History and THeory done
This commit is contained in:
parent
a9fa8a6f70
commit
3f885dc439
@ -1,7 +1,7 @@
|
||||
%% This BibTeX bibliography file was created using BibDesk.
|
||||
%% http://bibdesk.sourceforge.net/
|
||||
|
||||
%% Created for Pierre-Francois Loos at 2020-06-04 22:08:23 +0200
|
||||
%% Created for Pierre-Francois Loos at 2020-06-05 12:56:19 +0200
|
||||
|
||||
|
||||
%% Saved with string encoding Unicode (UTF-8)
|
||||
@ -722,18 +722,18 @@
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}
|
||||
|
||||
@article{Duchemin_2020,
|
||||
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}
|
||||
}
|
||||
Author = {Duchemin, Ivan and Blase, Xavier},
|
||||
Doi = {10.1021/acs.jctc.9b01235},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.9b01235},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 32023052},
|
||||
Number = {3},
|
||||
Pages = {1742-1756},
|
||||
Title = {Robust Analytic-Continuation Approach to Many-Body GW Calculations},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.9b01235},
|
||||
Volume = {16},
|
||||
Year = {2020},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b01235}}
|
||||
|
||||
@article{Dunning_1989,
|
||||
Author = {T. H. {Dunning, Jr.}},
|
||||
@ -2380,7 +2380,7 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235}
|
||||
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 = { J. Phys. Chem. Lett. },
|
||||
Journal = {J. Phys. Chem. Lett.},
|
||||
Note = {PMID: 29280376},
|
||||
Number = {2},
|
||||
Pages = {306-312},
|
||||
@ -12745,21 +12745,6 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235}
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4940139}}
|
||||
|
||||
@article{Boulanger_2014,
|
||||
Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
|
||||
Doi = {10.1021/ct401101u},
|
||||
File = {/Users/loos/Zotero/storage/KTW3SS9F/Boulanger_2014.pdf},
|
||||
Issn = {1549-9618, 1549-9626},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Language = {en},
|
||||
Month = mar,
|
||||
Number = {3},
|
||||
Pages = {1212--1218},
|
||||
Title = {Fast and {{Accurate Electronic Excitations}} in {{Cyanines}} with the {{Many}}-{{Body Bethe}}\textendash{}{{Salpeter Approach}}},
|
||||
Volume = {10},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct401101u}}
|
||||
|
||||
@article{Bruneval_2009,
|
||||
Author = {Bruneval, Fabien},
|
||||
Doi = {10.1103/PhysRevLett.103.176403},
|
||||
@ -14645,90 +14630,95 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235}
|
||||
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}
|
||||
}
|
||||
Author = {Nguyen, Ngoc Linh and Ma, He and Govoni, Marco and Gygi, Francois and Galli, Giulia},
|
||||
Doi = {10.1103/PhysRevLett.122.237402},
|
||||
Issue = {23},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Month = {Jun},
|
||||
Numpages = {6},
|
||||
Pages = {237402},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Finite-Field Approach to Solving the Bethe-Salpeter Equation},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402},
|
||||
Volume = {122},
|
||||
Year = {2019},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402},
|
||||
Bdsk-Url-2 = {https://doi.org/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}
|
||||
}
|
||||
Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
|
||||
Doi = {10.1021/ct401101u},
|
||||
Eprint = {https://doi.org/10.1021/ct401101u},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 26580191},
|
||||
Number = {3},
|
||||
Pages = {1212-1218},
|
||||
Title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe--Salpeter Approach},
|
||||
Url = {https://doi.org/10.1021/ct401101u},
|
||||
Volume = {10},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {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}
|
||||
}
|
||||
Author = {Spataru, Catalin D.},
|
||||
Doi = {10.1103/PhysRevB.88.125412},
|
||||
Issue = {12},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Sep},
|
||||
Numpages = {8},
|
||||
Pages = {125412},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Electronic and optical gap renormalization in carbon nanotubes near a metallic surface},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412},
|
||||
Volume = {88},
|
||||
Year = {2013},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412},
|
||||
Bdsk-Url-2 = {https://doi.org/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}
|
||||
}
|
||||
Author = {Rohlfing, Michael},
|
||||
Doi = {10.1103/PhysRevLett.108.087402},
|
||||
Issue = {8},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Month = {Feb},
|
||||
Numpages = {5},
|
||||
Pages = {087402},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Redshift of Excitons in Carbon Nanotubes Caused by the Environment Polarizability},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402},
|
||||
Volume = {108},
|
||||
Year = {2012},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402},
|
||||
Bdsk-Url-2 = {https://doi.org/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}
|
||||
}
|
||||
Author = {Yin, Huabing and Ma, Yuchen and Mu, Jinglin and Liu, Chengbu and Rohlfing, Michael},
|
||||
Doi = {10.1103/PhysRevLett.112.228301},
|
||||
Issue = {22},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Month = {Jun},
|
||||
Numpages = {5},
|
||||
Pages = {228301},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Charge-Transfer Excited States in Aqueous DNA: Insights from Many-Body Green's Function Theory},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301},
|
||||
Volume = {112},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301},
|
||||
Bdsk-Url-2 = {https://doi.org/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}
|
||||
}
|
||||
Author = {Li, Jing and D'Avino, Gabriele and Pershin, Anton and Jacquemin, Denis and Duchemin, Ivan and Beljonne, David and Blase, Xavier},
|
||||
Doi = {10.1103/PhysRevMaterials.1.025602},
|
||||
Issue = {2},
|
||||
Journal = {Phys. Rev. Materials},
|
||||
Month = {Jul},
|
||||
Numpages = {9},
|
||||
Pages = {025602},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Correlated electron-hole mechanism for molecular doping in organic semiconductors},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602},
|
||||
Volume = {1},
|
||||
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}}
|
||||
|
||||
@ -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 \xavier{singlet ?} 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 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.
|
||||
@ -304,13 +304,13 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.7\linewidth]{fig1/fig1}
|
||||
\includegraphics[width=0.55\linewidth]{fig1/fig1}
|
||||
\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}
|
||||
The path made of back arrow 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.
|
||||
Depending on the level of self-consistency of 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 neutral excitations.
|
||||
As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE neutral excitations of the system of interest.
|
||||
\label{fig:pentagon}}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
@ -429,7 +429,9 @@ 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. They are here taken to be real in the case of finite size systems.
|
||||
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.
|
||||
(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
|
||||
\begin{gather}
|
||||
R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab},
|
||||
@ -451,7 +453,7 @@ $(ia|jb)$ 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.
|
||||
\\
|
||||
@ -463,14 +465,14 @@ 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{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}
|
||||
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
|
||||
\begin{equation}
|
||||
\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
|
||||
\end{equation}
|
||||
with the experimental (photoemission) fundamental gap \cite{Bredas_2014}
|
||||
with the experimental (photoemission) fundamental gap
|
||||
\begin{equation}
|
||||
\EgFun = I^\Nel - A^\Nel,
|
||||
\end{equation}
|
||||
@ -505,8 +507,8 @@ where $\EB$ is the excitonic effect, that is, the stabilization implied by the a
|
||||
|
||||
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}
|
||||
As a result, BSE excitation singlet 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 more than hundred representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
|
||||
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 more than hundred 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 global hybrid functionals with various amounts of exact exchange.
|
||||
|
||||
|
||||
@ -557,7 +559,7 @@ 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. 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.
|
||||
\cite{Rocca_2010,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}
|
||||
|
||||
|
||||
Binary file not shown.
@ -35,8 +35,8 @@ decoration={snake,
|
||||
\node [Op4, align=center] (Sigma) at (-3*0.587785, 3*0.809017) {$\Sigma$};
|
||||
\node [Input, align=center] (In) [above=of G] {};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {};
|
||||
\node [Input, align=center] (In) [above=of G] {KS-DFT};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {BSE};
|
||||
\node [Input, align=center] (In) [above=of G, yshift=1cm] {KS-DFT};
|
||||
\node [Output, align=center] (Out) [above=of Sigma, yshift=1cm] {BSE};
|
||||
\path
|
||||
(G) edge [->,color=gray!50] node [above,sloped,black] {$\Gamma = 1 + \fdv{\Sigma}{G} GG \Gamma$} (Gamma)
|
||||
(Gamma) edge [->,color=gray!50] node [below,sloped,black] {$P = - i GG \Gamma$} (P)
|
||||
@ -45,7 +45,7 @@ decoration={snake,
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$G = G_\text{0} + G_\text{0} \Sigma G$} (G)
|
||||
(G) edge [->,color=black] node [above,sloped,black] {$P = - i GG \quad (\Gamma = 1)$} (P)
|
||||
(In) edge [->,color=black] node [above,sloped,black] {$\varepsilon^\text{KS}$} (G)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$\varepsilon^\text{GW}$} (Out)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$W(\omega)$ \& $\varepsilon^\text{GW}$} (Out)
|
||||
;
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
|
||||
Binary file not shown.
|
Before Width: | Height: | Size: 205 KiB After Width: | Height: | Size: 204 KiB |
Loading…
Reference in New Issue
Block a user