From 327d151f2da7c56ef17c3d272058f069573b17a4 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 11:24:19 +0100 Subject: [PATCH 1/4] ok with Sec III --- Manuscript/SRGGW.tex | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 1781b45..c30355f 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -488,9 +488,13 @@ and % \end{align} %\end{subequations} Equation \eqref{eq:F0_C0} implies +\begin{subequations} \begin{align} - \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, + \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & + \\ + \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \end{align} +\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} @@ -499,7 +503,7 @@ At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals de \begin{equation} \lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0. \end{equation} -Therefore, $W_{pq}^{\nu(1)}(s)$ are genuine renormalized two-electron screened integrals. +Therefore, $W_{pq}^{\nu(1)}(s)$ is a genuine renormalized two-electron screened integral. It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}). %///////////////////////////% @@ -620,7 +624,7 @@ The convergence properties and the accuracy of both static approximations are qu To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}. Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023} Is it then possible to rely on the SRG machinery to remove discontinuities? -Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite $s$ values are suitable to avoid intruder states in its static part. +Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part. However, performing the following bijective transformation \titou{\begin{align} e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t}, From 2f62bb78cceed5d148480a73f583dac224e61718 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 11:26:46 +0100 Subject: [PATCH 2/4] ok with Sec IV --- Manuscript/SRGGW.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index c30355f..91974e8 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -639,7 +639,7 @@ Note that, after this transformation, the form of the regularizer is actually cl %=================================================================% % Reference comp det -Our set is composed by closed-shell organic molecules that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015} +Our set of molecules is composed by closed-shell organic compounds that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015} Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR} The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively. @@ -651,7 +651,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$ The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively. In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme. However, in order to perform a black-box comparison, these parameters have been fixed to these default values. -\titou{The $\eta$ value used in the convetional $G_0W_0$ and $qsGW$ calculations corresponds to the largest value where one succesfully converges all systems.} +\titou{The $\eta$ value used in the convetional $G_0W_0$ and qs$GW$ calculations corresponds to the largest value where one successfully converges all systems.} %=================================================================% \section{Results} From ebd9ba49695679475d9e24cd4c75c60b73f48b2c Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 18:39:45 +0100 Subject: [PATCH 3/4] fix typo --- Manuscript/SRGGW.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 91974e8..1ce32fb 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -200,7 +200,7 @@ with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie witho with \begin{subequations} \begin{align} - A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, + A_{ia,jb} & = (\epsilon_a - \epsilon_i) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\ B_{ia,jb} & = \eri{ij}{ab}, \end{align} From 33410e512700d2564e4230fcb8b1b284e129c3a3 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 19:43:41 +0100 Subject: [PATCH 4/4] corrected one more typo --- Manuscript/SRGGW.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 1ce32fb..1a01819 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -195,7 +195,7 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as \begin{equation} \label{eq:full_dRPA} - \mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ), + \mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX & \bY \\ \bY & \bX ) = \mqty( \bX & \bY \\ \bY & \bX ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ), \end{equation} with \begin{subequations}