minor modifications

Manuscript/SRGGW.tex

@@ -131,7 +131,7 @@ The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hyberts
 These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
 
 In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction, in the quasiparticle equation, of a regularizer inspired by the similarity renormalization group (SRG). \cite{Monino_2022}
-Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022,Coveney_2023} the present work investigates the application of the SRG formalism in $GW$-based methods.
+Encouraged by \ant{this study and} the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022,Coveney_2023} the present work investigates the application of the SRG formalism in $GW$-based methods.
 In particular, we focus here on the possibility of curing the qs$GW$ convergence issues using the SRG.
 
 The SRG formalism has been developed independently by Wegner \cite{Wegner_1994} in the context of condensed matter systems and Glazek \& Wilson \cite{Glazek_1993,Glazek_1994} in light-front quantum field theory.
@@ -247,7 +247,7 @@ Then, the qs$GW$ equations are solved via a standard self-consistent field proce
 Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following Hermitian approximation
 \begin{equation}
   \label{eq:sym_qsgw}
-  \Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
+  \Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\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
@@ -255,7 +255,7 @@ The corresponding matrix elements are
   \label{eq:sym_qsGW}
   \Sigma_{pq}^{\qsGW} 
   = \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_ {pr}^{\nu} W_{qr}^{\nu}.
+  + \eta^2} +\frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2} ] W_ {pr}^{\nu} W_{qr}^{\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. 
@@ -282,7 +282,7 @@ This is one of the aims of the present work.
 The SRG method aims at continuously transforming a general Hamiltonian matrix to its diagonal form, or more often, to a block-diagonal form.
 Hence, the first step is to decompose this Hamiltonian matrix 
 \begin{equation}
-  \bH = \bH^\text{d} + \bH^\text{od}.
+  \bH = \bH^\text{d} + \bH^\text{od},
 \end{equation}
 into an off-diagonal part, $\bH^\text{od}$, that we aim at removing and the remaining diagonal part, $\bH^\text{d}$.
 
@@ -430,9 +430,9 @@ Then, the aim is to solve, order by order, the flow equation \eqref{eq:flowEquat
 where the supermatrices 
 \begin{subequations}
 \begin{align}
-  \bC & = \mqty( \bC^{\text{2h1p}}  &  \bO  \\  \bO  &  \bC^{\text{2p1h}}  )
+  \bC & = \mqty( \bC^{\text{2h1p}}  &  \bO  \\  \bO  &  \bC^{\text{2p1h}}  ),
   \\
-  \bW & = \mqty(  \bW^{\text{2h1p}} &  \bW^{\text{2p1h}} )
+  \bW & = \mqty(  \bW^{\text{2h1p}} &  \bW^{\text{2p1h}} ),
 \end{align}
 \end{subequations}
 collect the 2h1p and 2p1h channels.
@@ -499,7 +499,7 @@ Equation \eqref{eq:F0_C0} implies
 \end{subequations}
 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_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s}
+  W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s}.
 \end{equation}
 At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while,
 \begin{equation}
@@ -710,7 +710,7 @@ This behavior as a function of $s$ can be understood by applying matrix perturba
 Through second order in the coupling block, the principal IP is 
 \begin{equation}
   \label{eq:2nd_order_IP}
-  \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} 
+  \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu},
 \end{equation}
 where $\text{h}$ is the index of the highest occupied molecular orbital (HOMO).
 The first term of the right-hand side of Eq.~\eqref{eq:2nd_order_IP} is the zeroth-order IP and the following two terms originate from the 2h1p and 2p1h coupling, respectively.
@@ -882,7 +882,7 @@ These convergence problems are much more dramatic than for SRG-qs$GW$ because th
 For example, out of the 37 molecules that could be converged for $\eta=\num{e-2}$, the variation of the IP with respect to $\eta=\num{5e-2}$ can go up to \SI{0.1}{\eV}.
 
 This difference in behavior is due to the energy (in)dependence of the regularizers.
-Indeed, the SRG regularizer first includes the terms \titou{that are important for the energy} and finally adds the intruder states.
+Indeed, the SRG regularizer first includes the terms that are \ant{contributing to} the energy and finally adds the intruder states.
 On the other hand, the imaginary shift regularizer acts equivalently on all terms. 
 
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