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\begin{abstract}
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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.
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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.
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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.
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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 model UV/Vis spectra by catching excitonic effects, has shown to provide accurate singlet excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
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With a similar computational cost as time-dependent density-functional theory (TD-DFT), the BSE formalism is 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.
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In this \textit{Perspective} article, we provide a historical overview of the BSE formalism, with a particular focus on its condensed-matter roots.
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We also propose a critical review of its strengths and weaknesses in different chemical situations.
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Future directions of developments and improvements are also discussed.
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@ -229,7 +229,7 @@ Future directions of developments and improvements are also discussed.
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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 a tool for chemists as the test tube.
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Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003}
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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.
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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}
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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 and DNA damage under irradiation. \cite{Kippelen_2009,Improta_2016,Wu_2019}
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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}
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\\
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@ -246,7 +246,7 @@ where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function.
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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.
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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.
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This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", 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.
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This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", more commonly labeled as electronic energy levels, as obtained, \eg, in a direct or inverse photo-emission experiments where an electron is ejected or added to the $N$-electron system.
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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
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\begin{equation}\label{eq:IPAEgap}
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\EgFun = I^\Nel - A^\Nel,
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@ -266,7 +266,7 @@ where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\var
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Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{th} excited state of the $\Nel$-electron system.
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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).
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Unlike 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.
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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.
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In contrast 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.
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Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
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\begin{equation}\label{eq:Gmotion}
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@ -324,7 +324,7 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
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\end{figure}
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%%% %%% %%%
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In practice, the input $G$ and $\chi_0$ required to initially build $\Sigma^{\GW}$ are taken as the ``best'' Green's function and susceptibility that can be easily computed, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
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In practice, the input $G$ and $\chi_0$ required to initially build $\Sigma^{\GW}$ are chosen as the ``best'' Green's function and susceptibility that can be easily computed, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
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Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained by solving the so-called quasiparticle equation
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\begin{equation} \label{eq:QP-eq}
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\omega = \varepsilon_p^{\KS} +
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@ -359,7 +359,7 @@ In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983}
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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.
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For example, the quasiparticle energies, and in particular the HOMO-LUMO gap, depends on the input KS eigenvalues.
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Tuning the starting point functional or applying a self-consistent $GW$ scheme are two different approaches commonly employed to improve on this problem.
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Tuning the starting point functional or applying a self-consistent $GW$ scheme are two different approaches commonly employed to tackle this problem.
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We will comment further on this particular point below when addressing the quality of the BSE optical excitations.
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%Further improvements may be obtained via self-consistency of Hedin's equations (see Fig.~\ref{fig:pentagon}).
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%There exists two main types of self-consistent $GW$ methods:
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@ -484,7 +484,7 @@ where we notice that the two occupied (virtual) eigenstates are taken at the sam
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\end{equation}
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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 (TDA).
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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.
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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.
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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.
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This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its \emph{pros} and \emph{cons}.
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\\
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@ -498,7 +498,7 @@ This defines the standard (static) BSE@$GW$ scheme that we discuss in this \text
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Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} the BSE formalism has emerged in condensed-matter physics around the 1960's at the tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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Three decades later, the first \textit{ab initio} implementations, starting with small clusters, \cite{Onida_1995,Rohlfing_1998} extended semiconductors, and wide-gap insulators \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999b} paved the way to the popularization in the solid-state physics community of the BSE formalism.
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Following pioneering applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in quantum chemistry \cite{listofrefs} with, in particular, several benchmark calculations \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular sets performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations. \cite{Schreiber_2008} %such as CC3. \cite{Christiansen_1995}
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Following pioneering applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in quantum chemistry \cite{listofrefs} with, in particular, several benchmarks \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular sets performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations. \cite{Schreiber_2008} %such as CC3. \cite{Christiansen_1995}
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Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques \cite{Ren_2012b} were used.
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies is strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap
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@ -519,13 +519,13 @@ Such an underestimation of the fundamental gap leads to a similar underestimatio
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\begin{equation}
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\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
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\end{equation}
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where $\EB$ accounts for the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
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where $\EB$ accounts for the excitonic effect, that is, the stabilization induced by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
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%Because of this, we have $\EgOpt < \EgFun$.
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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,Gui_2018}
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Alternatively, self-consistent approaches such as eigenvalue self-consistent (ev$GW$) \cite{Hybertsen_1986} or quasiparticle self-consistent (qs$GW$) \cite{vanSchilfgaarde_2006} schemes, 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{Rostgaard_2010,Blase_2011,Ke_2011,Rangel_2016,Kaplan_2016,Caruso_2016}
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As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
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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}
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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}
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This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -539,8 +539,8 @@ This is equivalent to the best TD-DFT results obtained by scanning a large varie
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A very remarkable success of the BSE formalism lies in the description of charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT adopting standard (semi-)local functionals. \cite{Dreuw_2004}
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Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes (Fig.~\ref{fig:CTfig}), cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
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These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
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It can be cured through an exact exchange contribution, a solution that explains the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
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The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where the screening reduces the long-range electron-hole interactions.
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It can be cured through an exact exchange contribution, a solution that explains the success of (optimally-tuned) range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
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The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc.) where the screening reduces the long-range electron-hole interactions.
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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.\\
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%%% FIG 3 %%%
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@ -558,7 +558,7 @@ The success of the BSE formalism to treat CT excitations has been demonstrated i
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\subsection{Combining BSE with PCM and QM/MM models}
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%==========================================
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The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface, \ldots) is an important step towards the description of realistic systems.
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The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface\ldots) is an important step towards the description of realistic systems.
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Pioneering BSE studies demonstrated, for example, the large renormalization of charged and neutral excitations in molecular systems and nanotubes close to a metallic electrode or in bundles. \cite{Lastra_2011,Rohlfing_2012,Spataru_2013}
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Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
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\cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the screened Coulomb potential designed to capture polarization effects at all spatial ranges.
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@ -566,7 +566,7 @@ As a matter of fact, dressing the bare Coulomb potential with the reaction field
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$[
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v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
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]$
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in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment with the same complexity as in the gas phase.
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in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment at the same computational cost as the corresponding gas-phase calculation.
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The reaction field operator $v^{\text{reac}}(\br,\br'; \omega)$ describes the potential generated in $\br'$ by the charge rearrangements in the polarizable environment induced by a source charge located in $\br$, where $\br$ and $\br'$ lie in the quantum mechanical subsystem of interest.
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The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.
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Once the reaction field matrix is known, with typically $\order*{\Norb N_\text{MM}^2}$ operations (where $\Norb$ is the number of orbitals and $N_\text{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
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@ -575,7 +575,7 @@ A remarkable property \cite{Duchemin_2018} of the scheme described above, which
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This is an important advantage as compared to, \eg, TD-DFT where linear-response and state-specific effects have to be explored with different formalisms.
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To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg, considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant of water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range.
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Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
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Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, situations where neither the adiabatic limit nor the anti-adiabatic limits are expected to be valid (for a recent discussion, see Ref.
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~\citenum{Huu_2020}).
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We now leave the description of successes to discuss difficulties and future directions of developments and improvements.
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@ -603,7 +603,7 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
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\subsection{The triplet instability challenge}
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%==========================================
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The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission or thermally activated delayed fluorescence (TADF).
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From a more theoretical point of view, triplet instabilities, which hampers the applicability of TD-DFT for popular range-separated hybrids containing a large fraction of long-range exact exchange, are intimately linked to the stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and KS \cite{Bauernschmitt_1996} levels.
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From a more theoretical point of view, triplet instabilities that often plagues the applicability of TD-DFT are intimately linked to the stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and KS \cite{Bauernschmitt_1996} levels.
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While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter(s) as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
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Benchmark calculations \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably too low in energy within BSE and that the use of the TDA was able to partly reduce this error.
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@ -611,24 +611,6 @@ However, as it stands, the BSE accuracy for triplets remains rather unsatisfacto
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An alternative cure was offered by hybridizing TD-DFT and BSE, that is, by 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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\\
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%==========================================
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\subsection{The challenge of analytical nuclear gradients}
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%==========================================
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The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Olivucci_2010}
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For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011}
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Reliable predictions of these mechanisms, which have attracted much experimental and theoretical interest lately, require exploring the ground- and excited-state PES.
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From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Roos_1996}) cannot be afforded.
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For the last two decades, TD-DFT has been the go-to method to compute absorption and emission spectra in large molecular systems.
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In TD-DFT, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state.
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One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first- and second-order derivatives of the excited-state energy with respect to atomic displacements), which enables the exploration of excited-state PES. \cite{Furche_2002}
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A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients for both the ground and excited states, preventing efficient studies of excited-state processes.
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While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
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\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003}
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In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, with the approximation that the gradient of the screened Coulomb potential can be neglected, computing further the KS-LDA forces as its ground-state contribution.
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\\
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%==========================================
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\subsection{The challenge of the ground-state energy}
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@ -637,15 +619,34 @@ In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the
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In contrast to TD-DFT which relies on KS-DFT as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
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Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020,Loos_2020}
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A promising route, which closely follows RPA-type formalisms, \cite{Angyan_2011} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005}
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A promising route, which closely follows RPA-type formalisms, \cite{Angyan_2011} is to calculate the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005}
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Thanks to comparisons with both similar and state-of-art computational approaches, it was recently shown that the ACFDT@BSE@$GW$ approach yields extremely accurate PES around equilibrium, and can even compete with high-order coupled cluster methods in terms of absolute ground-state energies and equilibrium distances. \cite{Loos_2020}
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However, their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
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Indeed, in the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the ACFDT framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
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Moreover, it was also observed in Ref.~\citenum{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES show up due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
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Moreover, it was also observed in Ref.~\citenum{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES appear due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
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Such an unphysical behavior stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found [see Eq.~\eqref{eq:QP-eq}].
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We refer the interested reader to Refs.~\citenum{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} for detailed discussions.
|
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\\
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%==========================================
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\subsection{The challenge of analytical nuclear gradients}
|
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%==========================================
|
||||
|
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The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Olivucci_2010}
|
||||
For example, chemoluminescence and fluorescence are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011}
|
||||
Reliable predictions of these mechanisms, which have attracted much experimental and theoretical interest lately, require exploring the ground- and excited-state PES.
|
||||
From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Roos_1996}) cannot be afforded.
|
||||
For the last two decades, TD-DFT has been the go-to method to compute absorption and emission spectra in large molecular systems.
|
||||
|
||||
In TD-DFT, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state.
|
||||
One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first- and second-order derivatives of the excited-state energy with respect to atomic displacements), which enables the exploration of excited-state PES. \cite{Furche_2002}
|
||||
|
||||
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients for both the ground and excited states, preventing efficient studies of many key excited-state processes.
|
||||
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
|
||||
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003}
|
||||
In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, with the approximation that the gradient of the screened Coulomb potential can be neglected, computing further the KS-LDA forces as its ground-state contribution.
|
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\\
|
||||
|
||||
%==========================================
|
||||
\subsection{Beyond the static approximation}
|
||||
%==========================================
|
||||
@ -738,10 +739,10 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusions}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
Although far from being exhaustive, we hope to have provided, in the present \textit{Perspective}, a concise and fair assessment of the strengths and weaknesses of the BSE formalism of many-body perturbation theory.
|
||||
Although far from being exhaustive, we hope that this \textit{Perspective} provides a concise and fair assessment of the strengths and weaknesses of the BSE formalism of many-body perturbation theory.
|
||||
To do so, we have briefly reviewed the theoretical aspects behind BSE, and its intimate link with the underlying $GW$ calculation that one must perform to compute quasiparticle energies and the dynamically-screened Coulomb potential; two of the key input ingredients associated with the BSE formalism.
|
||||
We have then provided a succinct historical overview with a particular focus on its condensed-matter roots, and the lessons that the community has learnt from several systematic benchmark studies on large molecular systems.
|
||||
Several success stories are then discussed (charge-transfer excited states and combination with reaction field methods), before debating some of the challenges faced by the BSE formalism (computational cost, triplet instabilities, lack of analytical nuclear gradients, ambiguity in the definition of the ground-state energy, and limitations due to the static approximation).
|
||||
Several success stories are then discussed (charge-transfer excited states and combination with reaction field methods), before debating some of the challenges faced by the BSE formalism (computational cost, triplet instabilities, ambiguity in the definition of the ground-state energy, lack of analytical nuclear gradients, and limitations due to the static approximation).
|
||||
We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will motivate a larger community to participate to the development of this alternative to TD-DFT which, we believe, may become a very valuable computational tool for the physical chemistry community.
|
||||
\\
|
||||
|
||||
|
||||
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