From 00a7e5b2816070a8cf6119d42a6b6dc9b64dda2c Mon Sep 17 00:00:00 2001 From: pfloos Date: Sun, 5 Mar 2023 16:04:46 +0100 Subject: [PATCH] small modifs --- Manuscript/SRGGW.tex | 15 ++++++++------- 1 file changed, 8 insertions(+), 7 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 5a140d9..043cf4c 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -303,7 +303,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as \boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s). \end{equation} -\ant{The flow equation can be approximately solved by introduction of an approximate form of $\boldsymbol{\eta}(s)$.} +The flow equation can be approximately solved by introducing an approximate form of $\boldsymbol{\eta}(s)$. In this work, we consider Wegner's canonical generator \cite{Wegner_1994} \begin{equation} \boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}, @@ -338,7 +338,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t \label{sec:srggw} %%%%%%%%%%%%%%%%%%%%%% -\ant{Finally, the two previous subsections will be combined by applying the SRG method to the $GW$ formalism.} +Here, we combine the concepts of the two previous subsections and apply the SRG method to the $GW$ formalism. However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms. Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} @@ -654,7 +654,7 @@ Note that, after this transformation, the form of the regularizer is actually cl %=================================================================% % Reference comp det -Our set of molecules is composed by closed-shell compounds that correspond to the 50 smallest (wrt the number of electrons) atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015} +Our set of systems is composed by closed-shell compounds that correspond to the 50 smallest atoms and molecules (in terms of the number of electrons) 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 (without frozen core) \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. @@ -665,8 +665,8 @@ In all $GW$ calculations, we use the aug-cc-pVTZ cartesian basis set and self-co We use (restricted) HF guess orbitals and energies for all self-consistent $GW$ calculations. 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. -\ant{The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculation while for (SRG-)qs$GW$ calculations the $\eta$ value has been chosen as the largest value where one successfully converges the 50 systems of the test set.} +However, in order to perform black-box comparisons, these parameters have been fixed to these default values. +The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations while, for the (SRG-)qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set. %=================================================================% \section{Results} @@ -734,7 +734,6 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig2}. The TDA values are now underestimating the IP, unlike their RPA counterparts. For both static self-energies, the TDA leads to a slight increase in the absolute error. -\trashant{This trend is investigated in more detail in the next subsection.} Next, we investigate the flow parameter dependence of SRG-qs$GW$ for three more challenging molecular systems. The left panel of Fig.~\ref{fig:fig3} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, the HF approximation underestimates the reference IP. @@ -768,6 +767,7 @@ Therefore, it seems that the effect of the TDA cannot be systematically predicte \includegraphics[width=\linewidth]{fig5.pdf} \caption{ Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. + \PFL{Add MAE and MSE values to each figure.} \label{fig:fig4}} \end{figure*} %%% %%% %%% %%% @@ -890,7 +890,8 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder \includegraphics[width=\linewidth]{fig7.pdf} \caption{ Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. - \label{fig:fig6}} + \PFL{Add MAE and MSE values to each figure.} + \label{fig:fig6}} \end{figure*} %%% %%% %%% %%%