diff --git a/Fig/flow.tex b/Fig/flow.tex index 93e07ba..ef01982 100644 --- a/Fig/flow.tex +++ b/Fig/flow.tex @@ -6,6 +6,7 @@ \begin{tikzpicture}[] % frame + \draw[-,thick] (0,0) -- (4,0) node[right,sloped,below,yshift=-0.25cm]{flow}; \draw[-,thick] (0,0) node[anchor=north west]{} -- (5.5,0); \draw[-,dash pattern=on 20pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 20pt,thick] (5,0) -- (7,0); \draw[->,thick] (6.5,0) -- (8,0) node[anchor=west]{$s = \Lambda^{-2}$}; diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 7a47013..6ab7742 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -81,12 +81,12 @@ Despite this, self-consistent versions still pose challenges in terms of converg A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first principles, the expression of a new, naturally hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations. -The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations and slightly improves the overall accuracy. -%\bigskip -%\begin{center} -% \boxed{\includegraphics[width=0.5\linewidth]{TOC}} -%\end{center} -%\bigskip +The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code. +\bigskip +\begin{center} + \boxed{\includegraphics[width=0.5\linewidth]{flow}} +\end{center} +\bigskip \end{abstract} \maketitle @@ -551,7 +551,7 @@ which can be solved by simple integration along with the initial condition $\bF^ \centering \includegraphics[width=\linewidth]{flow} \caption{ - 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). \label{fig:flow}} \end{figure} %%% %%% %%% %%% @@ -600,7 +600,7 @@ This yields a $s$-dependent static self-energy which matrix elements read \\ \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ]. \end{multline} -Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization. +Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is straightforward to implement in existing code and is naturally hermitian as opposed to the usual case [see Eq.~\eqref{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 self-energy denominator. Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms. @@ -686,8 +686,8 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set. Figure \ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value. -The IP at the HF level (dashed black line) is overestimated; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019} -The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference. +The IP at the HF level (dashed black line) is too large; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019} +The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference value. Figure \ref{fig:fig2} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve). At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}. @@ -711,28 +711,30 @@ Therefore, for small $s$, only the last term of Eq.~\eqref{eq:2nd_order_IP} is p As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}. 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 IPs are now underestimated, unlike their RPA counterparts. +The TDA values are now underestimated the IP, unlike their RPA counterparts. For both static self-energies, the TDA leads to a slight increase in the absolute error. 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 IP underestimates the reference IP. +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. On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are too large. Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water. In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig2}. -Both TDA results are worse than their RPA counterparts but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$. +Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$. -We now turn to the lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2}). +We now turn to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2}). In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value. However, SRG-qs$GW$ improves slightly the accuracy as compared to HF. + Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021} -The SRG-qs$GW$ could be converged without any problem even for large values of $s$. +The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values. Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart. Note that, for \ce{LiH} and \ce{BeO}, the TDA actually improves the accuracy compared to the RPA-based qs$GW$ schemes. -However, as we shall see in Sec.~\ref{sec:SRG_vs_Sym}, these are special cases and, on average, the RPA polarizability performs better than the TDA one. -Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig3} but this is the other way around. -Therefore, it seems that the effect of the TDA can not be systematically predicted. +However, as we shall see in Sec.~\ref{sec:SRG_vs_Sym}, these are special cases, and, on average, the RPA polarizability performs better than its TDA version. +Also, SRG-qs$GW^\TDA$ is better than qs$GW^\TDA$ in the three cases of Fig.~\ref{fig:fig3}. +However, it is not a general rule. +Therefore, it seems that the effect of the TDA cannot be systematically predicted. \begin{table*} \caption{First ionization potential (left) and first electron attachment (right) in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference. \ANT{Maybe change the values of SRG with the one for s=1000}} diff --git a/Manuscript/flow.pdf b/Manuscript/flow.pdf index e07cd54..4bcb403 100644 Binary files a/Manuscript/flow.pdf and b/Manuscript/flow.pdf differ