more corrections in the intro

Manuscript/SRGGW.bib

@@ -1,13 +1,78 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2023-01-30 22:12:43 +0100 
+%% Created for Pierre-Francois Loos at 2023-02-03 22:09:44 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Biswas_2021,
+	author = {Biswas, T. and Singh, A.K.},
+	date-added = {2023-02-03 21:59:35 +0100},
+	date-modified = {2023-02-03 21:59:35 +0100},
+	doi = {10.1038/s41524-021-00640-3},
+	journal = {npj Comput. Mater.},
+	pages = {189},
+	title = {Excitonic effects in absorption spectra of carbon dioxide reduction photocatalysts},
+	volume = {7},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1038/s41524-021-00640-3}}
+
+@article{Friedrich_2019,
+	author = {Friedrich, Christoph},
+	date-added = {2023-02-03 21:59:17 +0100},
+	date-modified = {2023-02-03 21:59:17 +0100},
+	doi = {10.1103/PhysRevB.100.075142},
+	issue = {7},
+	journal = {Phys. Rev. B},
+	month = {Aug},
+	numpages = {15},
+	pages = {075142},
+	publisher = {American Physical Society},
+	title = {Tetrahedron integration method for strongly varying functions: Application to the $GT$ self-energy},
+	url = {https://link.aps.org/doi/10.1103/PhysRevB.100.075142},
+	volume = {100},
+	year = {2019},
+	bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.100.075142},
+	bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.100.075142}}
+
+@article{Muller_2019,
+	author = {M\"uller, Mathias C. T. D. and Bl\"ugel, Stefan and Friedrich, Christoph},
+	date-added = {2023-02-03 21:58:42 +0100},
+	date-modified = {2023-02-03 21:58:42 +0100},
+	doi = {10.1103/PhysRevB.100.045130},
+	issue = {4},
+	journal = {Phys. Rev. B},
+	month = {Jul},
+	numpages = {16},
+	pages = {045130},
+	publisher = {American Physical Society},
+	title = {Electron-magnon scattering in elementary ferromagnets from first principles: Lifetime broadening and band anomalies},
+	url = {https://link.aps.org/doi/10.1103/PhysRevB.100.045130},
+	volume = {100},
+	year = {2019},
+	bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.100.045130},
+	bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.100.045130}}
+
+@article{Muller_2001,
+	author = {M{\"u}ller, Thomas and Lischka, Hans},
+	date-added = {2023-02-03 21:58:42 +0100},
+	date-modified = {2023-02-03 21:58:42 +0100},
+	day = {01},
+	doi = {10.1007/s002140100286},
+	issn = {1432-2234},
+	journal = {Theor. Chem. Acc.},
+	month = {Oct},
+	number = {5},
+	pages = {369--378},
+	title = {Simultaneous Calculation of Rydberg and Valence Excited States of Formaldehyde},
+	url = {https://doi.org/10.1007/s002140100286},
+	volume = {106},
+	year = {2001},
+	bdsk-url-1 = {https://doi.org/10.1007/s002140100286}}
+
 @article{Zhang_2019,
 	author = {Zhang, Tianyuan and Li, Chenyang and Evangelista, Francesco A.},
 	date-added = {2023-01-30 22:12:16 +0100},
@@ -1442,21 +1507,6 @@
 	year = {2022},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00368}}
 
-@article{Forsberg_1997,
-	abstract = {In multiconfigurational perturbation theory, so-called intruders may cause singularities in the potential energy functions, at geometries where an energy denominator becomes zero. When the singularities are weak, they may be successfully removed by level shift techniques. When applied to excited states, a small shift merely moves the singularity. A large shift may cause new divergencies, and too large shifts are unacceptable since the potential function is affected in regions further away from the singularities. This Letter presents an alternative which may be regarded as an imaginary shift. The singularities are not moved, but disappear completely. They are replaced by a small distortion of the potential function. Applications to the N2 ground state, its A3/gEu+ state, and the Cr2 ground state show that the distortion caused by this procedure is small.},
-	author = {Niclas Forsberg and Per-{\AA}ke Malmqvist},
-	date-added = {2022-02-21 14:05:35 +0100},
-	date-modified = {2022-02-21 14:05:55 +0100},
-	doi = {https://doi.org/10.1016/S0009-2614(97)00669-6},
-	journal = {Chem. Phys. Lett.},
-	number = {1},
-	pages = {196-204},
-	title = {Multiconfiguration perturbation theory with imaginary level shift},
-	volume = {274},
-	year = {1997},
-	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0009261497006696},
-	bdsk-url-2 = {https://doi.org/10.1016/S0009-2614(97)00669-6}}
-
 @article{Golze_2018,
 	author = {Golze, Dorothea and Wilhelm, Jan and van Setten, Michiel J. and Rinke, Patrick},
 	date-added = {2022-02-21 11:30:10 +0100},
@@ -11858,23 +11908,6 @@
 	year = {1973},
 	bdsk-url-1 = {https://doi.org/10.1063/1.1680030}}
 
-@article{Muller_2001,
-	author = {M{\"u}ller, Thomas and Lischka, Hans},
-	date-added = {2020-01-01 21:36:51 +0100},
-	date-modified = {2020-01-01 21:36:52 +0100},
-	day = {01},
-	doi = {10.1007/s002140100286},
-	issn = {1432-2234},
-	journal = {Theor. Chem. Acc.},
-	month = {Oct},
-	number = {5},
-	pages = {369--378},
-	title = {Simultaneous Calculation of Rydberg and Valence Excited States of Formaldehyde},
-	url = {https://doi.org/10.1007/s002140100286},
-	volume = {106},
-	year = {2001},
-	bdsk-url-1 = {https://doi.org/10.1007/s002140100286}}
-
 @article{Nagy_1998,
 	author = {Nagy, \'A},
 	date-added = {2020-01-01 21:36:51 +0100},
@@ -14909,28 +14942,28 @@
 	bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.111.073003}}
 
 @article{Surjan_1996,
-  title = {Damping of Perturbation Corrections in Quasidegenerate Situations},
+	author = {Surj{\'a}n, P. R. and Szabados, {\'A}.},
-  author = {Surj{\'a}n, P. R. and Szabados, {\'A}.},
+	doi = {10.1063/1.471814},
-  year = {1996},
+	issn = {0021-9606},
-  journal = {The Journal of Chemical Physics},
+	journal = {The Journal of Chemical Physics},
-  volume = {104},
+	number = {9},
-  number = {9},
+	pages = {3320--3324},
-  pages = {3320--3324},
+	title = {Damping of Perturbation Corrections in Quasidegenerate Situations},
-  issn = {0021-9606},
+	volume = {104},
-  doi = {10.1063/1.471814}
+	year = {1996},
-}
+	bdsk-url-1 = {https://doi.org/10.1063/1.471814}}
 
 @article{Forsberg_1997,
-  title = {Multiconfiguration Perturbation Theory with Imaginary Level Shift},
+	author = {Forsberg, Niclas and Malmqvist, Per-{\AA}ke},
-  author = {Forsberg, Niclas and Malmqvist, Per-{\AA}ke},
+	doi = {10.1016/S0009-2614(97)00669-6},
-  year = {1997},
+	issn = {0009-2614},
-  journal = {Chemical Physics Letters},
+	journal = {Chemical Physics Letters},
-  volume = {274},
+	number = {1},
-  number = {1},
+	pages = {196--204},
-  pages = {196--204},
+	title = {Multiconfiguration Perturbation Theory with Imaginary Level Shift},
-  issn = {0009-2614},
+	volume = {274},
-  doi = {10.1016/S0009-2614(97)00669-6}
+	year = {1997},
-}
+	bdsk-url-1 = {https://doi.org/10.1016/S0009-2614(97)00669-6}}
 
 @article{Tew_2007,
 	author = {D. P. Tew and W. Klopper and C. Neiss and C. Hattig},

Manuscript/SRGGW.tex

@@ -117,18 +117,17 @@ Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of
 For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasi-particle 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. 
 
-\ant{Many-body perturbation theory also suffers from the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} 
+Many-body perturbation theory also suffers from the infamous intruder-state problem,\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} where they manifest themselves as solutions of the quasi-particle equation with non-negligible spectral weights.
-Within many-body perturbation theory intruder states manifest themselves as additional solutions of the quasi-particle equation with non-negligible spectral weights.
+In some cases, this transfer of spectral weight makes it difficult to distinguish between a quasiparticle and a satellite.
-In some cases, this splitting of the spectral weight even precludes the assignation of the quasi-particle character to a given solution.
+These multiple solutions hinder the convergence of partially self-consistent schemes such as quasiparticle self-consistent $GW$ (qs$GW$) and eigenvalue-only self-consistent $GW$ (ev$GW$). \cite{Veril_2018,Forster_2021,Monino_2022}
-These multiple solutions are known to hamper the convergence of partially self-consistent schemes such as quasi-particle self-consistent (qs) $GW$ and eigenvalue-only self-consistent (ev) $GW$. \cite{Veril_2018,Forster_2021,Monino_2022}
+The simpler one-shot $G_0W_0$ scheme is also impacted by these intruder states, leading to discontinuities in a variety of physical quantities including charged and neutral excitation energies, correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
-Even within the simpler one-shot $G_0W_0$ scheme, these intruder states lead to discontinuities in a plethora of physical quantities such as charged and neutral excitations energies as well as correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
+These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
-Even more worrying, these convergence problems and discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.}
 
-\ant{In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasi-particle equation. \cite{Monino_2022}
+In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasi-particle 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 $GW$.
+Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such 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 $GW$.
-In particular, the focus will be on the possibility of curing the qs$GW$ convergence problems using the SRG.}
+In particular, we focus here on the possibility of curing the qs$GW$ convergence problems using the SRG.
 
-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 formalism 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} 
@@ -142,8 +141,8 @@ By stopping the SRG transformation once all external configurations except the i
 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.
-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 \ant{partially self-consistent $GW$} 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 partially self-consistent $GW$ calculations.
-\ant{Note that throughout the manuscript we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) \cite{Casida_1989,Casida_1991,SzaboBook,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022}  or $T$-matrix.}
+Note that throughout the manuscript we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) \cite{Casida_1989,Casida_1991,SzaboBook,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} or $T$-matrix. \cite{Liebsch_1981,Bickers_1989a,Bickers_1991,Katsnelson_1999,Katsnelson_2002,Zhukov_2005,vonFriesen_2010,Romaniello_2012,Gukelberger_2015,Muller_2019,Friedrich_2019,Biswas_2021,Zhang_2017,Li_2021b,Loos_2022}
 
 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}.
@@ -562,6 +561,7 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
 This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
 Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value.
 The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well-understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
+\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
 The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.
 %The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
 
@@ -620,7 +620,7 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
 %%% %%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{Statistical performance \ANT{Need a better subsection title...}}
+\subsection{Statistical analysis}
 \label{sec:SRG_vs_Sym}
 %%%%%%%%%%%%%%%%%%%%%%