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Pierre-Francois Loos 2023-02-17 14:38:13 +01:00
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1
Fig/flow.tex
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@ -6,6 +6,7 @@
\begin{tikzpicture}[] \begin{tikzpicture}[]
% frame % 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[-,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[-,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]{$s = \Lambda^{-2}$}; \draw[->,thick] (6.5,0) -- (8,0) node[anchor=west]{$s = \Lambda^{-2}$};

38
Manuscript/SRGGW.tex
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@ -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. 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. 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 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. 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 \bigskip
%\begin{center} \begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}} \boxed{\includegraphics[width=0.5\linewidth]{flow}}
%\end{center} \end{center}
%\bigskip \bigskip
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -551,7 +551,7 @@ which can be solved by simple integration along with the initial condition $\bF^
\centering \centering
\includegraphics[width=\linewidth]{flow} \includegraphics[width=\linewidth]{flow}
\caption{ \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}} \label{fig:flow}}
\end{figure} \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} ]. \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ].
\end{multline} \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. 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. 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. 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. 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 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. 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). 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}. 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}. 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}. 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. 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. 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. 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. 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. 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}. 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. 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. 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} 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. 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. 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. 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, 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. Also, SRG-qs$GW^\TDA$ is better than qs$GW^\TDA$ in the three cases of Fig.~\ref{fig:fig3}.
Therefore, it seems that the effect of the TDA can not be systematically predicted. However, it is not a general rule.
Therefore, it seems that the effect of the TDA cannot be systematically predicted.
\begin{table*} \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}} \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}}

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Manuscript/flow.pdf
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