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@ -1,13 +1,269 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2023-01-20 10:08:25 +0100 %% Created for Pierre-Francois Loos at 2023-01-30 15:48:50 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Shirley_1996,
author = {Shirley, Eric L.},
date-added = {2023-01-30 15:47:29 +0100},
date-modified = {2023-01-30 15:47:36 +0100},
doi = {10.1103/PhysRevB.54.7758},
issue = {11},
journal = {Phys. Rev. B},
month = {Sep},
numpages = {0},
pages = {7758--7764},
publisher = {American Physical Society},
title = {Self-consistent GW and higher-order calculations of electron states in metals},
url = {https://link.aps.org/doi/10.1103/PhysRevB.54.7758},
volume = {54},
year = {1996},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.54.7758},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.54.7758}}
@article{Maggio_2017b,
author = {Maggio, Emanuele and Kresse, Georg},
date-added = {2023-01-30 15:45:22 +0100},
date-modified = {2023-01-30 15:45:39 +0100},
doi = {10.1021/acs.jctc.7b00586},
eprint = {https://doi.org/10.1021/acs.jctc.7b00586},
journal = {J. Chem. Theory Comput.},
note = {PMID: 28873298},
number = {10},
pages = {4765-4778},
title = {GW Vertex Corrected Calculations for Molecular Systems},
url = {https://doi.org/10.1021/acs.jctc.7b00586},
volume = {13},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.7b00586}}
@article{Gruneis_2014,
author = {Gr\"uneis, Andreas and Kresse, Georg and Hinuma, Yoyo and Oba, Fumiyasu},
date-added = {2023-01-30 15:41:07 +0100},
date-modified = {2023-01-30 15:41:14 +0100},
doi = {10.1103/PhysRevLett.112.096401},
issue = {9},
journal = {Phys. Rev. Lett.},
month = {Mar},
numpages = {5},
pages = {096401},
publisher = {American Physical Society},
title = {Ionization Potentials of Solids: The Importance of Vertex Corrections},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.096401},
volume = {112},
year = {2014},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.112.096401},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.112.096401}}
@article{Shishkin_2007b,
author = {Shishkin, M. and Marsman, M. and Kresse, G.},
date-added = {2023-01-30 15:39:45 +0100},
date-modified = {2023-01-30 15:39:56 +0100},
doi = {10.1103/PhysRevLett.99.246403},
issue = {24},
journal = {Phys. Rev. Lett.},
month = {Dec},
numpages = {4},
pages = {246403},
publisher = {American Physical Society},
title = {Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.99.246403},
volume = {99},
year = {2007},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.99.246403},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.99.246403}}
@article{DelSol_1994,
author = {Del Sole, R. and Reining, Lucia and Godby, R. W.},
date-added = {2023-01-30 15:38:03 +0100},
date-modified = {2023-01-30 15:38:11 +0100},
doi = {10.1103/PhysRevB.49.8024},
issue = {12},
journal = {Phys. Rev. B},
month = {Mar},
numpages = {0},
pages = {8024--8028},
publisher = {American Physical Society},
title = {GW\ensuremath{\Gamma} approximation for electron self-energies in semiconductors and insulators},
url = {https://link.aps.org/doi/10.1103/PhysRevB.49.8024},
volume = {49},
year = {1994},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.49.8024},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.49.8024}}
@article{Morris_2007,
author = {Morris, Andrew J. and Stankovski, Martin and Delaney, Kris T. and Rinke, Patrick and Garc\'{\i}a-Gonz\'alez, P. and Godby, R. W.},
date-added = {2023-01-30 15:36:17 +0100},
date-modified = {2023-01-30 15:36:24 +0100},
doi = {10.1103/PhysRevB.76.155106},
issue = {15},
journal = {Phys. Rev. B},
month = {Oct},
numpages = {9},
pages = {155106},
publisher = {American Physical Society},
title = {Vertex corrections in localized and extended systems},
url = {https://link.aps.org/doi/10.1103/PhysRevB.76.155106},
volume = {76},
year = {2007},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.76.155106},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.76.155106}}
@article{Bickers_1989a,
author = {Bickers, N. E. and Scalapino, D. J. and White, S. R.},
date-added = {2023-01-30 14:25:50 +0100},
date-modified = {2023-01-30 14:25:57 +0100},
doi = {10.1103/PhysRevLett.62.961},
issue = {8},
journal = {Phys. Rev. Lett.},
month = {Feb},
numpages = {0},
pages = {961--964},
publisher = {American Physical Society},
title = {Conserving Approximations for Strongly Correlated Electron Systems: Bethe-Salpeter Equation and Dynamics for the Two-Dimensional Hubbard Model},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.62.961},
volume = {62},
year = {1989},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.62.961},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.62.961}}
@article{Mejuto-Zaera_2022,
author = {Mejuto-Zaera, Carlos and Vl\ifmmode \check{c}\else \v{c}\fi{}ek, Vojt\ifmmode \check{e}\else \v{e}\fi{}ch},
date-added = {2023-01-30 14:21:17 +0100},
date-modified = {2023-01-30 14:21:26 +0100},
doi = {10.1103/PhysRevB.106.165129},
issue = {16},
journal = {Phys. Rev. B},
month = {Oct},
numpages = {8},
pages = {165129},
publisher = {American Physical Society},
title = {Self-consistency in $GW\mathrm{\ensuremath{\Gamma}}$ formalism leading to quasiparticle-quasiparticle couplings},
url = {https://link.aps.org/doi/10.1103/PhysRevB.106.165129},
volume = {106},
year = {2022},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.106.165129},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.106.165129}}
@article{DeDominicis_1964b,
author = {De Dominicis,Cyrano and Martin,Paul C.},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100},
doi = {10.1063/1.1704064},
journal = {J. Math. Phys.},
number = {1},
pages = {31-59},
title = {Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation},
volume = {5},
year = {1964},
bdsk-url-1 = {https://doi-org-s.docadis.univ-tlse3.fr/10.1063/1.1704064},
bdsk-url-2 = {https://doi.org/10.1063/1.1704064}}
@article{DeDominicis_1964a,
author = {De Dominicis,Cyrano and Martin,Paul C.},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100},
doi = {10.1063/1.1704062},
journal = {J. Math. Phys.},
number = {1},
pages = {14-30},
title = {Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation},
volume = {5},
year = {1964},
bdsk-url-1 = {https://doi-org-s.docadis.univ-tlse3.fr/10.1063/1.1704062},
bdsk-url-2 = {https://doi.org/10.1063/1.1704062}}
@inbook{Bickers_2004,
abstract = {Self-consistent field techniques for the many-electron problem are examined using the modern formalism of functional methods. Baym-Kadanoff, or $\Phi$-derivable, approximations are introduced first. After a brief review of functional integration results, the connection between conventional mean-field theory and higher-order Baym-Kadanoff approximations is established through the concept of the action functional. The $\Phi$-derivability criterion for thermodynamic consistency is discussed, along with the calculation of free-energy derivatives. Parquet, or crossing-symmetric, approximations are introduced next. The principal advantages of the parquet approach and its relationship to Baym-Kadanoff theory are outlined. A linear eigenvalue equation is derived to study instabilities of the electronic normal state within Baym-Kadanoff or parquet theory. Finally, numerical techniques for the solution of self-consistent field approximations are reviewed, with particular emphasis on renormalization group methods for frequency and momentum space.},
address = {New York, NY},
author = {Bickers, N. E.},
booktitle = {Theoretical Methods for Strongly Correlated Electrons},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100},
doi = {10.1007/0-387-21717-7_6},
editor = {S{\'e}n{\'e}chal, David and Tremblay, Andr{\'e}-Marie and Bourbonnais, Claude},
isbn = {978-0-387-21717-8},
pages = {237--296},
publisher = {Springer New York},
title = {Self-Consistent Many-Body Theory for Condensed Matter Systems},
url = {https://doi.org/10.1007/0-387-21717-7_6},
year = {2004},
bdsk-url-1 = {https://doi.org/10.1007/0-387-21717-7_6}}
@article{Bickers_1989b,
abstract = {We discuss the solution of nontrivial conserving approximations for electronic correlation functions in systems with strong collective fluctuations. The formal properties of conserving approximations have been well known for over twenty years, but numerical solutions have been limited to Hartree-Fock level. We extend the formal analysis of Baym and Kadanoff in order to derive the simplest self-consistent approximation based on exchange of fluctuations in the particle-hole and particle-particle channels. We then describe a practical technique for calculating self-consistent single-particle Green's functions and solving the finite-temperature Bethe-Salpeter equation for electrons on a lattice.},
author = {N.E Bickers and D.J Scalapino},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:26:01 +0100},
doi = {https://doi.org/10.1016/0003-4916(89)90359-X},
issn = {0003-4916},
journal = {Ann. Phys.},
number = {1},
pages = {206-251},
title = {Conserving approximations for strongly fluctuating electron systems. I. Formalism and calculational approach},
url = {https://www.sciencedirect.com/science/article/pii/000349168990359X},
volume = {193},
year = {1989},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/000349168990359X},
bdsk-url-2 = {https://doi.org/10.1016/0003-4916(89)90359-X}}
@article{Baym_1962,
author = {Baym, Gordon},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100},
doi = {10.1103/PhysRev.127.1391},
issue = {4},
journal = {Phys. Rev.},
month = {Aug},
numpages = {0},
pages = {1391--1401},
publisher = {American Physical Society},
title = {Self-Consistent Approximations in Many-Body Systems},
url = {https://link.aps.org/doi/10.1103/PhysRev.127.1391},
volume = {127},
year = {1962},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.127.1391},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.127.1391}}
@article{Baym_1961,
author = {Baym, Gordon and Kadanoff, Leo P.},
date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100},
doi = {10.1103/PhysRev.124.287},
issue = {2},
journal = {Phys. Rev.},
month = {Oct},
numpages = {0},
pages = {287--299},
publisher = {American Physical Society},
title = {Conservation Laws and Correlation Functions},
url = {https://link.aps.org/doi/10.1103/PhysRev.124.287},
volume = {124},
year = {1961},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.124.287},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.124.287}}
@article{Li_2022,
author = {Li, Jiachen and Jin, Ye and Rinke, Patrick and Yang, Weitao and Golze, Dorothea},
date-added = {2023-01-30 13:59:42 +0100},
date-modified = {2023-01-30 14:00:00 +0100},
doi = {10.1021/acs.jctc.2c00617},
eprint = {https://doi.org/10.1021/acs.jctc.2c00617},
journal = {J. Chem. Theory Comput.},
note = {PMID: 36322136},
number = {12},
pages = {7570-7585},
title = {Benchmark of GW Methods for Core-Level Binding Energies},
url = {https://doi.org/10.1021/acs.jctc.2c00617},
volume = {18},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00617}}
@incollection{CsanakBook, @incollection{CsanakBook,
author = {Csanak, Gy and Taylor, HS and Yaris, Robert}, author = {Csanak, Gy and Taylor, HS and Yaris, Robert},
booktitle = {Advances in atomic and molecular physics}, booktitle = {Advances in atomic and molecular physics},
@ -1394,23 +1650,6 @@
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.82.155108}, bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.82.155108},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.82.155108}} bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.82.155108}}
@article{Bickers_1989,
abstract = {We discuss the solution of nontrivial conserving approximations for electronic correlation functions in systems with strong collective fluctuations. The formal properties of conserving approximations have been well known for over twenty years, but numerical solutions have been limited to Hartree-Fock level. We extend the formal analysis of Baym and Kadanoff in order to derive the simplest self-consistent approximation based on exchange of fluctuations in the particle-hole and particle-particle channels. We then describe a practical technique for calculating self-consistent single-particle Green's functions and solving the finite-temperature Bethe-Salpeter equation for electrons on a lattice.},
author = {N.E Bickers and D.J Scalapino},
date-added = {2021-11-03 15:18:58 +0100},
date-modified = {2021-11-03 15:24:13 +0100},
doi = {https://doi.org/10.1016/0003-4916(89)90359-X},
issn = {0003-4916},
journal = {Ann. Phys.},
number = {1},
pages = {206-251},
title = {Conserving approximations for strongly fluctuating electron systems. I. Formalism and calculational approach},
url = {https://www.sciencedirect.com/science/article/pii/000349168990359X},
volume = {193},
year = {1989},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/000349168990359X},
bdsk-url-2 = {https://doi.org/10.1016/0003-4916(89)90359-X}}
@article{Bickers_1991, @article{Bickers_1991,
author = {Bickers, N. E. and White, S. R.}, author = {Bickers, N. E. and White, S. R.},
date-added = {2021-11-03 15:18:26 +0100}, date-added = {2021-11-03 15:18:26 +0100},
@ -1446,24 +1685,6 @@
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0003491684900939}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0003491684900939},
bdsk-url-2 = {https://doi.org/10.1016/0003-4916(84)90093-9}} bdsk-url-2 = {https://doi.org/10.1016/0003-4916(84)90093-9}}
@article{Baym_1961,
author = {Baym, Gordon and Kadanoff, Leo P.},
date-added = {2021-11-03 15:14:17 +0100},
date-modified = {2021-11-03 15:14:25 +0100},
doi = {10.1103/PhysRev.124.287},
issue = {2},
journal = {Phys. Rev.},
month = {Oct},
numpages = {0},
pages = {287--299},
publisher = {American Physical Society},
title = {Conservation Laws and Correlation Functions},
url = {https://link.aps.org/doi/10.1103/PhysRev.124.287},
volume = {124},
year = {1961},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.124.287},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.124.287}}
@article{Bethe_1957, @article{Bethe_1957,
abstract = {Using Brueckner's method for the treatment of complex nuclei, the effect of an infinite repulsive core in the interaction between nucleons is studied. The Pauli principle is taken into account from the beginning. A spatial wave function for two nucleons is defined, and an integro-differential equation for this function is derived. Owing to the Pauli principle, the wave function contains no outgoing spherical waves. A solution is given for the case when only a repulsive core potential acts. The effective-mass approximation is investigated for virtual states of very large momentum.}, abstract = {Using Brueckner's method for the treatment of complex nuclei, the effect of an infinite repulsive core in the interaction between nucleons is studied. The Pauli principle is taken into account from the beginning. A spatial wave function for two nucleons is defined, and an integro-differential equation for this function is derived. Owing to the Pauli principle, the wave function contains no outgoing spherical waves. A solution is given for the case when only a repulsive core potential acts. The effective-mass approximation is investigated for virtual states of very large momentum.},
author = {H. A. Bethe and J. Goldstone}, author = {H. A. Bethe and J. Goldstone},
@ -4381,22 +4602,6 @@
year = {1996}, year = {1996},
bdsk-url-1 = {https://doi.org/10.1063/1.471637}} bdsk-url-1 = {https://doi.org/10.1063/1.471637}}
@article{Baym_1962,
author = {Baym, Gordon},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
doi = {10.1103/PhysRev.127.1391},
issn = {0031-899X},
journal = {Phys. Rev.},
language = {en},
month = aug,
number = {4},
pages = {1391--1401},
title = {Self-{{Consistent Approximations}} in {{Many}}-{{Body Systems}}},
volume = {127},
year = {1962},
bdsk-url-1 = {https://doi.org/10.1103/PhysRev.127.1391}}
@article{Beigi_2003, @article{Beigi_2003,
author = {Ismail-Beigi, Sohrab and Louie, Steven G.}, author = {Ismail-Beigi, Sohrab and Louie, Steven G.},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
@ -5255,12 +5460,12 @@
@article{Holzer_2018a, @article{Holzer_2018a,
author = {Holzer,Christof and Klopper,Wim}, author = {Holzer,Christof and Klopper,Wim},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2021-01-11 09:18:51 +0100}, date-modified = {2023-01-30 14:05:30 +0100},
doi = {10.1063/1.5051028}, doi = {10.1063/1.5051028},
journal = {J. Chem. Phys.}, journal = {J. Chem. Phys.},
number = {10}, number = {10},
pages = {101101}, pages = {101101},
title = {Communication: A hybrid Bethe--Salpeter/time-dependent density-functional-theory approach for excitation energies}, title = {A hybrid Bethe--Salpeter/time-dependent density-functional-theory approach for excitation energies},
volume = {149}, volume = {149},
year = {2018}, year = {2018},
bdsk-url-1 = {https://doi.org/10.1063/1.5051028}} bdsk-url-1 = {https://doi.org/10.1063/1.5051028}}
@ -15289,10 +15494,10 @@
year = {2016}, year = {2016},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00871}} bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00871}}
@article{Maggio_2017, @article{Maggio_2017a,
author = {Maggio, Emanuele and Liu, Peitao and {van Setten}, Michiel J. and Kresse, Georg}, author = {Maggio, Emanuele and Liu, Peitao and {van Setten}, Michiel J. and Kresse, Georg},
date-added = {2018-04-22 16:22:34 +0000}, date-added = {2018-04-22 16:22:34 +0000},
date-modified = {2018-04-22 16:22:34 +0000}, date-modified = {2023-01-30 15:45:44 +0100},
doi = {10.1021/acs.jctc.6b01150}, doi = {10.1021/acs.jctc.6b01150},
issn = {1549-9618, 1549-9626}, issn = {1549-9618, 1549-9626},
journal = {J. Chem. Theory Comput.}, journal = {J. Chem. Theory Comput.},
@ -16147,8 +16352,9 @@
year = {2011}, year = {2011},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.83.115103}} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.83.115103}}
@article{Shishkin_2007, @article{Shishkin_2007a,
author = {Shishkin, M. and Kresse, G.}, author = {Shishkin, M. and Kresse, G.},
date-modified = {2023-01-30 15:40:02 +0100},
doi = {10.1103/PhysRevB.75.235102}, doi = {10.1103/PhysRevB.75.235102},
journal = {Physical Review B}, journal = {Physical Review B},
number = {23}, number = {23},

32
Manuscript/SRGGW.tex
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@ -86,7 +86,7 @@ Here comes the abstract.
%=================================================================% %=================================================================%
One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019} One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
The non-linear Hedin's equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965} The non-linear Hedin equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965}
Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations. Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations.
In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} yields accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021} In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} yields accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
@ -103,26 +103,26 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
\label{eq:gw_selfenergy} \label{eq:gw_selfenergy}
\Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2). \Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
\end{equation} \end{equation}
Diagrammatically, $GW$ corresponds to a resummation of the direct ring diagrams and is thus particularly well suited for weak correlation. Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013} The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}
Despite a wide range of successes, many-body perturbation theory is not flawless. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017,Duchemin_2020} Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
\ant{ For example, modelling core electron spectroscopy requires core ionisation energies which have been proved to be challenging for routine $GW$ calculations. \cite{Golze_2018} For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter. However, the accuracy is not yet satisfying for triplet excited states. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b} Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter equation. \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} However, the accuracy is not yet satisfying for triplet excited states, where instabilities often occur. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b,Holzer_2018a}
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require more accuracy. Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require higher precision.
Unfortunately, defining a systematic way to go beyond $GW$, the so-called vertex corrections, is a tricky task. Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of vertex corrections has been demonstrated to be a tricky task. \cite{Baym_1961,Baym_1962,DeDominicis_1964a,DeDominicis_1964b,Bickers_1989a,Bickers_1989b,Bickers_1991,Hedin_1999,Bickers_2004,Shirley_1996,DelSol_1994,Schindlmayr_1998,Morris_2007,Shishkin_2007b,Romaniello_2009a,Romaniello_2012,Gruneis_2014,Hung_2017,Maggio_2017b,Mejuto-Zaera_2022}
Lewis and Berkelbach have shown that naive vertex corrections can even worsen the results with respect to the initial $GW$ results. \cite{Lewis_2019} For example, Lewis and Berkelbach have recently shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers for an extensive list of current challenges in many-body perturbation theory (see Ref.~\onlinecite{Golze_2019}) and we will now focus on another flaw throughout this manuscript.} We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in many-body perturbation theory.
From hereon, we will focus on another flaw.
It has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibit some discontinuities. \cite{Veril_2018,Loos_2018b,Loos_2020e,Berger_2021,DiSabatino_2021} It has been shown that a variety of physical quantities, such as charged and neutral excitations energies or correlation and total energies, computed within many-body perturbation theory exhibit unphysical discontinuities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021}
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is supposed to be valid. Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022} These discontinuities have been traced back to a transfer of spectral weight between two solutions of the quasi-particle equation, \cite{Monino_2022} and is another occurrence of the infamous intruder-state problem.\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001}
This is another occurrence of the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} In addition, systems where the quasiparticle equation admits two solutions with similar spectral weights are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
In addition, systems \ant{whose quasi-particle equation admits two solutions with} a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regulariser inspired by the similarity renormalisation group (SRG) in the quasi-particle equation. \cite{Monino_2022} In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regulariser inspired by the similarity renormalization group (SRG) in the quasi-particle equation. \cite{Monino_2022}
Encouraged by the recent successes of regularisation schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants. Encouraged by the recent successes of regularisation schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively. The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a} This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a}
@ -218,7 +218,7 @@ One obvious flaw of the one-shot scheme mentioned above is its starting point de
Indeed, in Eq.~\eqref{eq:G0W0} we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example. Indeed, in Eq.~\eqref{eq:G0W0} we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
Therefore, one can \titou{optimize} the starting point to obtain the best one-shot energies possible, which is commonly done. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016} Therefore, one can \titou{optimize} the starting point to obtain the best one-shot energies possible, which is commonly done. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
\PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?} \PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?}
Alternatively, one could solve this set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016} Alternatively, one could solve this set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equation are solved for $\omega$ again. The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equation are solved for $\omega$ again.
This procedure is iterated until convergence for $\epsilon_p$ is reached. This procedure is iterated until convergence for $\epsilon_p$ is reached.
\PFL{This is not quite right. It is probably going to be easier to explain when you're going to introduce the explicit expressions of these quantities.} \PFL{This is not quite right. It is probably going to be easier to explain when you're going to introduce the explicit expressions of these quantities.}