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15
Manuscript/BSE_JPCL.bib
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@ -14735,3 +14735,18 @@
Year = {2017},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevMaterials.1.025602}}
@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},
journal = {Phys. Rev. B},
volume = {86},
issue = {24},
pages = {245127},
numpages = {16},
year = {2012},
month = {Dec},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.86.245127},
url = {https://link.aps.org/doi/10.1103/PhysRevB.86.245127}
}

104
Manuscript/BSE_JPCL.tex
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@ -244,7 +244,14 @@ While the one-body density stands as the basic variable in density-functional th
\end{equation}
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.
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.
\xavier{ This definition indicates that the one-body Green's function is well suited to obtain ``charged excitations", more commonly labeled the electronic energy levels, as obtained e.g. in a direct or inverse photo-emission experiment where an electron is ejected or added to the N-electron system. In particular, and as compared to Kohn-Sham (KS) eigenvalues, the Green's function formalism offers in a first step a more rigorous path to the ionization potential, the electronic affinity and the experimental (photoemission) fundamental gap:
\begin{equation}
\EgFun = I^\Nel - A^\Nel,
\label{eq:IPAEgap}
\end{equation}
where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system. Since these energy levels obtained thanks to Green's function MBPT serve as an input to Bethe-Salpeter calculations, we start by discussing them in some details. }
\\
%===================================
@ -322,42 +329,58 @@ Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V
\omega = \varepsilon_p^{\KS} +
\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\omega) - V^{\XC}}{\phi_p^{\KS}}.
\end{equation}
As a non-linear equation, the quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with spectral weights $Z(\varepsilon_{p,s}^{\GW})$, where
\begin{equation}
Z_p(\omega) = \qty[ 1 - \pdv{\Sigma^{\GW}(\omega)}{\omega} ]^{-1}.
\end{equation}
These solutions have different physical meanings.
In addition to the principal quasiparticle solution $\varepsilon_{p}^{\GW} \equiv \varepsilon_{p,0}^{\GW}$, which contains most of the spectral weight, there is a finite number of satellite resonances stemming from the poles of the self-energy with smaller spectral weights.
As a non-linear equation, the self-consistent quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with spectral weights $Z(\omega) = [ 1- \partial \Sigma^{\GW} / \partial {\omega} ]^{-1}$ taken at $\omega = \varepsilon_{p,s}^{\GW} }$. The existence of a well defined quasiparticle energy requires a solution with a large Z-factor, namely close to unity, a condition not always fulfilled for states far away from the gap.
\cite{Veril_2018}
Because, one is usually interested only by the quasiparticle solution, in practice, Eq.~\eqref{eq:QP-eq} is often linearized around $\omega = \varepsilon_p^{\KS}$ as follows:
\begin{equation}
\varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
Z_p(\varepsilon_p^{\KS}) \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\KS}) - V^{\XC}}{\phi_p^{\KS}}.
\end{equation}
%\begin{equation}
% Z_p = \qty[ 1 -
% % \pdv{\Sigma^{\GW}(\omega)}{\omega} ]^{-1}.
% \end{equation}
%These solutions have different physical meanings.
%In addition to the principal quasiparticle solution $\varepsilon_{p}^{\GW} \equiv \varepsilon_{p,0}^{\GW}$, which contains most of the spectral weight, there is a finite number of satellite resonances stemming from the poles of the self-energy with smaller spectral weights.
%Because, one is usually interested only by the quasiparticle solution, in practice, Eq.~\eqref{eq:QP-eq} is often linearized around $\omega = \varepsilon_p^{\KS}$ as follows:
%\begin{equation}
% \varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
% Z_p(\varepsilon_p^{\KS})
% \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\KS}) - V^{\XC}}{\phi_p^{\KS}}.
%\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} 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.
Further improvements may be obtained via self-consistency of Hedin's equations (see Fig.~\ref{fig:pentagon}).
There exists two main types of self-consistent $GW$ methods:
i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986,Blase_2011} where the quasiparticle energies are updated at each iteration, and
ii) \textit{``quasiparticle self-consistent''} $GW$ (qs$GW$), \cite{vanSchilfgaarde_2006} where one updates both the quasiparticle energies and the corresponding orbitals.
Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
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. Namely, the quasiparticle energies, and in particular the HOMO-LUMO gap, depends on the input Kohn-Sham eigenvalues. Tuning of the starting point functionals or self-consistency are two difference approaches to improve on this problem. We will comment on this below when addressing Bethe-Salpeter optical excitations.
%Further improvements may be obtained via self-consistency of Hedin's equations (see Fig.~\ref{fig:pentagon}).
%There exists two main types of self-consistent $GW$ methods:
%i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986} where the quasiparticle energies are updated at each iteration, and
%ii) \textit{``quasiparticle self-consistent''} $GW$ (qs$GW$), \cite{vanSchilfgaarde_2006} where one updates both the quasiparticle energies and the corresponding orbitals. \xavier{In the case of molecular systems, self-consistency has proven important in particular when starting with DFT Kohn-Sham eigenstates generated with local or semilocal functionals. \cite{Blase_2011,Marom_2012,Kaplan_2016,Rangel_2016} As an alternative, input DFT calculations with a functional tuned such that the Kohn-Sham gap is close to the one obtained by calculating the ionization potential (IP) and electronic affinity (EA) via $\Delta$SCF total energy differences between the anion, cation and neutral systems, lead to good accuracy already at the single-shot $G_0W_0$ level. \cite{Bruneval_2012,Rangel_2016} }
%Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006,Rostgaard_2010,Caruso_2013a,Koval_2014}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017,Loos_2018}
A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998}
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}
%However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook} Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006,Rostgaard_2010,Caruso_2013a,Koval_2014}
%In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017,Loos_2018}
%A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998}
%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}
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}
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}
\\
%%% FIG 2 %%%
\begin{figure*}[ht]
\includegraphics[width=0.7\linewidth]{gaps}
\caption{
Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
$\EB$ 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}}
\end{figure*}
%%% %%% %%%
%===================================
\subsection{Neutral excitations}
%===================================
@ -430,7 +453,7 @@ 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 typically 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.
%(In the case of complex orbitals, we refer the reader to Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation.)
The resonant and coupling parts of the BSE Hamiltonian read
@ -458,6 +481,8 @@ We emphasize that these equations can be solved at exactly the same cost as the
This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its \emph{pros} and \emph{cons}.
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Historical overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -465,30 +490,17 @@ 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}
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{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 Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. \cite{Ren_2012b}
Following pioneering applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in quantum chemistry 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.
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}
\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
\end{equation}
with the experimental (photoemission) fundamental gap
\begin{equation}
\EgFun = I^\Nel - A^\Nel,
\end{equation}
where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system (see Fig.~\ref{fig:gaps}).
of Eq.~\ref{eq:IPAEgap} (see Fig.~\ref{fig:gaps}).
%%% FIG 2 %%%
\begin{figure*}[ht]
\includegraphics[width=0.7\linewidth]{gaps}
\caption{
Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
$\EB$ 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}}
\end{figure*}
%%% %%% %%%
Standard $G_0W_0$ calculations starting with KS eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input KS gap
\begin{equation}
@ -502,11 +514,11 @@ Such an underestimation of the fundamental gap leads to a similar underestimatio
\begin{equation}
\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
\end{equation}
where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
where $\EB$ accounts for the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
%Because of this, we have $\EgOpt < \EgFun$.
Such a residual gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016}
Alternatively, self-consistent schemes such as ev$GW$ and qs$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011} where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016}
Such a residual gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016,Gui_2018}
Alternatively, self-consistent approaches such as eigenvalue self-consistent (ev$GW$) \cite{Hybertsen_1986} or quasiparticle self-consistent (qs$GW$) \cite{vanSchilfgaarde_2006} schemes, where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rostgaard_2010,Blase_2011,Ke_2011,Rangel_2016,Kaplan_2016,Caruso_2016}
As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering roughly ca.~200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange.
@ -530,7 +542,7 @@ The success of the BSE formalism to treat CT excitations has been demonstrated i
\begin{figure}[ht]
\includegraphics[width=0.6\linewidth]{CTfig}
\caption{
Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface (bottom).
Symbolic representation of (a) extended Wannier exciton with large electron-hole average distance, and (b) Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface.
Wannier and CT excitations require long-range electron-hole interaction accounting for the host dielectric constant.
In the case of Wannier excitons, the binding energy $\EB$ can be well approximated by the standard hydrogenoid model where $\mu$ is the effective mass and $\epsilon$ is the dielectric constant.
\label{fig:CTfig}}

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