starting messing around with notations but I need to talk to Antoine

Manuscript/SRGGW.rty

@@ -126,9 +126,10 @@
 \newcommand{\GW}{GW}
 \newcommand{\GF}{\text{GF(2)}}
 \newcommand{\GT}{GT}	
-\newcommand{\evGW}{\text{ev}$GW$}	
+\newcommand{\evGW}{\text{ev}GW}	
 \newcommand{\qsGW}{\text{qs}GW}
-\newcommand{\qs}{\text{qs}}
+\newcommand{\SRGGW}{\text{SRG-}GW}
+\newcommand{\SRGqsGW}{\text{SRG-qs}GW}
 \newcommand{\GOWO}{G_0W_0}
 
 %%% Notations %%%

Manuscript/SRGGW.tex

@@ -170,20 +170,20 @@ Unless otherwise stated, atomic units are used throughout.
 The central equation of many-body perturbation theory based on Hedin's equations is the so-called dynamical and non-hermitian quasiparticle 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),
+  \qty[ \bF + \bSig^{\GW}(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
 \end{equation}
-where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy. 
+where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig^{\GW}(\omega)$ is (the correlation part of) the $GW$ self-energy. 
 Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
 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 (single) excitations.
+The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations.
 
 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 but the dynamical and non-hermitian nature of $\bSig(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view.
+Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently but the dynamical and non-hermitian nature of $\bSig^{\GW}(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view.
 
-The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
+The matrix elements of $\bSig^{\GW}(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
 \begin{equation}
   \label{eq:GW_selfenergy}
-  \Sigma_{pq}(\omega) 
+  \Sigma_{pq}^{\GW}(\omega) 
   = \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
   + \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
 \end{equation}
@@ -220,14 +220,14 @@ The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where
 Assuming a HF starting point, this results in $K$ quasiparticle equations that read
 \begin{equation}
   \label{eq:G0W0}
-  \epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
+  \epsilon_p^{\HF} + \Sigma_{pp}^{\GW}(\omega) - \omega = 0,
 \end{equation}
-where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
+where $\Sigma_{pp}^{\GW}(\omega)$ are the diagonal elements of $\bSig^{\GW}$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
 The previous equations are non-linear with respect to $\omega$ and therefore 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
 \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.
+  0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}^{\GW}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
 \end{equation}
 The solution with the largest weight is referred to as the quasiparticle 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, hence the quasiparticle is not well-defined.
@@ -241,19 +241,19 @@ Alternatively, one may solve iteratively the set of quasiparticle equations \eqr
 However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible. 
 Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
 
-In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
+In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig^{\GW}(\omega)$.
-To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
+To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig^{\GW}(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$.
-Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qs}$.
+Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$.
-Various choices for $\bSig^\qs$ are possible but the most popular is the following Hermitian approximation
+Various choices for $\bSig^{\qsGW}$ are possible but the most popular 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) ].
+  \Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}^{\GW}(\epsilon_p) + \Sigma_{pq}^{\GW}(\epsilon_q) ].
 \end{equation}
 which was first 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 functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
 The corresponding matrix elements are
 \begin{equation}
   \label{eq:sym_qsGW}
-  \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2}  ) W_ {p,r\nu} W_{q,r\nu}.
+  \Sigma_{pq}^{\qsGW}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2}  ) W_ {p,r\nu} W_{q,r\nu}.
 \end{equation}
 with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
 One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. 
@@ -379,7 +379,7 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
 \begin{equation}
   \begin{split}
     \label{eq:downfolded_sigma}
-  \bSig(\omega)
+  \bSig^{\GW}(\omega)
   & = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1}  (\bW^{\hhp})^\dag
   \\
   & + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1}  (\bW^{\pph})^\dag, 
@@ -492,7 +492,7 @@ Equation \eqref{eq:F0_C0} implies
 \end{align}
 and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
 \begin{equation}
-  W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
+  W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) \titou{e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}}
 \end{equation}
 At $s=0$ the elements  $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
 Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
@@ -506,7 +506,7 @@ The second-order renormalized quasiparticle equation is given by
 \begin{equation}
   \label{eq:GW_renorm}
 %  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
-  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
+  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}^{\GW}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
 \end{equation}
 with a regularized Fock matrix of the form
 \begin{equation}
@@ -515,13 +515,13 @@ with a regularized Fock matrix of the form
 and a regularized dynamical self-energy
 \begin{equation}
   \label{eq:srg_sigma}
-  \widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger},
+  \widetilde{\bSig}^{\GW}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger},
 \end{equation}
 with elements
 \begin{equation}
   \label{eq:SRG-GW_selfenergy}
   \begin{split}
-    \widetilde{\bSig}_{pq}(\omega; s) 
+    \widetilde{\bSig}_{pq}^{\GW}(\omega; s) 
   &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ 
   &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
   \end{split}
@@ -544,7 +544,7 @@ which can be solved by simple integration along with the initial condition $\bF^
   \centering
   \includegraphics[width=\linewidth]{flow}
   \caption{
-    \ant{Schematic} evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). \ANT{Maybe we should replace dynamic by full?}
+    Schematic evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). \ANT{Maybe we should replace dynamic by full?}
     \label{fig:flow}}
 \end{figure}
 %%% %%% %%% %%%
@@ -560,9 +560,9 @@ For $s\to\infty$, it tends towards the following static limit
 \end{equation}
 while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
 \begin{equation}
-  \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO.
+  \lim_{s\to\infty} \widetilde{\bSig}^{\GW}(\omega; s) = \bO.
 \end{equation}
-Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
+Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}^{\GW}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
 As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
 %%% FIG 2 %%% 
@@ -582,10 +582,16 @@ Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and her
 Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
 However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}.
 This yields a $s$-dependent static self-energy which matrix elements read
-\begin{multline}
+\begin{equation}
   \label{eq:SRG_qsGW}
-   \Sigma_{pq}^{\text{SRG}}(s) = F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  \\ \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
+	\begin{split}
-\end{multline}
+   \Sigma_{pq}^{\SRGqsGW}(s) 
+   & = F_{pq}^{(2)}(s) 
+   \\
+   & = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  
+   \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
+	\end{split}
+\end{equation}
 Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization.
 Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
 Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.