loos/BSE_JPCLperspective
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

blush

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-06-05 12:41:20 +02:00
parent f30ce1331a
commit a9fa8a6f70
2 changed files with 138 additions and 42 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

124
Manuscript/BSE_JPCL.bib
View File

@ -722,13 +722,18 @@
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}
@article{Duchemin_2020,
Author = {Ivan Duchemin and Xavier Blase},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Journal = {J. Chem. Theory Comput (accepted)},
Title = {Robust Analytic Continuation Approach to Many-Body GW Calculations},
Year = {2020},
Bdsk-Url-1 = {}}
author = {Duchemin, Ivan and Blase, Xavier},
title = {Robust Analytic-Continuation Approach to Many-Body GW Calculations},
journal = { J. Chem. Theory Comput. },
volume = {16},
number = {3},
pages = {1742-1756},
year = {2020},
doi = {10.1021/acs.jctc.9b01235},
note ={PMID: 32023052},
URL = { https://doi.org/10.1021/acs.jctc.9b01235},
eprint = { https://doi.org/10.1021/acs.jctc.9b01235}
}
@article{Dunning_1989,
Author = {T. H. {Dunning, Jr.}},
@ -1183,8 +1188,8 @@
Month = aug,
Number = {8},
Pages = {085125},
Shorttitle = {Excitation Spectra of Aromatic Molecules within a Real-Space {{G W}} -{{BSE}} Formalism},
Title = {Excitation Spectra of Aromatic Molecules within a Real-Space {{G W}} -{{BSE}} Formalism: {{Role}} of Self-Consistency and Vertex Corrections},
Shorttitle = {Excitation Spectra of Aromatic Molecules within a Real-Space {{GW}} -{{BSE}} Formalism},
Title = {Excitation Spectra of Aromatic Molecules within a Real-Space {{GW}} -{{BSE}} Formalism: {{Role}} of Self-Consistency and Vertex Corrections},
Volume = {94},
Year = {2016},
Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.94.085125}}
@ -2133,7 +2138,7 @@
Month = jun,
Number = {23},
Pages = {235102},
Title = {Self-Consistent {{G W}} Calculations for Semiconductors and Insulators},
Title = {Self-Consistent {{GW}} Calculations for Semiconductors and Insulators},
Volume = {75},
Year = {2007},
Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.75.235102}}
@ -2375,7 +2380,7 @@
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.1021/acs.jpclett.7b02740},
Eprint = {https://doi.org/10.1021/acs.jpclett.7b02740},
Journal = {The Journal of Physical Chemistry Letters},
Journal = { J. Phys. Chem. Lett. },
Note = {PMID: 29280376},
Number = {2},
Pages = {306-312},
@ -12981,7 +12986,7 @@
Month = sep,
Number = {12},
Pages = {126406},
Shorttitle = {All-{{Electron Self}}-{{Consistent G W Approximation}}},
Shorttitle = {All-{{Electron Self}}-{{Consistent GW Approximation}}},
Title = {All-{{Electron Self}}-{{Consistent G W Approximation}}: {{Application}} to {{Si}}, {{MnO}}, and {{NiO}}},
Volume = {93},
Year = {2004},
@ -13024,7 +13029,7 @@
Month = jun,
Number = {22},
Pages = {226402},
Title = {Quasiparticle {{Self}}-{{Consistent G W Theory}}},
Title = {Quasiparticle {{Self}}-{{Consistent GW Theory}}},
Volume = {96},
Year = {2006},
Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevLett.96.226402}}
@ -13250,7 +13255,7 @@
Month = nov,
Number = {20},
Pages = {205415},
Shorttitle = {All-Electron {{G W}} Methods Implemented in Molecular Orbital Space},
Shorttitle = {All-Electron {{GW}} Methods Implemented in Molecular Orbital Space},
Title = {All-Electron {{G W}} Methods Implemented in Molecular Orbital Space: {{Ionization}} Energy and Electron Affinity of Conjugated Molecules},
Volume = {84},
Year = {2011},
@ -13335,7 +13340,7 @@
Month = apr,
Number = {16},
Pages = {163001},
Title = {Helium {{Atom Excitations}} by the {{G W}} and {{Bethe}}-{{Salpeter Many}}-{{Body Formalism}}},
Title = {Helium {{Atom Excitations}} by the {{GW}} and {{Bethe}}-{{Salpeter Many}}-{{Body Formalism}}},
Volume = {118},
Year = {2017},
Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevLett.118.163001}}
@ -14638,3 +14643,92 @@
Year = 2012,
Bdsk-Url-1 = {https://doi.org/10.1088%2F1367-2630%2F14%2F5%2F053020},
Bdsk-Url-2 = {https://doi.org/10.1088/1367-2630/14/5/053020}}
@article{Nguyen_2019,
title = {Finite-Field Approach to Solving the Bethe-Salpeter Equation},
author = {Nguyen, Ngoc Linh and Ma, He and Govoni, Marco and Gygi, Francois and Galli, Giulia},
journal = {Phys. Rev. Lett.},
volume = {122},
issue = {23},
pages = {237402},
numpages = {6},
year = {2019},
month = {Jun},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.122.237402},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402}
}
@article{Boulanger_2014,
author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe–Salpeter Approach},
journal = {J. Chem. Theory Comput. },
volume = {10},
number = {3},
pages = {1212-1218},
year = {2014},
doi = {10.1021/ct401101u},
note ={PMID: 26580191},
URL = { https://doi.org/10.1021/ct401101u},
eprint = { https://doi.org/10.1021/ct401101u}
}
@article{Spataru_2013,
title = {Electronic and optical gap renormalization in carbon nanotubes near a metallic surface},
author = {Spataru, Catalin D.},
journal = {Phys. Rev. B},
volume = {88},
issue = {12},
pages = {125412},
numpages = {8},
year = {2013},
month = {Sep},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.88.125412},
url = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412}
}
@article{Rohlfing_2012,
title = {Redshift of Excitons in Carbon Nanotubes Caused by the Environment Polarizability},
author = {Rohlfing, Michael},
journal = {Phys. Rev. Lett.},
volume = {108},
issue = {8},
pages = {087402},
numpages = {5},
year = {2012},
month = {Feb},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.108.087402},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402}
}
@article{Yin_2014,
title = {Charge-Transfer Excited States in Aqueous DNA: Insights from Many-Body Green's Function Theory},
author = {Yin, Huabing and Ma, Yuchen and Mu, Jinglin and Liu, Chengbu and Rohlfing, Michael},
journal = {Phys. Rev. Lett.},
volume = {112},
issue = {22},
pages = {228301},
numpages = {5},
year = {2014},
month = {Jun},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.112.228301},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301}
}
@article{Li_2017b,
title = {Correlated electron-hole mechanism for molecular doping in organic semiconductors},
author = {Li, Jing and D'Avino, Gabriele and Pershin, Anton and Jacquemin, Denis and Duchemin, Ivan and Beljonne, David and Blase, Xavier},
journal = {Phys. Rev. Materials},
volume = {1},
issue = {2},
pages = {025602},
numpages = {9},
year = {2017},
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/PhysRevMaterials.1.025602},
url = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602}
}

56
Manuscript/BSE_JPCL.tex
View File

@ -199,7 +199,7 @@
\begin{abstract}
The many-body Green's function Bethe-Salpeter equation (BSE) 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.
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, able 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.
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, able to catch excitonic effects, has shown to provide accurate \xavier{singlet ?} excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
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.
In this \textit{Perspective} article, we provide a historical overview of the BSE formalism, with a particular focus on its condensed-matter roots.
We also propose a critical review of its strengths and weaknesses for different chemical situations.
@ -252,7 +252,7 @@ For $t > t'$, $G$ provides the amplitude of probability of finding, on top of th
%===================================
A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies of the system
\begin{equation}\label{eq:spectralG}
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \, \text{sgn}(\varepsilon_s - \mu ) },
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) },
\end{equation}
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{\Nel+1} - E_0^{\Nel}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{\Nel} - E_s^{\Nel-1}$ for $\varepsilon_s < \mu$.
Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
@ -335,8 +335,8 @@ Because, one is usually interested only by the quasiparticle solution, in practi
\end{equation}
Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique.
This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988}
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} and
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the local-density approximation (LDA) KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV with a computational cost scaling quartically with the system size (see below). A compilation of data for $G_0W_0$ applied to extended inorganic semiconductors can be found in Ref.~\citenum{Shishkin_2007}.
Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature.
@ -375,7 +375,7 @@ The equation of motion for $G$ [see Eq.~\eqref{eq:Gmotion}] can be reformulated
G = G_0 + G_0 ( v_H + U + \Sigma ) G,
\end{equation}
that relates the full (interacting) Green's function, $G$, to its non-interacting version, $G_0$, where $v_H$ and $U$ are the Hartree and external potentials, respectively.
The derivative with respect to $U$ of this Dyson equation yields
The derivative with respect to $U$ of this Dyson equation yields the self-consistent Bethe-Salpeter equation
\begin{multline}\label{eq:DysonL}
L(1,2;1',2') = L_0(1,2;1',2') +
\\
@ -402,7 +402,7 @@ Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that w
= v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
\end{multline}
where it is customary to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. \cite{Hanke_1980,Strinati_1982,Strinati_1984}
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied spatial orbitals and $(a,b)$ are unoccupied spatial orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT: \cite{Casida_1995}
At that stage, the BSE kernel is fully dynamical. Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied spatial orbitals and $(a,b)$ are unoccupied spatial orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT: \cite{Casida_1995}
\begin{equation} \label{eq:BSE-eigen}
\begin{pmatrix}
R & C
@ -429,7 +429,7 @@ with electron-hole ($eh$) eigenstates written as
+ Y_{ia}^{m} \phi_i(\br_e) \phi_a(\br_h) ],
\end{equation}
where $m$ indexes the electronic excitations.
The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy.
The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy. They are here taken to be real in the case of finite size systems.
The resonant and coupling parts of the BSE Hamiltonian read
\begin{gather}
R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab},
@ -451,7 +451,7 @@ $(ia|jb)$ bare Coulomb term defined as
\end{equation}
Neglecting the coupling term $C$ between the resonant term $R$ and anti-resonant term $-R^*$ in Eq.~\eqref{eq:BSE-eigen}, leads to the well-known Tamm-Dancoff approximation.
As compared to TD-DFT, 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.
As compared to TD-DFT, \; i) the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues, and ii) the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
We emphasize that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations.
This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its pros and cons.
\\
@ -463,7 +463,7 @@ This defines the standard (static) BSE@$GW$ scheme that we discuss in this \text
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 semi-empirical tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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.
Following early applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems 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}
Following early applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in the quantum chemistry community 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 systems 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}
Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by well-documented correlation-consistent Gaussian basis sets, \cite{Dunning_1989} together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. \cite{Ren_2012b}
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
@ -523,24 +523,25 @@ Similar difficulties emerge in solid-state physics for semiconductors where exte
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}
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_2011,Baumeier_2012a,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{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to important applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
%==========================================
\subsection{Combining BSE with PCM and QM/MM models}
%==========================================
The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface, etc.) is an important step towards the description of realistic systems. Pioneering $BSE$ studies demonstrated e.g. the large renormalization of charge and neutral excitations of molecular systems and nanotubes close to a metallic electrode or in bundles. \cite{Lastra_2011,Rohlfing_2012,Spataru_2013}
Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
\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. As a matter of fact,
dressing the bare Coulomb potential with the reaction field matrix
\begin{equation}
$ \;[
v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
\end{equation}
\titou{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 (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase.
]$
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 (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase.
The reaction field matrix $v^{\text{reac}}(\br,\br'; \omega)$ 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.
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.
Once the reaction field matrix is known, with typically $\order*{N^3}$ operations (where $\Norb$ is the number of orbitals), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
Once the reaction field matrix is known, with typically $\order*{\Norb N_{MM}^2}$ operations (where $\Norb$ is the number of orbitals and $N_{MM}$ 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.
A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field allows to capture both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations.
A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations.
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.
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 for water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range.
@ -555,15 +556,16 @@ We now leave the description of successes to discuss difficulties and future dir
%==========================================
As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT.
Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques.
At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{\Norb^3}$ operations using iterative techniques. \cite{Ljungberg_2015}
Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. Explicit calculation of the full BSE Hamiltonian in transition space can be further avoided using density matrix perturbation theory,
\cite{Rocca_10,Nguyen_2019} not reducing though the $\order*{\Norb^4}$ scaling, sacrificing further the knowledge of the eigenvectors.
Exploiting further the locality of localized atomic basis orbitals, the BSE absorption spectrum could be obtained with $\order*{\Norb^3}$ operations using such 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 $GW$ calculation that scales as $\order{\Norb^4}$ 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.
%%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.
\xavier{While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit,} \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}
While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit, \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 spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in random-phase approximation (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 containing several hundred atoms on standard laboratory clusters.
@ -573,11 +575,11 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
\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
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels. \titou{[REFS]}
\titou{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}}
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, \titou{[REFS]}
contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.
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.
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 Tamm-Dancoff approximation was able to partly reduce this error. However, the error remains rather unsatisfactory for reliable applications.
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 Tamm-Dancoff approximation was able to partly reduce this error. However, as it stands, the BSE inaccuracy for triplets remains rather unsatisfactory for reliable applications.
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}
\\
@ -590,7 +592,7 @@ An alternative cure was offered by hybridizing TD-DFT and BSE, that is, by addin
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{Bernardi_1996,Olivucci_2010,Robb_2007}
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}
Reliable predictions of these mechanisms which have attracted much experimental and theoretical interest lately require exploring the ground- and excited-state PES.
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{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) 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.
@ -599,8 +601,8 @@ One of the strongest assets of TD-DFT is the availability of first- and second-o
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.
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,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, while computing the KS-LDA forces as its ground-state contribution.
\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.
\\
%==========================================
@ -614,7 +616,7 @@ A promising route, which closely follows RPA-type formalisms, \cite{Furche_2008,
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}
Their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
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}
Moreover, it was also observed in Ref.~\citenum{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
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.
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.
This shortcoming has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
\\
@ -644,7 +646,7 @@ This shortcoming has been thoroughly described in several previous studies.\cite
%==========================================
\subsection{Beyond the static approximation}
%==========================================
Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups around the world.\cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups.\cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
As mentioned earlier in this \textit{Perspective}, most of BSE calculations are performed within the so-called static approximation, which substitutes the dynamically-screened (\ie, frequency-dependent) Coulomb potential $W(\omega)$ by its static limit $W(\omega = 0)$.
It is important to mention that diagonalizing the BSE Hamiltonian in the static approximation corresponds to solving a \textit{linear} eigenvalue problem in the space of single excitations, while it is, in its dynamical form, a non-linear eigenvalue problem (in the same space) which is much harder to solve from a numerical point of view.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE spectrum, which obviously hampers the applicability of BSE as double excitation may play, indirectly, a key role in photochemistry mechanisms.
@ -749,7 +751,7 @@ Although far from being exhaustive, we hope to have provided, in the present \te
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).
We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will inspire the next generation of theoretical and computational chemists to roll up their sleeves and embrace this fascinating formalism, which, we believe, has a bright future within the physical chemistry community.
We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will inspire the next generation of theoretical and computational chemists to roll up their sleeves and embrace this fascinating formalism, which, we believe, has a bright future within the physical chemistry community. \xavier{ Je serais un peu moins emphatique ... 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 be extremely valuable within the physical chemistry community.}
\\
%%%%%%%%%%%%%%%%%%%%%%%%