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@ -90,14 +90,12 @@
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date-added = {2023-01-30 22:10:49 +0100},
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date-added = {2023-01-30 22:10:49 +0100},
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date-modified = {2023-01-30 22:11:03 +0100},
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date-modified = {2023-01-30 22:11:03 +0100},
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doi = {10.1063/5.0059362},
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doi = {10.1063/5.0059362},
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eprint = {https://doi.org/10.1063/5.0059362},
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journal = {J. Chem. Phys.},
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journal = {J. Chem. Phys.},
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number = {11},
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number = {11},
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pages = {114111},
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pages = {114111},
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title = {Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics},
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title = {Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics},
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volume = {155},
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volume = {155},
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year = {2021},
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year = {2021}}
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bdsk-url-1 = {https://doi.org/10.1063/5.0059362}}
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@article{Wang_2021,
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@article{Wang_2021,
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author = {Wang, Shuhe and Li, Chenyang and Evangelista, Francesco A.},
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author = {Wang, Shuhe and Li, Chenyang and Evangelista, Francesco A.},
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@ -1321,9 +1319,8 @@
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author = {Hergert, H. and Bogner, S. K. and Morris, T. D. and Schwenk, A. and Tsukiyama, K.},
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author = {Hergert, H. and Bogner, S. K. and Morris, T. D. and Schwenk, A. and Tsukiyama, K.},
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doi = {10.1016/j.physrep.2015.12.007},
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doi = {10.1016/j.physrep.2015.12.007},
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issn = {0370-1573},
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issn = {0370-1573},
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journal = {Physics Reports},
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journal = {Phys. Rep.},
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pages = {165--222},
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pages = {165--222},,
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series = {Memorial {{Volume}} in {{Honor}} of {{Gerald E}}. {{Brown}}},
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title = {The {{In-Medium Similarity Renormalization Group}}: {{A}} Novel Ab Initio Method for Nuclei},
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title = {The {{In-Medium Similarity Renormalization Group}}: {{A}} Novel Ab Initio Method for Nuclei},
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volume = {621},
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volume = {621},
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year = {2016},
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year = {2016},
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@ -17106,7 +17103,6 @@ note={Gaussian Inc. Wallingford CT}
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@article{Tiago_2006,
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@article{Tiago_2006,
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author = {Tiago, Murilo L. and Chelikowsky, James R.},
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author = {Tiago, Murilo L. and Chelikowsky, James R.},
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doi = {10.1103/PhysRevB.73.205334},
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issn = {1098-0121, 1550-235X},
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issn = {1098-0121, 1550-235X},
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journal = {Phys. Rev. B},
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journal = {Phys. Rev. B},
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language = {en},
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language = {en},
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@ -17220,14 +17216,16 @@ note={Gaussian Inc. Wallingford CT}
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year = {1998}}
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year = {1998}}
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@article{Schindlmayr_1998,
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@article{Schindlmayr_1998,
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author = {Schindlmayr, Arno and Godby, Rex William},
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title = {Systematic {{Vertex Corrections}} through {{Iterative Solution}} of {{Hedin}}'s {{Equations Beyond}} the \$\textbackslash mathit\{\vphantom\}{{GW}}\vphantom\{\}\$ {{Approximation}}},
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file = {/Users/loos/Zotero/storage/S32MIQEF/Schindlmayr_1998b.pdf},
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author = {Schindlmayr, Arno and Godby, R. W.},
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year = {1998},
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journal = {Phys. Rev. Lett.},
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journal = {Phys. Rev. Lett.},
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number = {8},
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pages = {1702},
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title = {Systematic Vertex Corrections through Iterative Solution of {{Hedin}}'s Equations beyond the {{GW}} Approximation},
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volume = {80},
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volume = {80},
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year = {1998}}
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number = {8},
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pages = {1702--1705},
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doi = {10.1103/PhysRevLett.80.1702}
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}
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@article{Schindlmayr_2013,
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@article{Schindlmayr_2013,
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author = {Schindlmayr, Arno},
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author = {Schindlmayr, Arno},
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@ -716,7 +716,7 @@ Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~
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Therefore, it seems that the effect of the TDA can not be systematically predicted.
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Therefore, it seems that the effect of the TDA can not be systematically predicted.
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\begin{table}
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\begin{table}
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\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.}
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\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. \ANT{Maybe change the values of SRG with the one for s=1000}}
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\label{tab:tab1}
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\label{tab:tab1}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{lddddd}
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\begin{tabular}{lddddd}
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@ -781,7 +781,6 @@ In addition, the MSE and MAE (\SI{0.24}{\electronvolt}/\SI{0.25}{\electronvolt})
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Now turning to the new results of this manuscript, \ie the alternative self-consistent scheme SRG-qs$GW$.
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Now turning to the new results of this manuscript, \ie the alternative self-consistent scheme SRG-qs$GW$.
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Table~\ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
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Table~\ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
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For this value of the flow parameter, the MAE is converged to \SI{d-3}{\electronvolt} (see Supplementary Material).
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The statistical descriptors corresponding to the alternative static self-energy are all improved with respect to qs$GW$.
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The statistical descriptors corresponding to the alternative static self-energy are all improved with respect to qs$GW$.
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Of course these are slight improvements but this is done with no additional computational cost and can be implemented really quickly just by changing the form of the static approximation.
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Of course these are slight improvements but this is done with no additional computational cost and can be implemented really quickly just by changing the form of the static approximation.
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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}.
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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}.
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@ -797,6 +796,8 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
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\end{figure}
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\end{figure}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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The difference in
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In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
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In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
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Indeed, up to $s=10^3$ self-consistency can be attained without any problems (mean and max number of iterations = n for s=100).
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Indeed, up to $s=10^3$ self-consistency can be attained without any problems (mean and max number of iterations = n for s=100).
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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 $s$.
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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 $s$.
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@ -849,12 +850,9 @@ The values of the IP that could be converged for $\eta=0.01$ can vary between $1
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% \end{ruledtabular}
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% \end{ruledtabular}
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% \end{table}
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% \end{table}
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Part on EA:
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MgO- does not converge yet but when we have it same analysis as Table 1 and Fig 4 but for the EA
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\begin{table}
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\begin{table}
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\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.}
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\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.}
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\label{tab:tab1}
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\label{tab:tab2}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{lddddd}
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\begin{tabular}{lddddd}
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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$GW$} \\
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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$GW$} \\
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@ -897,10 +895,17 @@ MgO- does not converge yet but when we have it same analysis as Table 1 and Fig
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\includegraphics[width=\linewidth]{fig6.pdf}
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\includegraphics[width=\linewidth]{fig6.pdf}
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\caption{
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\caption{
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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$GW$.
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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$GW$.
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\label{fig:fig4}}
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\label{fig:fig6}}
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\end{figure*}
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\end{figure*}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal electron attachement (EA) energies.
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The raw results are given in Tab.~\ref{tab:tab2} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
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The HF EA are understimated in averaged with some large outliers while $G_0W_0$@HF mitigates the average error there are still large outliers.
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The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt} and the error of the outliers is reduced with respect to $G_0W_0$@HF.
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\ANT{Maybe we should mention that some EA are not chemically meaningful.}
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%=================================================================%
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%=================================================================%
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\section{Conclusion}
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\section{Conclusion}
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\label{sec:conclusion}
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\label{sec:conclusion}
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