small changes

Manuscript/SRGGW.tex

@@ -122,8 +122,8 @@ These discontinuities are due to a transfer of spectral weight between two solut
 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 \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 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} this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
+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}
+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}
 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.
@@ -205,7 +205,7 @@ This results in $K$ quasi-particle equations that read
 \end{equation}
 where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
 The previous equations are non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
-These solutions can be characterized by their spectral weight given by the renormalization factor $Z_{p,s}$
+These solutions can be characterized by their spectral weight given by the renormalisation factor $Z_{p,s}$
 \begin{equation}
   \label{eq:renorm_factor}
   0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
@@ -244,10 +244,10 @@ Therefore, by suppressing this dependence the static approximation relies on the
 If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
 
 The satellites causing convergence problems are the so-called intruder states.
-The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
-The $\ii \eta$ term that is usually added in the denominators of the self-energy [see Eq.~(\ref{eq:GW_selfenergy})] is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
-Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages over the imaginary shift. \cite{Monino_2022}
-But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
+The intruder state problem can be dealt with by introducing \textit{ad hoc} regularisers.
+The $\ii \eta$ term that is usually added in the denominators of the self-energy [see Eq.~(\ref{eq:GW_selfenergy})] is the usual imaginary-shift regulariser used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
+Various other regularisers are possible and in particular one of us has shown that a regulariser inspired by the SRG had some advantages over the imaginary shift. \cite{Monino_2022}
+But it would be more rigorous, and more instructive, to obtain this regulariser from first principles by applying the SRG formalism to many-body perturbation theory.
 This is the aim of the rest of this work.
 
 Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
@@ -303,11 +303,11 @@ As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $
 Therefore, these blocks will be the target of the SRG transformation but before going into more detail we will review the SRG formalism.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\section{The similarity renormalization group}
+\section{The similarity renormalisation group}
 \label{sec:srg}
 %%%%%%%%%%%%%%%%%%%%%%
 
-The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
+The similarity renormalisation group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
 Therefore, the transformed Hamiltonian
 \begin{equation}
   \label{eq:SRG_Ham}
@@ -364,7 +364,7 @@ Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as w
 Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\section{Regularized $GW$ approximation}
+\section{Regularised $GW$ approximation}
 \label{sec:srggw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
@@ -547,7 +547,7 @@ However, doing a change of variable such that
   e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t} 
 \end{align}
 hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
-In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
+In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regulariser introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
 
 %=================================================================%
 \section{Computational details}
@@ -579,30 +579,36 @@ Then the accuracy of the IP yielded by the Sym and SRG schemes will be statistic
   \centering
   \includegraphics[width=\linewidth]{fig1.pdf}
   \caption{
-    Add caption
+    Add caption \ANT{Should we add $G_0W_0$?}
     \label{fig:fig1}}
 \end{figure*}
 %%% %%% %%% %%%
 
 This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
-The HF values (orange lines) lie below the reference CCSD(T) ones, a result which is now well-understood.
-Indeed, this is due to an over(under? \ant{I will check this...}) screening of the interactions in the mean-field treatment. \cite{Lewis_2019}
-The usual Sym-qs$GW$ scheme (blue lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
-The Neon atom is a well-behaved system and could be converged without regularization parameter while for water it was set to 0.01 to help convergence.
+Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to CCSD(T).
+The HF IPs (cyan lines) are 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.}
+The usual Sym-qs$GW$ scheme (green lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\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.
 
-Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP energies as a function of the flow parameter.
-At $s=0$, the IPs are equal to their HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
-For $s\to\infty$ both IPs reach a plateau that are significantly better than their $s=0$ starting point.
+Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IPs as a function of the flow parameter (blue curves).
+At $s=0$, the IPs are equal to their HF counterparts as expected from the discussion of Sec.~\ref{sec:srggw}.
+For $s\to\infty$ both IPs reach a plateau at an error that is significantly smaller than their $s=0$ starting point.
 Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
-The SRG-qs$GW$ IPs do not increase smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
+However, the SRG-qs$GW$ error do not decrease smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
 
+In addition, we also considered the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening.
+The TDA IPs are now understimated unlike their RPA counterparts.
+The SRG-qs$GW$ absolute error for the IPs clearly deteriorates when going from RPA to TDA.
+On the other hand, for qs$GW$ the TDA-based IP is better than the RPA one for Neon while it is the other way around for water.
+This trend will be investigated in more details in the next subsection. 
+
+\ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.}
 The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
 Add sentence about $GW$ better than GF2 when the results will be here.
 The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
 We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
 The GF(2) IP admits the following perturbation expansion...
 
-
 Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
 But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.