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Manuscript/SRGGW.bib
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@ -1,13 +1,90 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2023-01-30 15:48:50 +0100
%% Created for Pierre-Francois Loos at 2023-01-30 22:12:43 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Zhang_2019,
author = {Zhang, Tianyuan and Li, Chenyang and Evangelista, Francesco A.},
date-added = {2023-01-30 22:12:16 +0100},
date-modified = {2023-01-30 22:12:32 +0100},
doi = {10.1021/acs.jctc.9b00353},
eprint = {https://doi.org/10.1021/acs.jctc.9b00353},
journal = {J. Chem. Theory Comput.},
note = {PMID: 31268704},
number = {8},
pages = {4399-4414},
title = {Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation},
url = {https://doi.org/10.1021/acs.jctc.9b00353},
volume = {15},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.9b00353}}
@article{ChenyangLi_2021,
author = {Li,Chenyang and Evangelista,Francesco A.},
date-added = {2023-01-30 22:10:49 +0100},
date-modified = {2023-01-30 22:11:03 +0100},
doi = {10.1063/5.0059362},
eprint = {https://doi.org/10.1063/5.0059362},
journal = {J. Chem. Phys.},
number = {11},
pages = {114111},
title = {Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics},
url = {https://doi.org/10.1063/5.0059362},
volume = {155},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1063/5.0059362}}
@article{Wang_2021,
author = {Wang, Shuhe and Li, Chenyang and Evangelista, Francesco A.},
date-added = {2023-01-30 22:09:49 +0100},
date-modified = {2023-01-30 22:10:02 +0100},
doi = {10.1021/acs.jctc.1c00980},
eprint = {https://doi.org/10.1021/acs.jctc.1c00980},
journal = {J. Chem. Theory Comput.},
note = {PMID: 34839660},
number = {12},
pages = {7666-7681},
title = {Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory},
url = {https://doi.org/10.1021/acs.jctc.1c00980},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00980}}
@article{Wang_2023,
author = {Wang, Meng and Fang, Wei-Hai and Li, Chenyang},
date-added = {2023-01-30 22:07:40 +0100},
date-modified = {2023-01-30 22:07:50 +0100},
doi = {10.1021/acs.jctc.2c00966},
eprint = {https://doi.org/10.1021/acs.jctc.2c00966},
journal = {J. Chem. Theory Comput.},
note = {PMID: 36534617},
number = {1},
pages = {122-136},
title = {Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases},
url = {https://doi.org/10.1021/acs.jctc.2c00966},
volume = {19},
year = {2023},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00966}}
@misc{Scott_2023,
author = {Scott, Charles J. C. and Backhouse, Oliver J. and Booth, George H.},
copyright = {Creative Commons Attribution 4.0 International},
date-added = {2023-01-30 22:00:55 +0100},
date-modified = {2023-01-30 22:01:05 +0100},
doi = {10.48550/ARXIV.2301.09107},
keywords = {Chemical Physics (physics.chem-ph), Strongly Correlated Electrons (cond-mat.str-el), Computational Physics (physics.comp-ph), FOS: Physical sciences, FOS: Physical sciences},
publisher = {arXiv},
title = {A 'moment-conserving' reformulation of GW theory},
url = {https://arxiv.org/abs/2301.09107},
year = {2023},
bdsk-url-1 = {https://arxiv.org/abs/2301.09107},
bdsk-url-2 = {https://doi.org/10.48550/ARXIV.2301.09107}}
@article{Shirley_1996,
author = {Shirley, Eric L.},
date-added = {2023-01-30 15:47:29 +0100},

21
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}
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}
The non-linear Hedin equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, to a wealth of properties such as 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.
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}
@ -113,32 +113,31 @@ For example, modeling core electron spectroscopy requires core ionization energi
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 higher precision.
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}
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}
For example, Lewis and Berkelbach have 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 (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 unphysical discontinuities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021}
Recently, it has been shown that a variety of physical quantities, such as charged and neutral excitations energies as well as correlation and total energies, computed within many-body perturbation theory exhibit unphysical discontinuities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
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}
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, 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{Veril_2018,Forster_2021,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.
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}
In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
Encouraged by the recent successes of regularization 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} the present work investigates 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.
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
\cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016,Frosini_2022a,Frosini_2022b,Frosini_2022c}
See Ref.~\onlinecite{Hergert_2016a} for a recent review in this field.
The SRG transformation aims at decoupling an internal (or reference) space from an external space while incorporating information about their coupling in the reference space.
This process can often result in the appearance of intruder states. \cite{Evangelista_2014b,ChenyangLi_2019a}
This process often results in the appearance of intruder states. \cite{Evangelista_2014b,ChenyangLi_2019a}
However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space.
By definition, intruder states have energies that are close to the reference energy, and therefore are the last to be decoupled.
By stopping the SRG transformation once all external configurations except the intruder states have been decoupled,
correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder states.
The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
\ant{This open question will lead us to an \textit{intruder-state-free first-principle static approximation of the self-energy} that can be used for qs$GW$ calculations.}
This open question will lead us to an intruder-state-free static approximation of the self-energy derived from first-principles that can be employed in \titou{qs$GW$} calculations.
The manuscript is organized as follows.
We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.