titou saving work
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\usepackage{mathtools}
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\usepackage[dvipsnames]{xcolor}
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\usepackage{xspace}
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\usepackage{ifthen}
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\usepackage{qcircuit}
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\usepackage{graphicx,longtable,dcolumn,mhchem}
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\usepackage{rotating,color}
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\newcommand{\cmark}{\color{green}{\text{\ding{51}}}}
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\newcommand{\xmark}{\color{red}{\text{\ding{55}}}}
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\newcommand{\mc}{\multicolumn}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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%matrices
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bh}{\boldsymbol{h}}
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\newcommand{\bc}{\boldsymbol{c}}
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\renewcommand{\tr}[1]{{#1}^{\dag}}
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% methods
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\newcommand{\DFT}{\text{DFT}}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\GW}{GW}
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\newcommand{\XC}{\text{xc}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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% sets
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\usepackage[colorlinks = true,
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linkcolor = blue,
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urlcolor = black,
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\let\oldmaketitle\maketitle
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\let\maketitle\relax
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\title{A Chemist Guide to the Bethe--Salpeter Formalism}
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\title{A Chemist Guide to the Bethe-Salpeter Formalism}
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\date{\today}
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\begin{tocentry}
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@ -162,7 +179,10 @@
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%%%%%%%%%%%%%%%%
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\begin{abstract}
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Here goes the abstract
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The many-body Green's function Bethe-Salpeter formalism is steadily asserting itself as a new efficient and accurate tool in the armada of computational methods available to chemists in order to predict neutral electronic excitations in molecular systems.
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In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable charged excitations and quasiparticle energies, and the Bethe-Salpeter formalism, allowing to catch excitonic effects, has shown to provide accurate excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
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In this \textit{Perspective} article, we provide a historical overview of the Bethe-Salpeter formalism, with a particular focus on its condensed-matter roots, and we propose a critical review of its strengths and weaknesses for different chemical situations, such as \titou{bla bla bla}.
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Future directions of developments and improvements are also discussed.
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\end{abstract}
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@ -183,11 +203,11 @@ Here goes the abstract
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%%% INTRODUCTION %%%
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%%%%%%%%%%%%%%%%%%%%
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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 that ``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." Martin Karplus Nobel lecture moderated this bold statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are "much too complicated to be soluble", urging the scientist to develop "approximate practical methods." This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of complex systems.
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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 \titou{as} a tool for chemists as the test tube. Simulations are so realistic that they predict the outcome of traditional experiments.''} Martin Karplus Nobel lecture moderated this bold 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 the scientist to develop \textit{``approximate practical methods''}. This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
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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 and stones for jewellery, to the understanding e.g. of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties.
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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 and stones for jewellery, to the understanding, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties.
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The Bethe-Salpeter equation formalism [REFS] belongs to the family of Green's function many-body perturbation theories (MBPT) [REFS] to which belong as well the Algebraic Diagrammatic Construction (ADC) techniques in quantum chemistry. [REFS] While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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The Bethe-Salpeter equation 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} to which belong as well the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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\begin{equation}
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G(xt,x't') = -i \langle N | T \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle
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\end{equation}
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@ -197,14 +217,14 @@ where $| N \rangle $ is the N-electron ground-state wavefunction. The operators
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\begin{equation}
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G(x,x'; \omega ) = \sum_n \frac{ f_s(x) f^*_s(x') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) } \label{spectralG}
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\end{equation}
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where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the s-th excited state of the (N+1) and (N-1)-electron systems, while $E_0(N)$ is the N-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper charging energies of the N-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. The $\lbrace f_s \rbrace$ are called the Lehmann amplitudes that reduce to one-body orbitals in the case of mono-determinental many-body wavefunctions [more ??].
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where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the $s$-th excited state of the $(N+1)$ and $(N-1)$-electron systems, while $E_0(N)$ is the $N$-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper charging energies of the N-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. The $\lbrace f_s \rbrace$ are called the Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinent many-body wave functions [more ??].
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Using the equation of motion for the creation/destruction operators, it can be shown formally that $G$ verifies :
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\begin{equation}
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\left( \frac{\partial }{\partial t_1} - h({\bf r}_1) \right) G(1,2) - \int d3 \; \Sigma(1,3) G(3,2)
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\qty[ \pdv{}{t_1} - h({\bf r}_1) ] G(1,2) - \int d3 \; \Sigma(1,3) G(3,2)
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= \delta(1,2) \label{Gmotion}
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\end{equation}
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where we use the notation $1 = (x_1,t_1)$. Here $h$ is the one-body Hartree Hamiltonian and $\Sigma$ the so-called exchange-correlation self-energy operator. Using the spectral representation of $G$, one obtains a familiar eigenvalue equation:
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where we introduce the usual composite index, \eg, $1 \equiv (x_1,t_1)$. Here, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ the so-called exchange-correlation self-energy operator. Using the spectral representation of $G$, one obtains a familiar eigenvalue equation:
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\begin{equation}
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h({\bf r}) f_s({\bf r}) + \int d{\bf r}' \; \Sigma({\bf r},{\bf r}'; \varepsilon_s ) f_s({\bf r}) = \varepsilon_s f_s({\bf r})
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\end{equation}
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@ -221,10 +241,10 @@ where $W$ is the dynamically screened Coulomb potential and $\Gamma$ a ``vertex"
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W(1,2) &= v(1,2) + \int d34 \; v(1,2) \chi_0(3,4) W(4,2) \\
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\chi_0(1,2) &= -i \int d34 \; G(2,3) G(4,2)
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\end{align}
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with $\chi_0$ the well-known independent electron susceptibility and $v$ the bare Coulomb potential. In practice, the input $G$ and $\chi_0$ needed to buld $\Sigma$ are taken to be the best Green's function and susceptibility that can be easily calculated, namely the DFT (or HF) ones where the $\lbrace \varepsilon_s, f_s \rbrace$ of equation~\ref{spectralG} are taken to be DFT Kohn-Sham (or HF) eigenstates. Taking then $( \Sigma^{GW}-V^{XC} )$ as a correction to the $V^{XC}$ DFT exchange correlation potential, a first-order correction to the input Kohn-Sham $\lbrace \varepsilon_n^{KS} \rbrace$ energies is obtained as follows:
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with $\chi_0$ the well-known independent electron susceptibility and $v$ the bare Coulomb potential. In practice, the input $G$ and $\chi_0$ needed to buld $\Sigma$ are taken to be the best Green's function and susceptibility that can be easily calculated, namely the DFT (or HF) ones where the $\lbrace \varepsilon_s, f_s \rbrace$ of equation~\ref{spectralG} are taken to be DFT Kohn-Sham (or HF) eigenstates. Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the $V^{\XC}$ DFT exchange correlation potential, a first-order correction to the input Kohn-Sham $\lbrace \varepsilon_n^{KS} \rbrace$ energies is obtained as follows:
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\begin{equation}
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\varepsilon_n^{GW} = \varepsilon_n^{KS} +
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\langle \phi_n^{KS} | \Sigma^{GW}(\varepsilon_n^{GW}) -V^{XC} | \phi_n^{KS} \rangle
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\varepsilon_n^{\GW} = \varepsilon_n^{\KS} +
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\langle \phi_n^{\KS} | \Sigma^{\GW}(\varepsilon_n^{\GW}) -V^{\XC} | \phi_n^{\KS} \rangle
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\end{equation}
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Such an approach, where input Kohn-Sham energies are corrected to yield better electronic energy levels, is labeled the single-shot, or perturbative, $G_0W_0$ technique. This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, [REFS]
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surfaces [REFs], and 2D systems, [REFS] allowing to dramatically reduced the errors associated with Kohn-Sham eigenvalues in conjunction with the common LDA approximation.
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@ -232,9 +252,9 @@ In particular, the well-known ``band gap" problem, [REFS] namely the underestima
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\noindent {\textbf{Neutral excitations.}} While TD-DFT starts with the variation of the charge density $\rho$ with respect to an external local perturbation, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation:
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\begin{equation}
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\chi(1,2) \stackrel{DFT}{=} \frac{ \partial \rho(1) }{\partial U(2) }
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\chi(1,2) \stackrel{\DFT}{=} \frac{ \partial \rho(1) }{\partial U(2) }
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\;\; \rightarrow \;\;
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L(12,34) \stackrel{BSE}{=} -i \frac{ \partial G(1,2) } { \partial U(3,4) }
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L(12,34) \stackrel{\BSE}{=} -i \frac{ \partial G(1,2) } { \partial U(3,4) }
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\end{equation}
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%with the relation $\chi(1,2) = L(11,22)$ since $\rho(1) = -iG(1,1^{+})$, as a first bridge between the TD-DFT and BSE worlds.
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The equation of motion for $G$ (Eqn.~\ref{Gmotion}) can be reformulated in the form of a Dyson equation:
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@ -254,11 +274,11 @@ with $L_0 = \partial G_0 / \partial U$ the Hartree 4-point susceptibility and:
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\end{align*}
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This equation can be compared to its TD-DFT analog:
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\begin{equation}
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\chi(1,2) = \chi_0(1,2) + \int d34 \; \chi_0(1,3) \Xi^{DFT}(3,4) \chi(4,2)
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\chi(1,2) = \chi_0(1,2) + \int d34 \; \chi_0(1,3) \Xi^{\DFT}(3,4) \chi(4,2)
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\end{equation}
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with $\Xi^{DFT}$ the TD-DFT kernel :
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with $\Xi^{\DFT}$ the TD-DFT kernel :
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\begin{equation}
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\Xi^{DFT}(3,4) = v(3,4) + \frac{ \partial V^{XC}(3) }{ \partial \rho(4) }
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\Xi^{\DFT}(3,4) = v(3,4) + \frac{ \partial V^{\XC}(3) }{ \partial \rho(4) }
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\end{equation}
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Plugging now the $GW$ self-energy, in a scheme that we label the BSE/$GW$ approach, leads to an approximation to the BSE kernel:
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\begin{align*}
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\end{equation}
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where $\lambda$ index the electronic excitations. The $\lbrace \phi_{i/a} \rbrace$ are the input (Kohn-Sham) eigenstates used to build the $GW$ self-energy. The resonant part of the BSE Hamiltonian reads:
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\begin{align*}
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R_{ai,bj} = \left( \varepsilon_a^{GW} - \varepsilon_i^{GW} \right) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj}
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R_{ai,bj} = \left( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} \right) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj}
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\end{align*}
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with $\eta=2,0$ for singlets/triplets and:
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\begin{equation}
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@ -305,7 +325,7 @@ W_{ai,bj} = \int d{\bf r} d{\bf r}'
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where we notice that the 2 occupied (virtual) eigenstates are taken at the same space position, in contrast with the
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$(ai|bj)$ bare Coulomb term. As compared to TD-DFT :
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\begin{itemize}
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\item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{GW} \rbrace$ replace the Kohn-Sham eigenvalues
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\item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the Kohn-Sham eigenvalues
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\item the non-local screened Coulomb matrix elements replaces the DFT exchange-correlation kernel.
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\end{itemize}
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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 Perspective, emphasizing its pros and cons. \\
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@ -356,7 +376,7 @@ Another approach, that proved very fruitful, lies in the so-called space-time ap
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An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require
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This points to another direction for the BSE foramlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018,Li_2020,Loos_2020}
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This points to another direction for the BSE foramlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}
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\noindent {\textbf{The Triplet Instability Challenge.}} \\
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@ -367,7 +387,7 @@ While TD-DFT with RSH can benefit from tuning the range-separation parameter as
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benchmarks \cite{Jacquemin_2017b,Rangel_2017}
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a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018}
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a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}
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\noindent {\textbf{Dynamical kernels and multiple excitations.}} \\
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@ -381,11 +401,35 @@ XANES,
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diabatization and conical intersections \cite{Kaczmarski_2010}
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\noindent {\textbf{The Concept of dynamical properties.}}
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As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
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Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes.
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To do so, let us consider we want to solve a hard problem given by the Schr{\"o}dinger-like equation of the form $\bH \bc = \omega \bc$.
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If we assume that the Hamiltonian $\bH$ is of size $N \times N$, this \textit{linear} set of equations yields $K$ solutions.
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However, in practice, $K$ can be very large.
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Therefore, it is usually convenient to recast it as
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\begin{equation}
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\begin{pmatrix}
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\bH_0 & \tr{\bh} \\
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\bh & \bH_1 \\
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\end{pmatrix}
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\begin{pmatrix}
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\bc_0\\
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\bc_1 \\
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\end{pmatrix}
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= \omega
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\begin{pmatrix}
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\bc_0 \\
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\bc_1 \\
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\end{pmatrix}
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\end{equation}
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This system of equation has exactly the same number of solutions.
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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%%%%%%%%%%%%%%%%%%
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\noindent {\textbf{Conclusion.}}
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Here goes the conclusion.
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%%%%%%%%%%%%%%%%%%%%%%%%
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