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@ -224,12 +224,12 @@ Future directions of developments and improvements are also discussed.
\section{Introduction} \section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 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. 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.
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 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 \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Dreuw_2005} including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018} The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Dreuw_2005} including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{History} %\section{History}
@ -254,7 +254,10 @@ where $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ are the ionization
\begin{figure*} \begin{figure*}
\includegraphics[width=0.7\linewidth]{gaps} \includegraphics[width=0.7\linewidth]{gaps}
\caption{ \caption{
Optical and fundamental gaps. Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
$\EB$ is the excitonic effect, while $I^N$ and $A^N$ are the ionization potential and the electron affinity of the $N$-electron system.
$\Eg^{\KS}$ and $\Eg^{\GW}$ are the KS and $GW$ HOMO-LUMO gaps.
See main text for the definitions of the other quantities
\label{fig:gaps}} \label{fig:gaps}}
\end{figure*} \end{figure*}
@ -388,9 +391,9 @@ where
\end{equation} \end{equation}
is the TD-DFT kernel. is the TD-DFT kernel.
Plugging now the $GW$ self-energy, in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel Plugging now the $GW$ self-energy, in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{align*}\label{eq:BSEkernel} \begin{equation}\label{eq:BSEkernel}
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ), \Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ),
\end{align*} \end{equation}
where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$.
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces thus the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT: Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces thus the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT:
\begin{equation} \begin{equation}
@ -453,8 +456,8 @@ A very remarkable success of the BSE formalism lies in the description of the ch
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004} Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals. These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012} It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
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 screening reduces the long-range electron-hole interactions. 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 screening reduces the long-range electron-hole interactions.
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011b,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\ The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\
%========================================== %==========================================
\subsection{Solvent effects} \subsection{Solvent effects}
@ -465,21 +468,27 @@ We now leave the description of successes to discuss difficulties and Perspectiv
%========================================== %==========================================
\subsection{The computational challenge} \subsection{The computational challenge}
%========================================== %==========================================
As emphasized above, the BSE eigenvalue equation in the occupied-to-virtual product space is formally equivalent to that of TD-DFT or TD-HF. Searching iteratively for the lowest eigenstates presents the same $\mathcal{O}(N^4)$ matrix-vector multiplication computational cost within BSE and TD-DFT. Concerning the construction of the BSE Hamiltonian, it is no more expensive than building the TD-DFT one with hybrid functionals, reducing again to $\mathcal{O}(N^4)$ operations with standard resolution-of-identity techniques. At the price of sacrifying the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\mathcal{O}(N^3)$ operations using iterative techniques. \cite{Ljungberg_2015} With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015} As emphasized above, the BSE eigenvalue equation in the occupied-to-virtual product space is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
Searching iteratively for the lowest eigenstates presents the same $\order*{N^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT.
Concerning the construction of the BSE Hamiltonian, it is no more expensive than building the TD-DFT one with hybrid functionals, reducing again to $\order*{N^4}$ operations with standard RI techniques.
At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{N^3}$ operations using iterative techniques. \cite{Ljungberg_2015}
With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015}
In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\mathcal{O}(N^4)$ with system size using PWs or RI techniques, but with a rather large prefactor. In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\order{N^4}$ with system size using plane-wave basis sets or RI techniques, but with a rather large prefactor.
%%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states. %%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states.
%%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019} %%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019}
The field of low-scaling $GW$ calculations is however witnessing significant advances. While the sparcity of ..., \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992} The field of low-scaling $GW$ calculations is however witnessing significant advances.
Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017} The separability of occupied and virtual states summations lying at the heart of these approaches are now blooming in quantum chemistry withing the Interpolative Separable Density Fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in RPA and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020} These ongoing developments pave the way to applying the $GW$/BSE formalism to systems comprising several hundred atoms on standard laboratory clusters. \\ While the sparsity of ..., \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017}
The separability of occupied and virtual states summations lying at the heart of these approaches are now blooming in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in RPA and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020} These ongoing developments pave the way to applying the $GW$/BSE formalism to systems comprising several hundred atoms on standard laboratory clusters. \\
%========================================== %==========================================
\subsection{The triplet instability challenge} \subsection{The triplet instability challenge}
%========================================== %==========================================
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels. stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels.
contaminating as well TD-DFT calculations with popular range-separated hybrids (RSH) that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011} contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011}
While TD-DFT with RSH can benefit from tuning the range-separation parameter 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. While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter 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.
benchmarks \cite{Jacquemin_2017b,Rangel_2017} benchmarks \cite{Jacquemin_2017b,Rangel_2017}
@ -490,7 +499,7 @@ a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE
%========================================== %==========================================
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 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
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}\\ This points to another direction for the BSE formlism, 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}\\
%========================================== %==========================================
\subsection{The double excitation challenge} \subsection{The double excitation challenge}
@ -561,6 +570,8 @@ In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its
This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$. This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\ Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\
Beyond the static approximation \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
%========================================== %==========================================
\subsection{Core-level spectroscopy} \subsection{Core-level spectroscopy}
%========================================== %==========================================

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Manuscript/xbbib.bib
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@ -1,13 +1,37 @@
%% 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 2020-05-14 13:56:12 +0200 %% Created for Pierre-Francois Loos at 2020-05-14 22:49:58 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Lettmann_2019,
Author = {Tobias Lettmann and Michael Rohlfing},
Date-Added = {2020-05-14 22:27:01 +0200},
Date-Modified = {2020-05-14 22:35:55 +0200},
Doi = {10.1021/acs.jctc.9b00223},
Journal = {J. Chem. Theory Comput.},
Pages = {4547--4554},
Title = {Electronic Excitations of Polythiophene within Many-Body Perturbation Theory with and without the Tamm-Dancoff Approximation},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00223}}
@article{Sottile_2003,
Author = {Francesco Sottile and Valerio Olevano and Lucia Reining},
Date-Added = {2020-05-14 22:24:37 +0200},
Date-Modified = {2020-05-14 22:25:32 +0200},
Doi = {10.1103/PhysRevLett.91.056402},
Journal = {Phys. Rev. Lett.},
Pages = {056402},
Title = {Parameter-Free Calculation of Response Functions in Time-Dependent Density-Functional Theory},
Volume = {91},
Year = {2003},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.91.056402}}
@article{Ke_2011, @article{Ke_2011,
Author = {Ke, San-Huang}, Author = {Ke, San-Huang},
Date-Added = {2020-05-14 13:55:32 +0200}, Date-Added = {2020-05-14 13:55:32 +0200},
@ -1228,19 +1252,6 @@
Year = {2004}, Year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1021/ja039556n}} Bdsk-Url-1 = {https://doi.org/10.1021/ja039556n}}
@article{Blase_2011,
Author = {Blase,X. and Attaccalite,C.},
Doi = {10.1063/1.3655352},
Eprint = {https://doi.org/10.1063/1.3655352},
Journal = {Applied Physics Letters},
Number = {17},
Pages = {171909},
Title = {Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach},
Url = {https://doi.org/10.1063/1.3655352},
Volume = {99},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3655352}}
@article{Duchemin_2012, @article{Duchemin_2012,
Author = {Duchemin, I. and Deutsch, T. and Blase, X.}, Author = {Duchemin, I. and Deutsch, T. and Blase, X.},
Doi = {10.1103/PhysRevLett.109.167801}, Doi = {10.1103/PhysRevLett.109.167801},
@ -1840,8 +1851,9 @@
Year = {2014}, Year = {2014},
Bdsk-Url-1 = {https://doi.org/10.1021/ct5003658}} Bdsk-Url-1 = {https://doi.org/10.1021/ct5003658}}
@article{Blase_2011b, @article{Blase_2011,
Author = {Blase,X. and Attaccalite,C.}, Author = {Blase,X. and Attaccalite,C.},
Date-Modified = {2020-05-14 22:23:02 +0200},
Doi = {10.1063/1.3655352}, Doi = {10.1063/1.3655352},
Eprint = {https://doi.org/10.1063/1.3655352}, Eprint = {https://doi.org/10.1063/1.3655352},
Journal = {Appl. Phys. Lett.}, Journal = {Appl. Phys. Lett.},