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127
Manuscript/BSE_JPCL.bib
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@ -14545,3 +14545,130 @@
Year = {2020},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.124.107401},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.124.107401}}
@article{Prandini_2019,
Author = { Prandini, Gianluca and Rignanese, Gian-Marco and Marzari, Nicola},
Journal = { npj Comput. Mater. },
Issue ={ 5 },
Pages ={ 129 },
Year = { 2019 },
Title = { Photorealistic Modelling of Metals from First Principles },
Url = { https://doi.org/10.1038/s41524-019-0266-0 }
}
@article{Improta_2016,
author = {Improta, Roberto and Santoro, Fabrizio and Blancafort, Lluís},
title = {Quantum Mechanical Studies on the Photophysics and the Photochemistry of Nucleic Acids and Nucleobases},
journal = { Chem. Rev. },
volume = {116},
number = {6},
pages = {3540-3593},
year = {2016},
doi = {10.1021/acs.chemrev.5b00444},
note ={PMID: 26928320},
URL = { https://doi.org/10.1021/acs.chemrev.5b00444},
eprint = { https://doi.org/10.1021/acs.chemrev.5b00444}
}
@Article{Kippelen_2009,
author ="Kippelen, Bernard and Brédas, Jean-Luc",
title ="Organic photovoltaics",
journal ="Energy Environ. Sci.",
year ="2009",
volume ="2",
issue ="3",
pages ="251-261",
publisher ="The Royal Society of Chemistry",
doi ="10.1039/B812502N",
url ="http://dx.doi.org/10.1039/B812502N",
abstract ="Organic photovoltaics{,} the technology to convert sun light into electricity by employing thin films of organic semiconductors{,} has been the subject of active research over the past 20 years and has received increased interest in recent years by the industrial sector. This technology has the potential to spawn a new generation of low-cost{,} solar-powered products with thin and flexible form factors. Here{,} we introduce the energy and environmental science community to the basic concepts of organic photovoltaics and discuss some recent science and engineering results and future challenges."}
@article{Rohlfing_1999,
title = {Optical Excitations in Conjugated Polymers},
author = {Rohlfing, Michael and Louie, Steven G.},
journal = {Phys. Rev. Lett.},
volume = {82},
issue = {9},
pages = {1959--1962},
numpages = {0},
year = {1999},
month = {Mar},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.82.1959},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.82.1959}
}
@article{Horst_1999,
title = {Ab Initio Calculation of the Electronic and Optical Excitations in Polythiophene: Effects of Intra- and Interchain Screening},
author = {van der Horst, J.-W. and Bobbert, P. A. and Michels, M. A. J. and Brocks, G. and Kelly, P. J.},
journal = {Phys. Rev. Lett.},
volume = {83},
issue = {21},
pages = {4413--4416},
numpages = {0},
year = {1999},
month = {Nov},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.83.4413},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.4413}
}
@article{Puschnig_2002,
title = {Suppression of Electron-Hole Correlations in 3D Polymer Materials},
author = {Puschnig, Peter and Ambrosch-Draxl, Claudia},
journal = {Phys. Rev. Lett.},
volume = {89},
issue = {5},
pages = {056405},
numpages = {4},
year = {2002},
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.89.056405},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.89.056405}
}
@article{Sai_2008,
title = {Optical Spectra and Exchange-Correlation Effects in Molecular Crystals},
author = {Sai, Na and Tiago, Murilo L. and Chelikowsky, James R. and Reboredo, Fernando A.},
journal = {Phys. Rev. B},
volume = {77},
issue = {16},
pages = {161306},
numpages = {4},
year = {2008},
month = {Apr},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.77.161306},
url = {https://link.aps.org/doi/10.1103/PhysRevB.77.161306}
}
@article{Tiago_2003,
title = {Ab initio calculation of the electronic and optical properties of solid pentacene},
author = {Tiago, Murilo L. and Northrup, John E. and Louie, Steven G.},
journal = {Phys. Rev. B},
volume = {67},
issue = {11},
pages = {115212},
numpages = {6},
year = {2003},
month = {Mar},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.67.115212},
url = {https://link.aps.org/doi/10.1103/PhysRevB.67.115212}
}
@article{Ren_2012,
doi = {10.1088/1367-2630/14/5/053020},
url = {https://doi.org/10.1088%2F1367-2630%2F14%2F5%2F053020},
year = 2012,
month = {may},
publisher = {{IOP} Publishing},
volume = {14},
number = {5},
pages = {053020},
author = {Xinguo Ren and Patrick Rinke and Volker Blum and Jürgen Wieferink and Alexandre Tkatchenko and Andrea Sanfilippo and Karsten Reuter and Matthias Scheffler},
title = {Resolution-of-identity Approach to Hartree{\textendash}Fock, hybrid density functionals, {RPA}, {MP}2 and {GW} with numeric atom-centered orbital basis functions},
journal = {New J. Phys.},
abstract = {}
}

21
Manuscript/BSE_JPCL.tex
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@ -230,7 +230,7 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi
Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003}
Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
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. \titou{[REFS]}
The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016} \xavier{[Xav: Good ref on theory for photocatalysis still needed.]}
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 scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
\\
@ -267,13 +267,14 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
\end{equation}
where we introduce the usual composite index, \eg, $1 \equiv (\bx_1 t_1)$.
Here, $\delta$ is Dirac's delta function, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one gets the familiar eigenvalue equation, \ie,
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}],
dropping spin-variables for simplicity, one gets the familiar eigenvalue equation, \ie,
\begin{equation}
h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
\end{equation}
which resembles formally the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.
\titou{The spin variable has disappear. How do we deal with this?}
%% \titou{The spin variable has disappear. How do we deal with this?}
\\
%===================================
@ -336,7 +337,7 @@ Because, one is usually interested only by the quasiparticle solution, in practi
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.
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 LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV \titou{[REFS]} with a computational cost scaling quartically with the number of basis functions (see below).
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 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.
Further improvements may be obtained via self-consistency of the Hedin's equations (see Fig.~\ref{fig:pentagon}).
@ -378,7 +379,7 @@ The derivative with respect to $U$ of the Dyson equation yields
\begin{multline}\label{eq:DysonL}
L(12,34) = L_0(12,34)
\\
\int d5678 \, L_0(12,34) \Xi^{\BSE}(5,6,7,8) L(78,34),
\int d5678 \, L_0(12,56) \Xi^{\BSE}(5,6,7,8) L(78,34),
\end{multline}
where $L_0 = \partial G_0 / \partial U$ is the Hartree 4-point susceptibility and
\begin{equation}
@ -463,8 +464,8 @@ 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_1999} paved the way to the popularization in the solid-state physics community of the BSE formalism.
Following early applications to periodic polymers and molecules, \titou{[REFS]} 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}
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. \titou{[REFS]}
Following early applications to periodic polymers and molecules, \cite{Rohlfing_1999,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}
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_2012]}
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
\begin{equation}
@ -481,7 +482,7 @@ where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$
\includegraphics[width=0.7\linewidth]{gaps}
\caption{
Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
$\EB$ is the excitonic effect, while $I^\Nel$ and $A^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system.
$\EB$ is the electron-hole or excitonic binding energy, while $I^\Nel$ and $A^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system.
$\Eg^{\KS}$ and $\Eg^{\GW}$ are the KS and $GW$ HOMO-LUMO gaps.
See main text for the definition of the other quantities
\label{fig:gaps}}
@ -611,10 +612,10 @@ In contrast to TD-DFT which relies on KS-DFT as its ground-state analog, the gro
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}
A promising route, which closely follows RPA-type formalisms, \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005,Olsen_2014,Maggio_2016,Holzer_2018,Loos_2020}
Thanks to comparisons with both similar and state-of-art computational approaches, we have recently showed 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}
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, we also observe 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 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}
\\