lots of small changes in intro

Manuscript/SRGGW.rty

@@ -8,9 +8,9 @@
 
 %%% Latin %%%
 
-\newcommand{\ie}{\textit{i.e.}~}
-\newcommand{\eg}{\textit{e.g.}~}
-\newcommand{\etal}{\textit{et al.}~}
+\newcommand{\ie}{\textit{i.e.}\xspace}
+\newcommand{\eg}{\textit{e.g.}\xspace}
+\newcommand{\etal}{\textit{et al.}\xspace}
 
 %%% Operators %%%
 
@@ -19,7 +19,7 @@
 \newcommand{\Hsim}{\hat{\bar{H}}} % Similarity transformed Hamiltonian
 \newcommand{\hC}{\Hat{C}} % CI operator
 \newcommand{\hT}{\Hat{T}} % Cluster operator
-\newcommand{\T}[1]{\Hat{\mathnormal{T}}_{#1}} % Cluster operator of a given excitation number
+\newcommand{\T}[1]{#1^{\intercal}}
 \newcommand{\hsig}{\Hat{\sigma}} % Unitary cluster operator
 \newcommand{\hK}{\Hat{K}} % Anti-hermitian orbital rotation operator
 \newcommand{\hS}{\Hat{S}} % Anti-hermitian CI coefficients rotation operator

Manuscript/SRGGW.tex

@@ -1,5 +1,5 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
 \usepackage[version=4]{mhchem}
 
 \usepackage[utf8]{inputenc}
@@ -105,8 +105,8 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
 \end{equation}
 Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
 
-Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} 
-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}
+\titou{Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} 
+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 has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
 For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
@@ -122,7 +122,7 @@ These discontinuities have been traced back to a transfer of spectral weight bet
 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 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.
+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 $GW$ \titou{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.
@@ -138,6 +138,7 @@ correlation effects between the internal and external spaces can be incorporated
 
 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 \titou{qs$GW$} calculations.
+\titou{Here, we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) or $T$-matrix.}
 
 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}.
@@ -155,49 +156,50 @@ This section starts by
 \label{sec:gw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
-\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.}
-
-The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation
+The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation which, within the $GW$ approximation, reads
 \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 (the correlation part of) the self-energy. 
+where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ 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 \ant{correction to the HF problem (represented by $\bF$) accounting for dynamical screening effects}.
+The self-energy can be physically understood as a correction to the Hartree-Fock (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 $\bSig(\omega)$ 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.
+\titou{Note that $\bSig(\omega)$ is dynamical, \ie it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
 
 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},
+  = \sum_{i\nu} \frac{W_{pi\nu} W_{qi\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
+  + \sum_{a\nu} \frac{W_{pa\nu}W_{qa\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
 \end{equation}
-with the screened two-electron integrals defined as
+where $\eta$ is a positive infinitesimal and the screened two-electron integrals are
 \begin{equation}
   \label{eq:GW_sERI}
-  W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
+  W_{pq\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu}+\bY_{\nu})_{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
+where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as 
 \begin{equation}
-  \bA \bX = \boldsymbol{\Omega} \bX,
+  \mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) =  \mqty( \bX \\ \bY ) \boldsymbol{\Omega},
 \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 $v=(ia)$.
+\begin{subequations}
+\begin{align}
+  A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
+  \\
+  B_{ia,jb} & = \eri{ij}{ab}.
+\end{align}
+\end{subequations}
+The diagonal matrix $\boldsymbol{\Omega}$ contains the eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
+\titou{The TDA case is discussed in Appendix \ref{sec:nonTDA}.}
+Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
+The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral excitations.
 
 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.
-This results in $K$ quasi-particle equations that read
+\titou{Assuming a HF starting point,} this results in $K$ quasi-particle equations that read
 \begin{equation}
   \label{eq:G0W0}
   \epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
@@ -257,8 +259,8 @@ The upfolded $GW$ quasi-particle equation is
   \label{eq:GWlin}
   \begin{pmatrix}
     \bF				&	\bV^{\text{2h1p}}		&	\bV^{\text{2p1h}}	\\
-    (\bV^{\text{2h1p}})^{\mathrm{T}}	&	\bC^{\text{2h1p}}		&	\bO					\\
-    (\bV^{\text{2p1h}})^{\mathrm{T}}	&	\bO				&	\bC^{\text{2p1h}}	\\
+    \T{(\bV^{\text{2h1p}})}	&	\bC^{\text{2h1p}}		&	\bO					\\
+    \T{(\bV^{\text{2p1h}})}	&	\bO				&	\bC^{\text{2p1h}}	\\
   \end{pmatrix}
   \begin{pmatrix}
     \bX	\\
@@ -288,10 +290,14 @@ and the corresponding coupling blocks read
     V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
 \end{align}
 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}
+\begin{equation}
+\begin{split}
+  \bSig(\omega) 
+  & = \bV^{\hhp} \qty(\omega \mathbb{1} - \bC^{\hhp})^{-1}  \T{(\bV^{\hhp})}
+  \\
+  & + \bV^{\pph} \qty(\omega \mathbb{1} - \bC^{\pph})^{-1}  \T{(\bV^{\pph})}, 
+\end{split}
+\end{equation}
 which can be further developed to give exactly Eq.~(\ref{eq:GW_selfenergy}).
 
 Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not.