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@ -78,14 +78,12 @@
date-added = {2023-01-30 22:12:16 +0100},
date-modified = {2023-01-30 22:12:32 +0100},
doi = {10.1021/acs.jctc.9b00353},
eprint = {https://doi.org/10.1021/acs.jctc.9b00353},
journal = {J. Chem. Theory Comput.},
number = {8},
pages = {4399-4414},
title = {Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation},
volume = {15},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.9b00353}}
year = {2019}}
@article{ChenyangLi_2021,
author = {Li,Chenyang and Evangelista,Francesco A.},
@ -106,28 +104,24 @@
date-added = {2023-01-30 22:09:49 +0100},
date-modified = {2023-01-30 22:10:02 +0100},
doi = {10.1021/acs.jctc.1c00980},
eprint = {https://doi.org/10.1021/acs.jctc.1c00980},
journal = {J. Chem. Theory Comput.},
number = {12},
pages = {7666-7681},
title = {Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00980}}
year = {2021}}
@article{Wang_2023,
author = {Wang, Meng and Fang, Wei-Hai and Li, Chenyang},
date-added = {2023-01-30 22:07:40 +0100},
date-modified = {2023-01-30 22:07:50 +0100},
doi = {10.1021/acs.jctc.2c00966},
eprint = {https://doi.org/10.1021/acs.jctc.2c00966},
journal = {J. Chem. Theory Comput.},
number = {1},
pages = {122-136},
title = {Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases},
volume = {19},
year = {2023},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00966}}
year = {2023}}
@misc{Scott_2023,
author = {Scott, Charles J. C. and Backhouse, Oliver J. and Booth, George H.},
@ -166,7 +160,6 @@
date-added = {2023-01-30 15:45:22 +0100},
date-modified = {2023-01-30 15:45:39 +0100},
doi = {10.1021/acs.jctc.7b00586},
eprint = {https://doi.org/10.1021/acs.jctc.7b00586},
journal = {J. Chem. Theory Comput.},
number = {10},
pages = {4765-4778},
@ -386,14 +379,12 @@
date-added = {2023-01-30 13:59:42 +0100},
date-modified = {2023-01-30 14:00:00 +0100},
doi = {10.1021/acs.jctc.2c00617},
eprint = {https://doi.org/10.1021/acs.jctc.2c00617},
journal = {J. Chem. Theory Comput.},
number = {12},
pages = {7570-7585},
title = {Benchmark of GW Methods for Core-Level Binding Energies},
volume = {18},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00617}}
year = {2022}}
@incollection{CsanakBook,
author = {Csanak, Gy and Taylor, HS and Yaris, Robert},
@ -471,50 +462,42 @@
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0009261494011834},
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(94)01183-4}}
@article{Frosini_2022c,
author = {M. Frosini and T. Duguet and J.-P. Ebran and B. Bally and H. Hergert and T. R. Rodr{\'{\i}}guez and R. Roth and J. M. Yao and V. Som{\`{a}}},
date-added = {2023-01-20 09:45:17 +0100},
date-modified = {2023-01-20 09:45:37 +0100},
doi = {10.1140/epja/s10050-022-00694-x},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00694-x},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00694-x}}
@article{Frosini_2022b,
author = {M. Frosini and T. Duguet and J.-P. Ebran and B. Bally and H. Hergert and T. R. Rodr{\'{\i}}guez and R. Roth and J. M. Yao and V. Som{\`{a}}},
date-added = {2023-01-20 09:44:49 +0100},
date-modified = {2023-01-20 09:45:46 +0100},
doi = {10.1140/epja/s10050-022-00694-x},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00694-x},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00694-x}}
@article{Frosini_2022,
title = {Multi-Reference Many-Body Perturbation Theory for Nuclei},
author = {Frosini, M. and Duguet, T. and Ebran, J.-P. and Som{\`a}, V.},
year = {2022},
journal = {Eur. Phys. J. A},
volume = {58},
number = {4},
pages = {62},
issn = {1434-601X},
doi = {10.1140/epja/s10050-022-00692-z}
}
@article{Frosini_2022a,
author = {M. Frosini and T. Duguet and J.-P. Ebran and V. Som{\`{a}}},
date-added = {2023-01-20 09:44:22 +0100},
date-modified = {2023-01-20 09:45:43 +0100},
doi = {10.1140/epja/s10050-022-00692-z},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00692-z},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00692-z}}
title = {Multi-Reference Many-Body Perturbation Theory for Nuclei},
author = {Frosini, M. and Duguet, T. and Ebran, J.-P. and Bally, B. and Mongelli, T. and Rodr{\'i}guez, T. R. and Roth, R. and Som{\`a}, V.},
year = {2022},
journal = {Eur. Phys. J. A},
volume = {58},
number = {4},
pages = {63},
issn = {1434-601X},
doi = {10.1140/epja/s10050-022-00693-y}
}
@article{Frosini_2022b,
title = {Multi-Reference Many-Body Perturbation Theory for Nuclei},
author = {Frosini, M. and Duguet, T. and Ebran, J.-P. and Bally, B. and Hergert, H. and Rodr{\'i}guez, T. R. and Roth, R. and Yao, J. M. and Som{\`a}, V.},
year = {2022},
journal = {Eur. Phys. J. A},
volume = {58},
number = {4},
pages = {64},
issn = {1434-601X},
doi = {10.1140/epja/s10050-022-00694-x}
}
@misc{Tolle_2022,
archiveprefix = {arXiv},
@ -1920,8 +1903,7 @@
pages = {6203-6210},
title = {Renormalized Singles Green's Function in the T-Matrix Approximation for Accurate Quasiparticle Energy Calculation},
volume = {12},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.1c01723}}
year = {2021}}
@article{Scuseria_2013,
author = {Scuseria,Gustavo E. and Henderson,Thomas M. and Bulik,Ireneusz W.},
@ -1945,9 +1927,7 @@
pages = {012809},
title = {Hydrogen-molecule spectrum by the many-body $GW$ approximation and the Bethe-Salpeter equation},
volume = {103},
year = {2021},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.103.012809},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevA.103.012809}}
year = {2021}}
@article{Vitale_2020,
author = {Vitale, Eugenio and Alavi, Ali and Kats, Daniel},
@ -2192,7 +2172,7 @@ note={Gaussian Inc. Wallingford CT}
title = {Coupled-Cluster Techniques for Computational Chemistry: {{The CFOUR}} Program Package},
author = {Matthews, Devin A. and Cheng, Lan and Harding, Michael E. and Lipparini, Filippo and Stopkowicz, Stella and Jagau, Thomas-C. and Szalay, P{\'e}ter G. and Gauss, J{\"u}rgen and Stanton, John F.},
year = {2020},
journal = {The Journal of Chemical Physics},
journal = {J. Chem. Phys.},
volume = {152},
number = {21},
pages = {214108},
@ -14955,7 +14935,7 @@ note={Gaussian Inc. Wallingford CT}
author = {Surj{\'a}n, P. R. and Szabados, {\'A}.},
doi = {10.1063/1.471814},
issn = {0021-9606},
journal = {The Journal of Chemical Physics},
journal = {J. Chem. Phys.},
number = {9},
pages = {3320--3324},
title = {Damping of Perturbation Corrections in Quasidegenerate Situations},
@ -16464,7 +16444,7 @@ note={Gaussian Inc. Wallingford CT}
author = {{Ismail-Beigi}, Sohrab},
doi = {10.1088/1361-648X/aa7803},
issn = {0953-8984},
journal = {Journal of Physics: Condensed Matter},
journal = {J. Phys. Cond. Mat.},
number = {38},
pages = {385501},
title = {Justifying Quasiparticle Self-Consistent Schemes via Gradient Optimization in {{Baym}}\textendash{{Kadanoff}} Theory},
@ -16476,7 +16456,7 @@ note={Gaussian Inc. Wallingford CT}
author = {Bruneval, Fabien and Marques, Miguel A. L.},
doi = {10.1021/ct300835h},
issn = {1549-9618},
journal = {Journal of Chemical Theory and Computation},
journal = {J. Chem. Theory Comput.},
number = {1},
pages = {324--329},
title = {Benchmarking the {{Starting Points}} of the {{GW Approximation}} for {{Molecules}}},
@ -17127,7 +17107,6 @@ note={Gaussian Inc. Wallingford CT}
@article{Tiago_2006,
author = {Tiago, Murilo L. and Chelikowsky, James R.},
doi = {10.1103/PhysRevB.73.205334},
file = {/Users/loos/Zotero/storage/3YTWEDNW/Tiago_2006.pdf},
issn = {1098-0121, 1550-235X},
journal = {Phys. Rev. B},
language = {en},
@ -17135,8 +17114,8 @@ note={Gaussian Inc. Wallingford CT}
number = {20},
title = {Optical Excitations in Organic Molecules, Clusters, and Defects Studied by First-Principles {{Green}}'s Function Methods},
volume = {73},
year = {2006},
bdsk-url-1 = {https://dx.doi.org/10.1103/PhysRevB.73.205334}}
year = {2006}
}
@article{Ou_2016,
author = {Ou, Qi and Subotnik, Joseph E.},

192
Manuscript/SRGGW.tex
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@ -44,7 +44,9 @@
showstringspaces=false,
showtabs=false,
tabsize=2
}
}
\newcolumntype{d}{D{.}{.}{-1}}
\lstset{style=mystyle}
@ -134,7 +136,7 @@ In particular, we focus here on the possibility of curing the qs$GW$ convergence
The SRG formalism has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
\cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016,Frosini_2022a,Frosini_2022b,Frosini_2022c}
\cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016,Frosini_2022,Frosini_2022a,Frosini_2022b}
See Ref.~\onlinecite{Hergert_2016a} for a recent review in this field.
The SRG transformation aims at decoupling an internal (or reference) space from an external space while incorporating information about their coupling in the reference space.
@ -500,15 +502,6 @@ It is worth noting the close similarity of the first-order elements with the one
\subsection{Second-order matrix elements}
% ///////////////////////////%
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=0.8\linewidth]{fig1.pdf}
\caption{
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.
\label{fig:plot}}
\end{figure*}
%%% %%% %%% %%%
The second-order renormalized quasiparticle equation is given by
\begin{equation}
\label{eq:GW_renorm}
@ -529,8 +522,8 @@ with elements
\label{eq:SRG-GW_selfenergy}
\begin{split}
\widetilde{\bSig}_{pq}(\omega; s)
&= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\
&+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
&= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\
&+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
\end{split}
\end{equation}
@ -546,6 +539,16 @@ which can be solved by simple integration along with the initial condition $\bF^
\times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
\end{multline}
%%% FIG 1 %%%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{flow}
\caption{
\ant{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?}
\label{fig:flow}}
\end{figure}
%%% %%% %%% %%%
At $s=0$, the second-order correction vanishes, hence giving
\begin{equation}
\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
@ -563,13 +566,12 @@ Therefore, the SRG flow continuously transforms the dynamical self-energy $\wide
As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
%%% FIG 2 %%%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{flow}
\begin{figure*}
\includegraphics[width=0.8\linewidth]{fig1.pdf}
\caption{
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}
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.
\label{fig:plot}}
\end{figure*}
%%% %%% %%% %%%
%///////////////////////////%
@ -717,9 +719,9 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
\caption{First ionization potential 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.}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{lccccc}
Mol. & $\Delta$CCSD(T) & HF & $G_0W_0$@HF & qs$GW$ & SRG-qs$GW$ \\
& & & $\eta=\num{e-3}$ & $\eta=\num{e-1}$ & $s=\num{e2}$ \\
\begin{tabular}{lddddd}
Mol. & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} \\
& & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
\hline
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.54 \\
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 \\
@ -785,65 +787,119 @@ Of course these are slight improvements but this is done with no additional comp
The evolution of the statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively.
%%% FIG 5 %%%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig5.pdf}
\caption{
Temporary figure about convergence
\label{fig:fig5}}
\end{figure}
%%% %%% %%% %%%
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=10^3$ self-consistency of the SRG-qs$GW$ scheme can be converged without any problems.
For $s=10^4$, convergence could not be attained for the following molecules \ANT{waiting for calculation}.
On the other hand, the qs$GW$ convergence is much more erratic.
Indeed, up to $s=10^3$ self-consistency can be attained without any problems (mean and max number of iterations = n for s=100).
For $s=10^4$, convergence could not be attained for 12 molecules out of 22, meaning that some intruder states were included in the static correction for this value of $s$.
This is illustrated in the case of \ce{H2O} in the upper panel Fig.
However, this is not a problem as the MAE is already well converged before the intruder states are added to the SRG-qs$GW$ static self-energy (lower panel).
On the other hand, the qs$GW$ convergence with respect to $\eta$ is more difficult to evaluate.
The whole set considered in this work could be converged for $\eta=0.1$.
However, if we decrease $\eta$ then self-consistency could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
Unfortunately, the convergence of the IP is not as tight as in the SRG case because for $\eta=0.01$ the values of the IP that could be converged can vary between $10^{-3}$ and $10^{-1}$ with respect to $\eta=0.1$.
However, as soon as we decrease $\eta$ self-consistency could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
Unfortunately, the convergence of the IP is not as tight as in the SRG case.
The values of the IP that could be converged for $\eta=0.01$ can vary between $10^{-3}$ and $10^{-1}$ with respect to $\eta=0.1$.
We will now gauge the effect of the TDA for the screening on the accuracy of the various methods considered previously.
% \begin{table}
% \caption{First ionization potential in eV calculated using $G_0W_0^{\text{TDA}}$@HF, qs$GW^{\text{TDA}}$ and SRG-qs$GW^{\text{TDA}}$. The statistical descriptors are computed for the errors with respect to the reference.}
% \label{tab:tab1}
% \begin{ruledtabular}
% \begin{tabular}{lddd}
% Mol. & \multicolumn{1}{c}{$G_0W_0^{\text{TDA}}$@HF} & \multicolumn{1}{c}{qs$GW^{\text{TDA}}$} & \multicolumn{1}{c}{SRG-qs$GW^{\text{TDA}}$} \\
% & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{5e-2}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
% \hline
% \ce{He} & 24.45 & 24.48 & 24.39 \\
% \ce{Ne} & 20.85 & 21.23 & 20.92 \\
% \ce{H2} & 16.53 & 16.46 & 16.50 \\
% \ce{Li2} & 5.45 & 5.50 & 5.46 \\
% \ce{LiH} & 8.14 & 8.17 & 8.05 \\
% \ce{HF} & 15.64 & 15.79 & 15.66 \\
% \ce{Ar} & 15.60 & 15.42 & 15.46 \\
% \ce{H2O} & 12.42 & 12.40 & 12.31 \\
% \ce{LiF} & 10.75 & 11.02 & 10.85 \\
% \ce{HCl} & 12.70 & 12.65 & 12.59 \\
% \ce{BeO} & 9.33 & 10.21 & 10.05 \\
% \ce{CO} & 14.60 & 13.82 & 13.84 \\
% \ce{N2} & 17.36 & 15.15 & 15.21 \\
% \ce{CH4} & 14.67 & 14.50 & 14.47 \\
% \ce{BH3} & 13.66 & 13.57 & 13.54 \\
% \ce{NH3} & 10.91 & 10.75 & 10.68 \\
% \ce{BF} & 11.38 & 11.11 & 11.12 \\
% \ce{BN} & 11.85 & 12.05 & 12.04 \\
% \ce{SH2} & 10.47 & 10.44 & 10.38 \\
% \ce{F2} & 15.55 & 15.23 & 15.22 \\
% \ce{MgO} & 8.10 & 7.76 & 7.58 \\
% \ce{O3} & 13.68 & 12.22 & 12.22 \\
% \hline
% MSE & 0.07 & -0.12 & -0.18 \\
% MAE & 0.37 & 0.22 & 0.25 \\
% SDE & 0.55 & 0.26 & 0.27 \\
% Min & -0.72 & -0.63 & -0.63 \\
% Max & 1.82 & 0.26 & 0.22 \\
% \end{tabular}
% \end{ruledtabular}
% \end{table}
Part on approximation and correction for W:
TDHF G0W0 not that bad in GW100 but bad in GW22, qsGW TDHF even worse even with SRG,
Maybe that would be nice to add SRG G0W0 to see if it mitigates the outliers of GW20 cf Bruneval 2021,
That would be nice to understand clearly why qsGWTDHF is worse (screening, gap, etc)
Part on EA:
MgO- does not converge yet but when we have it same analysis as Table 1 and Fig 4 but for the EA
\begin{table}
\caption{First ionization potential in eV calculated using $G_0W_0^{\text{TDA}x}$@HF, qs$GW^{\text{TDA}}$ and SRG-qs$GW^{\text{TDA}}$. The statistical descriptors are computed for the errors with respect to the reference.}
\caption{First electron attachment 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.}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{lccc}
Mol. & $G_0W_0^{\text{TDA}}$@HF & qs$GW^{\text{TDA}}$ & SRG-qs$GW^{\text{TDA}}$ \\
& $\eta=\num{e-3}$ & $\eta=\num{5e-2}$ & $s=\num{e2}$ \\
\hline
\ce{He} & 24.45 & 24.48 & 24.39 \\
\ce{Ne} & 20.85 & 21.23 & 20.92 \\
\ce{H2} & 16.53 & 16.46 & 16.50 \\
\ce{Li2} & 5.45 & 5.50 & 5.46 \\
\ce{LiH} & 8.14 & 8.17 & 8.05 \\
\ce{HF} & 15.64 & 15.79 & 15.66 \\
\ce{Ar} & 15.60 & 15.42 & 15.46 \\
\ce{H2O} & 12.42 & 12.40 & 12.31 \\
\ce{LiF} & 10.75 & 11.02 & 10.85 \\
\ce{HCl} & 12.70 & 12.65 & 12.59 \\
\ce{BeO} & 9.33 & 10.21 & 10.05 \\
\ce{CO} & 14.60 & 13.82 & 13.84 \\
\ce{N2} & 17.36 & 15.15 & 15.21 \\
\ce{CH4} & 14.67 & 14.50 & 14.47 \\
\ce{BH3} & 13.66 & 13.57 & 13.54 \\
\ce{NH3} & 10.91 & 10.75 & 10.68 \\
\ce{BF} & 11.38 & 11.11 & 11.12 \\
\ce{BN} & 11.85 & 12.05 & 12.04 \\
\ce{SH2} & 10.47 & 10.44 & 10.38 \\
\ce{F2} & 15.55 & 15.23 & 15.22 \\
\ce{MgO} & 8.10 & 7.76 & 7.58 \\
\ce{O3} & 13.68 & 12.22 & 12.22 \\
\hline
MSE & 0.07 & -0.12 & -0.18 \\
MAE & 0.37 & 0.22 & 0.25 \\
SDE & 0.55 & 0.26 & 0.27 \\
Min & -0.72 & -0.63 & -0.63 \\
Max & 1.82 & 0.26 & 0.22 \\
\hline
\begin{tabular}{lddddd}
Mol. & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} \\
& & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
\hline
\ce{He} & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
\ce{Ne} & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\
\ce{H2} & 1.35 & 1.33 & 1.28 & 1.28 & 1.28 \\
\ce{Li2} & -0.34 & 0.08 & -0.17 & -0.18 & -0.21 \\
\ce{LiH} & 0.29 & -0.20 & -0.27 & -0.27 & -0.27 \\
\ce{HF} & 0.66 & 0.81 & 0.71 & 0.70 & 0.70 \\
\ce{Ar} & 2.55 & 2.97 & 2.68 & 2.64 & 2.65 \\
\ce{H2O} & 0.61 & 0.80 & 0.68 & 0.65 & 0.66 \\
\ce{LiF} & -0.35 & -0.29 & -0.33 & -0.32 & -0.33 \\
\ce{HCl} & 0.57 & 0.79 & 0.64 & 0.63 & 0.63 \\
\ce{BeO} & -2.17 & -1.80 & -2.28 & -2.10 & -2.13 \\
\ce{CO} & 1.57 & 1.80 & 1.66 & 1.61 & 1.62 \\
\ce{N2} & 2.37 & 2.20 & 2.10 & 2.10 & 2.10 \\
\ce{CH4} & 0.65 & 0.79 & 0.70 & 0.68 & 0.68 \\
\ce{BH3} & 0.09 & 0.81 & 0.46 & 0.29 & 0.30 \\
\ce{NH3} & 0.61 & 0.80 & 0.68 & 0.66 & 0.66 \\
\ce{BF} & 0.80 & 1.06 & 0.90 & 0.87 & 0.87 \\
\ce{BN} & -3.02 & -2.97 & -3.90 & -3.41 & -3.44 \\
\ce{SH2} & 0.52 & 0.76 & 0.60 & 0.58 & 0.59 \\
\ce{F2} & -0.32 & 1.71 & 0.53 & -0.10 & -0.07 \\
\ce{MgO} & -1.54 & -1.40 & -1.64 & -1.72 & -1.71 \\
\ce{O3} & -1.82 & -1.32 & -2.19 & -2.22 & -2.17 \\
\hline
MSE & & -0.30 & -0.02 & 0.00 & 0.00 \\
MAE & & 0.32 & 0.19 & 0.11 & 0.12 \\
SDE & & 0.43 & 0.31 & 0.17 & 0.17 \\
Min & & -2.03 & -0.85 & -0.22 & -0.25 \\
Max & & 0.17 & 0.88 & 0.41 & 0.42 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Part on EA:
MgO- does not converge yet but when we have it same analysis as Table 1 and Fig 4 but for the EA
%%% FIG 6 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig6.pdf}
\caption{
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
\label{fig:fig4}}
\end{figure*}
%%% %%% %%% %%%
%=================================================================%
\section{Conclusion}

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