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Manuscript/BSE_JPCL.bib
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@ -1,13 +1,17 @@
%% 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 2020-06-08 22:10:57 +0200 %% Created for Pierre-Francois Loos at 2020-06-16 09:59:55 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{cite-key,
Date-Added = {2020-06-16 09:59:45 +0200},
Date-Modified = {2020-06-16 09:59:45 +0200}}
@article{Packer_1996, @article{Packer_1996,
Author = {Packer, M. K. and Dalskov, E. K. and Enevoldsen, T. and Jensen, H. J. and Oddershede, J.}, Author = {Packer, M. K. and Dalskov, E. K. and Enevoldsen, T. and Jensen, H. J. and Oddershede, J.},
Date-Added = {2020-06-08 21:57:16 +0200}, Date-Added = {2020-06-08 21:57:16 +0200},
@ -17,7 +21,8 @@
Pages = {5886--5900}, Pages = {5886--5900},
Title = {A New Implementation of the Second-Order Polarization Propagator Approximation (SOPPA): The Excitation Spectra of Benzene and Naphthalene}, Title = {A New Implementation of the Second-Order Polarization Propagator Approximation (SOPPA): The Excitation Spectra of Benzene and Naphthalene},
Volume = {105}, Volume = {105},
Year = {1996}} Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1063/1.472430}}
@article{Wu_2019, @article{Wu_2019,
Author = {Xin‐Ping Wu and Indrani Choudhuri and Donald G. Truhlar}, Author = {Xin‐Ping Wu and Indrani Choudhuri and Donald G. Truhlar},
@ -14737,20 +14742,21 @@
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevMaterials.1.025602}} Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevMaterials.1.025602}}
@article{Marom_2012, @article{Marom_2012,
title = {Benchmark of $GW$ methods for azabenzenes}, Author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K\"orzd\"orfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick},
author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K\"orzd\"orfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick}, Doi = {10.1103/PhysRevB.86.245127},
journal = {Phys. Rev. B}, Issue = {24},
volume = {86}, Journal = {Phys. Rev. B},
issue = {24}, Month = {Dec},
pages = {245127}, Numpages = {16},
numpages = {16}, Pages = {245127},
year = {2012}, Publisher = {American Physical Society},
month = {Dec}, Title = {Benchmark of $GW$ methods for azabenzenes},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevB.86.245127},
doi = {10.1103/PhysRevB.86.245127}, Volume = {86},
url = {https://link.aps.org/doi/10.1103/PhysRevB.86.245127} Year = {2012},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.86.245127},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.86.245127}}
@misc{listofrefs, @misc{listofrefs,
note ={ For a list of applications to molecular systems, see e.g. Table 1 of Ref.~\citenum{Blase_2018}. } note ={See Table 1 of Ref.~\citenum{Blase_2018} for an exhaustive list of applications to molecular systems.}
} }

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Manuscript/BSE_JPCL.tex
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@ -246,7 +246,7 @@ where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function.
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. 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.
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. 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.
This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", \titou{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. 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.
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 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
\begin{equation}\label{eq:IPAEgap} \begin{equation}\label{eq:IPAEgap}
\EgFun = I^\Nel - A^\Nel, \EgFun = I^\Nel - A^\Nel,
@ -265,7 +265,7 @@ A central property of the one-body Green's function is that its frequency-depend
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 $\varepsilon_s < \mu$. 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 $s$\textsuperscript{th} excited state of the $\Nel$-electron system. Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{th} excited state of the $\Nel$-electron system.
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). 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).
Unlike Kohn-Sham (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. 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.
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. 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.
Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
@ -316,7 +316,7 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
\includegraphics[width=0.55\linewidth]{fig1/fig1} \includegraphics[width=0.55\linewidth]{fig1/fig1}
\caption{ \caption{
Hedin's pentagon connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965} Hedin's pentagon connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$ (gray arrows). The path made of black arrows shows the $GW$ process which bypasses the computation of $\Gamma$ (gray arrows).
As input, one must provide KS (or HF) orbitals and their corresponding energies. As input, one must provide KS (or HF) orbitals and their corresponding energies.
Depending on the level of self-consistency in the $GW$ calculation, only the orbital energies or both the orbitals and their energies are corrected. Depending on the level of self-consistency in the $GW$ calculation, only the orbital energies or both the orbitals and their energies are corrected.
As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE optical excitations of the system of interest. As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE optical excitations of the system of interest.
@ -373,8 +373,8 @@ We will comment further on this particular point below when addressing the quali
%These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations. %These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
%For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2013a,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017} %For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2013a,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999} Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999}
However, remaining a low-order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995} However, remaining a low-order perturbative approach starting with a single-determinant mean-field solution, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}
\\ \\
%%% FIG 2 %%% %%% FIG 2 %%%
@ -496,7 +496,7 @@ 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 tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000} 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}
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. 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 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} 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}
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. 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.
@ -603,10 +603,10 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
\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 or thermally activated delayed fluorescence (TADF). The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission or thermally activated delayed fluorescence (TADF).
From a more theoretical point of view, triplet instabilities are intimately linked to the stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and KS \cite{Bauernschmitt_1996} levels, hampering the applicability of TD-DFT for popular range-separated hybrids containing a large fraction of long-range exact exchange. 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.
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. 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. 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.
However, as it stands, the BSE accuracy for triplets remains rather unsatisfactory for reliable applications. However, as it stands, the BSE accuracy 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} 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}
\\ \\