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$. Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
@ -508,7 +509,7 @@ where $\EB$ is the excitonic effect, that is, the stabilization implied by the a
Such a residual gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} Such a residual gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016}
Alternatively, self-consistent schemes such as ev$GW$ and qs$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011} where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016} Alternatively, self-consistent schemes such as ev$GW$ and qs$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011} where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016}
As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations. As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering ca. 200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering roughly ca.~200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange. This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -527,7 +528,7 @@ The analysis of the screened Coulomb potential matrix elements in the BSE kernel
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to the modeling of key applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\ The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to the modeling of key applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
%%% FIG 3 %%% %%% FIG 3 %%%
\begin{figure}[h] \begin{figure}[ht]
\includegraphics[width=0.6\linewidth]{CTfig} \includegraphics[width=0.6\linewidth]{CTfig}
\caption{ \caption{
Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface (bottom). Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface (bottom).