Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

Manuscript/SRGGW.tex

@@ -195,12 +195,12 @@ 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}
 \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}
@@ -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},
@@ -635,7 +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.
 
@@ -647,7 +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}