figures F2

Manuscript/ufGW.bib

@@ -1,13 +1,27 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% https://bibdesk.sourceforge.io/
+%% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-02-24 15:41:54 +0100 
+%% Created for Pierre-Francois Loos at 2022-04-21 13:21:26 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Hedin_1999,
+	abstract = {The GW approximation (GWA) extends the well-known Hartree-Fock approximation (HFA) for the self-energy (exchange potential), by replacing the bare Coulomb potential v by the dynamically screened potential W, e.g. Vex = iGv is replaced by GW = iGW. Here G is the one-electron Green's function. The GWA like the HFA is self-consistent, which allows for solutions beyond perturbation theory, like say spin-density waves. In a first approximation, iGW is a sum of a statically screened exchange potential plus a Coulomb hole (equal to the electrostatic energy associated with the charge pushed away around a given electron). The Coulomb hole part is larger in magnitude, but the two parts give comparable contributions to the dispersion of the quasi-particle energy. The GWA can be said to describe an electronic polaron (an electron surrounded by an electronic polarization cloud), which has great similarities to the ordinary polaron (an electron surrounded by a cloud of phonons). The dynamical screening adds new crucial features beyond the HFA. With the GWA not only bandstructures but also spectral functions can be calculated, as well as charge densities, momentum distributions, and total energies. We will discuss the ideas behind the GWA, and generalizations which are necessary to improve on the rather poor GWA satellite structures in the spectral functions. We will further extend the GWA approach to fully describe spectroscopies like photoemission, x-ray absorption, and electron scattering. Finally we will comment on the relation between the GWA and theories for strongly correlated electronic systems. In collecting the material for this review, a number of new results and perspectives became apparent, which have not been published elsewhere.},
+	author = {Lars Hedin},
+	date-added = {2022-04-21 13:21:07 +0200},
+	date-modified = {2022-04-21 13:21:24 +0200},
+	doi = {10.1088/0953-8984/11/42/201},
+	journal = {J. Phys. Condens. Matter},
+	number = {42},
+	pages = {R489--R528},
+	title = {On correlation effects in electron spectroscopies and the$\less$i$\greater${GW}$\less$/i$\greater$approximation},
+	volume = {11},
+	year = 1999,
+	bdsk-url-1 = {https://doi.org/10.1088/0953-8984/11/42/201}}
+
 @article{Marie_2021,
 	abstract = {We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr{\"o}dinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pad{\'e} and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.},
 	author = {Antoine Marie and Hugh G A Burton and Pierre-Fran{\c{c}}ois Loos},
@@ -232,17 +246,6 @@
 	year = {2020},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.9b03423}}
 
-@article{Hedin_1999,
-	author = {Lars Hedin},
-	date-added = {2022-02-19 13:51:59 +0100},
-	date-modified = {2022-02-19 13:51:59 +0100},
-	journal = {J Phys.: Cond. Mat.},
-	number = {42},
-	pages = {R489-R528},
-	title = {On correlation effects in electron spectroscopies and the GW approximation},
-	volume = {11},
-	year = {1999}}
-
 @article{Bruneval_2006,
 	author = {Bruneval, Fabien and Vast, Nathalie and Reining, Lucia},
 	date-added = {2022-02-19 13:51:49 +0100},

Response_Letter/Response_Letter.tex

@@ -32,23 +32,40 @@ We look forward to hearing from you.
 {This in an interesting article and definitely deserves publication in JCP. 
 However, I think that there are few issues the authors need to address before publication of this article. 
 In my opinion addressing of these issues would be helpful to both quantum chemistry audience and even Green's function practitioners. 
+}
+\\
+\alert{
+Thank you for supporting publication of the present manuscript.
+As detailed below, we have taken into account the comments and suggestions of the reviewers that we believe have overall improved the quality of the present paper.}
 
-In my understanding methods that are solved through non-linear equations display multiple solutions due to the dependence on the starting point and presence of multiple local minima. 
+{In my understanding methods that are solved through non-linear equations display multiple solutions due to the dependence on the starting point and presence of multiple local minima. 
 While the presence of these multiple solutions may be connected to the disappearance of denominators, the problem is much more general. 
 The problem of multiple solutions exists even if fully self-consistent GW is solved in a finite temperature formalism (on imaginary axis - which effectively removes the problem with vanishing denominators). 
 Different solutions can be then illustrated by considering their physical properties for example $S^2$. 
-In such cases, the potential energy curves are smooth within a region where one solution is dominant and one can show properties of these solutions as a function of $S^2$, see J. Chem. Phys. 155, 024101, and J. Chem. Phys. 155, 024119. 
+In such cases, the potential energy curves are smooth within a region where one solution is dominant and one can show properties of these solutions as a function of $S^2$, see J. Chem. Phys. 155, 024101, and J. Chem. Phys. 155, 024119.}
+\\
+\alert{
+The reviewer is correct; these issues are more general and do not originate solely from the vanishing denominator.
+These multiple solutions issues stems from the fact that, in most $GW$ methods, one only consider the quasiparticle solutions at each iteration, discarding the other solutions known as satellites.
+In a fully self-consistent scheme where one takes into account all these solutions at each iteration, these issues do not appear, as we have recently discussed in Ref.~[Front. Chem. (9, 751054 (2021). ]. }
 
-Convergence accelerators such as DIIS, KAIN, LCIS, can be successfully used and smooth PES can be produced when one is relatively close to a local minima, one should not expect that DIIS will help with convergence where multiple close lying solutions exists and when two solutions are competing, see J. Chem. Phys. 156, 094101 (2022). 
 
-It is my understanding that the calculation on the illustrative example H2 in 6-31G basis illustrates the existence of multiple such solutions. 
+{Convergence accelerators such as DIIS, KAIN, LCIS, can be successfully used and smooth PES can be produced when one is relatively close to a local minima, one should not expect that DIIS will help with convergence where multiple close lying solutions exists and when two solutions are competing, see J. Chem. Phys. 156, 094101 (2022).}
+\\
+\alert{
+Indeed, convergence accelerators such as DIIS can be used to ease convergence but they will not make these discontinuitius disappear as their origin is more profond.
+This particular case is discussed in Ref.~[. Chem. Theory Comput. 14, 5220 (2018)] where we have provided our implementation of DIIS within $GW$ methods (see Appendix).}
+
+
+{It is my understanding that the calculation on the illustrative example H2 in 6-31G basis illustrates the existence of multiple such solutions. 
 The issue of introducing a regularizer to avoid vanishing denominators is more of an issue for GW on real axis. 
-
 Of course these issues of vanishing denominators and multiple solutions due to the non-linear character of the Dyson equation are connected but I think it is worth mentioning explicitly how different versions of GW are affected by these issues. 
-
 Frankly, I find it infuriating that in the GW community many times it is not clearly spelled out which flavor of the so-called GW is used and how the equations are solved. 
+}
+\\
+\alert{We agree with the reviewer that, historically, the $GW$ literature has been unclear on how to solve these equations, but we would like to stress that our group has made a clear effort to provide all the working equations and details necessary to solve these equations in the different cases.}
 
-Here, I will describe more specific issues that I noticed during the reading of the manuscript:}
+{Here, I will describe more specific issues that I noticed during the reading of the manuscript:}
 \\
 \alert{
 }
@@ -61,20 +78,26 @@ The convergence characteristics on imaginary or real axis is very different.
 Please, specify it. It may seem trivial to a regular GW practitioner, it is not trivial for a regular theoretical chemist not very familiar with Green's functions.}
 \\
 \alert{
+The evaluation of all the quantities of interest ares performed on the real axis, i.e., for real values of $\omega$.
+This is now stated clearly in the revised version of the manuscript. 
 }
 
 \item 
-{It would be reassuring if the authors mentioned that they use $O(K^9)$ scheme just to illustrate multiple solutions and this should not be done in regular calculations. A reader not familiar with GW algorithms may conclude that this what they advocate that should be done. 
+{It would be reassuring if the authors mentioned that they use $O(K^9)$ scheme just to illustrate multiple solutions and this should not be done in regular calculations. 
+A reader not familiar with GW algorithms may conclude that this what they advocate that should be done. 
 On the same note, it would be good if the authors mentioned that the scheme that scales as $O(K^6)$ is again not what one does in most GW schemes that are in use for calculations of larger systems.}
 \\
 \alert{
+Thank you for pointing this out. 
+We now clearly state that $O(K^9)$ scheme is illustrative and that state-of-the-art $GW$ calculations scales as $O(K^3)$ instead of $O(K^6)$.
 }
 
 \item 
-{In expressions such as $\epsilon^{GW}_{p,s}$ s is numbering solutions. 
+{In expressions such as $\epsilon^{GW}_{p,s}$ $s$ is numbering solutions. 
 I do not think that it is clearly mentioned in the article when the notation appears first time.}
 \\
 \alert{
+We now clearly state that $s$ numbers solutions.
 }
 
 \item