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modif in intro + first draft of the reworked section II

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Manuscript/SRGGW.bib
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@ -14961,7 +14961,7 @@
year = {2012},
bdsk-url-1 = {https://doi.org/10.1016/j.cpc.2011.12.006}}
@article{Lewis_2019a,
@article{Lewis_2019,
author = {Alan M. Lewis and Timothy C. Berkelbach},
date-added = {2019-10-12 14:31:33 +0200},
date-modified = {2019-10-12 14:32:30 +0200},

157
Manuscript/SRGGW.tex
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@ -51,7 +51,8 @@
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\ant}[1]{\textcolor{green}{#1}}
\newcommand{\ant}[1]{\textcolor{teal}{#1}}
\newcommand{\ANT}[1]{\ant{(\underline{\bf ANT}: #1)}}
% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
@ -99,6 +100,7 @@ The self-energy encapsulates all the Hartree-exchange-correlation effects which
%Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016}
\begin{equation}
\label{eq:gw_selfenergy}
\Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
\end{equation}
Diagrammatically, $GW$ corresponds to a resummation of the direct ring diagrams and is thus particularly well suited for weak correlation.
@ -107,12 +109,18 @@ Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in
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}
\PFL{to be expanded as discussed.}
In particular, 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}
\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}
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}
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require more accuracy.
Unfortunately, defining a systematic way to go beyond $GW$, the so-called vertex corrections, is a tricky task.
Lewis and Berkelbach have shown that naive vertex corrections can even worsen the results with respect to the initial $GW$ results. \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.}
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}
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ 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}
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 for which \titou{two quasi-particle solutions} have a similar spectral weight 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 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.
@ -130,7 +138,7 @@ 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.
\PFL{I think we should also mention that it may provide static approximations of the self-energy from first principles via this downfolding. What do you think?}
\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.}
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}.
@ -148,79 +156,104 @@ This section starts by
\label{sec:gw}
%%%%%%%%%%%%%%%%%%%%%%
\PFL{Antoine, please move the various expressions related to the $GW$ quantities in this section.}
\ant{The self-energy consider in this work will always be the $GW$ one [Eq.~\eqref{eq:gw_selfenergy}] but the subsequent derivations can be straightforwardly transposed to other approximations such as GF(2) or $GT$.
In addition, we assume a Hartree-Fock (HF) starting point throughout the manuscript.}
\titou{Here and in the following, we assume a Hartree-Fock (HF) starting point.}
The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation
\begin{equation}
\label{eq:quasipart_eq}
\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
\end{equation}
where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is \titou{(the correlation part of)} the self-energy.
where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the self-energy.
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
The self-energy can be physically understood as a dynamical \titou{screening} correction to the HF problem represented by $\bF$.
The self-energy can be physically understood as a \ant{correction to the HF problem (represented by $\bF$) accounting for dynamical screening effects}.
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
Note that $\bSig$ is dynamical, \titou{\ie} it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.
Note that $\bSig(\omega)$ is dynamical, \titou{\ie} it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.
Because of this frequency dependence, fully solving this equation is a rather complicated task.
The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
\begin{equation}
\label{eq:GW_selfenergy}
\Sigma_{pq}(\omega)
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
\end{equation}
with the screened two-electron integrals defined as
\begin{equation}
\label{eq:GW_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
\end{equation}
where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the Tamm-Dancoff approximation (TDA). This problem is defined as
\begin{equation}
\bA \bX = \boldsymbol{\Omega} \bX,
\end{equation}
with
\begin{equation}
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
\end{equation}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
The non-TDA case is discussed in Appendix~\ref{sec:nonTDA}.
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task.
Hence, several approximate schemes have been developed to bypass self-consistency.
The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
In this case, one gets $K$ quasi-particle equations that read
tr\begin{equation}
This results in $K$ quasi-particle equations that read
\begin{equation}
\label{eq:G0W0}
\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
\end{equation}
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
The previous equation is 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).
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 $Z_{p,s}$
\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.
\end{equation}
The solution with the largest weight is referred to as the quasi-particle while the others are known as satellites (or shake-up transitions).
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights and the quasi-particle is not well-defined.
In fact, these cases are related to the discontinuities and convergence problems discussed earlier (see Sec.~\ref{sec:intro}) because the additional solutions with large weights are the previously mentioned intruder states.
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasi-particle is not well-defined.
These additional solutions with large weights are the previously mentioned intruder states.
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
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}
\PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?}
Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~\eqref{eq:G0W0} and then this equation is solved for $\omega$ again until convergence is reached.
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}
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.
\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.}
However, if the quasi-particle solution is not well-defined, reaching self-consistency can be quite difficult, if not impossible.
Even at convergence, the starting point dependence is not totally removed as the results still depend on the starting orbitals. \cite{Marom_2012}
\ANT{Is it better now ?}
However, if one of the quasi-particle equations does not have a well-defined quasi-particle solution, reaching self-consistency can be quite difficult, if not impossible.
Even at convergence, the starting point dependence is not totally removed as the results still depend on the initial molecular orbitals. \cite{Marom_2012}
To update both energies and orbitals, one must take into account the off-diagonal elements in Eq.~\eqref{eq:quasipart_eq}.
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
In order to update both the orbital energies and coefficients, one must consider the off-diagonal elements in $\bSig(\omega)$.
To take into account the off-diagonal elements without solving the dynamic quasi-particle equation [Eq.~\eqref{eq:quasipart_eq}], one can resort to the quasi-particle self-consistent (qs) $GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
Then the qs$GW$ problem is solved using the usual HF algorithm with $\bF$ replaced by $\bF + \bSig^{\qs}$.
Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
\begin{equation}
\label{eq:sym_qsgw}
\Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
\end{equation}
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the \titou{form of the} effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form.
This will be done in the next section.
This will be done in the next sections.
In this case, as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
Multiple solutions arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
If it is not the case, the qs scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach.
Multiple solutions of Eq.~(\ref{eq:G0W0}) arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasi-particle solution.
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
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 denominator of the self-energy (see below) 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, in the $GW$ case, over the imaginary shift one. \cite{Monino_2022}
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.
This is the aim of this work.
This is the aim of the rest of this work.
Therefore if we apply it, the SRG would gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process.
From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies.
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
The upfolded $GW$ quasi-particle equation is the following
The upfolded $GW$ quasi-particle equation is
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
@ -244,9 +277,9 @@ The upfolded $GW$ quasi-particle equation is the following
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i}^{\GW} + \epsilon_{j}^{\GW} - \epsilon_{a}^{\GW}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i} + \epsilon_{j} - \epsilon_{a} ) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
\\
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a}^{\GW} + \epsilon_{b}^{\GW} - \epsilon_{i}^{\GW}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a} + \epsilon_{b} - \epsilon_{i} ) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
\end{align}
\end{subequations}
and the corresponding coupling blocks read
@ -255,46 +288,19 @@ and the corresponding coupling blocks read
&
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
\end{align}
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
The usual $GW$ non-linear equation
\begin{equation}
\label{eq:GWnonlin}
\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
\end{equation}
can be obtained by applying L\"odwin partitioning technique to Eq.~\eqref{eq:GWlin} \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy
The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} which gives the following expression for the self-energy \cite{Bintrim_2021}
\begin{align}
\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
\end{align}
which can be further developed as
\begin{equation}
\label{eq:GW_selfenergy}
\Sigma_{pq}(\omega)
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
\end{equation}
with the screened integrals defined as
\begin{equation}
\label{eq:GW_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
\end{equation}
where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the TDA defined as
\begin{equation}
\bA \bX = \boldsymbol{\Omega} \bX,
\end{equation}
with
\begin{equation}
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
\end{equation}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
which can be further developed to give exactly Eq.~(\ref{eq:GW_selfenergy}).
Equations \eqref{eq:GWlin} and \eqref{eq:GWnonlin} have exactly the same solutions but one is linear and the other not.
The price to pay for this linearity is that the size of the matrix in the former equation is $\order{K^3}$ while it is $\order{K}$ in the latter one.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
Therefore, these blocks will be the target of our SRG transformation but before going into more detail we will review the SRG formalism.
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}
@ -636,6 +642,7 @@ Here comes the conclusion.
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
The authors thank Francesco Evangelista for inspiring discussions.
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -650,11 +657,11 @@ The data that supports the findings of this study are available within the artic
\appendix
%%%%%%%%%%%%%%%%%%%%%%
\section{Non-TDA $GW$ equations}
\section{Non-TDA $GW$ and $GW_{\text{TDHF}}$ equations}
\label{sec:nonTDA}
%%%%%%%%%%%%%%%%%%%%%%
The $GW$ self-energy without TDA is the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
The matrix elements of the $GW$ self-energy without TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
\begin{equation}
\label{eq:GWnonTDA_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
@ -726,7 +733,7 @@ and the corresponding coupling blocks read
Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
%%%%%%%%%%%%%%%%%%%%%%
\section{GF(2) equations}
\section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}
\label{sec:GF2}
%%%%%%%%%%%%%%%%%%%%%%