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saving work because i need to leave, the conclusion is really bad for the moment

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Antoine Marie 2023-03-05 13:31:19 +01:00
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Manuscript/SRGGW.tex
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@ -819,9 +819,9 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
\ce{MgO} & 7.97 & 8.75 & 8.40 & 8.54 & 8.36 & 1.54 & 1.40 & 1.64 & 1.72 & 1.71 \\
\ce{O3} & 12.85 & 13.29 & 13.56 & 13.34 & 13.27 & 1.82 & 1.32 & 2.19 & 2.23 & 2.17 \\
\ce{C2H2} & 11.45 & 11.16 & 11.57 & 11.46 & 11.43 & -0.80 & -0.80 & -0.71 & -0.71 & -0.71 \\
\ce{NCH} & 13.76 & 13.50 & 13.86 & 13.75 & 13.73 & -0.53 & -0.61 & -0.52 & -0.55 & -0.54 \\
\ce{HCN} & 13.76 & 13.50 & 13.86 & 13.75 & 13.73 & -0.53 & -0.61 & -0.52 & -0.55 & -0.54 \\
\ce{B2H6} & 12.27 & 12.84 & 12.81 & 12.67 & 12.64 & -0.52 & -0.64 & -0.56 & -0.55 & -0.55 \\
\ce{H2CO} & 10.93 & 12.09 & 11.39 & 11.33 & 11.25 & -0.60 & -0.70 & -0.61 & -0.62 & -0.62 \\
\ce{CH2O} & 10.93 & 12.09 & 11.39 & 11.33 & 11.25 & -0.60 & -0.70 & -0.61 & -0.62 & -0.62 \\
\ce{C2H4} & 10.69 & 10.26 & 10.74 & 10.70 & 10.67 & -1.90 & -0.86 & -0.75 & -0.73 & -0.74 \\
\ce{SiH4} & 12.79 & 13.23 & 13.22 & 13.15 & 13.11 & -0.53 & -0.69 & -0.59 & -0.57 & -0.58 \\
\ce{PH3} & 10.60 & 10.60 & 10.79 & 10.76 & 10.73 & -0.51 & -0.71 & -0.58 & -0.56 & -0.57 \\
@ -842,7 +842,7 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
\ce{C3H8} & 12.13 & 12.73 & 12.61 & 12.51 & 12.46 & -0.63 & -0.83 & -0.70 & -0.67 & -0.67 \\
\ce{NaCl} & 9.10 & 9.60 & 9.20 & 9.25 & 9.16 & 0.67 & 0.56 & 0.64 & 0.64 & 0.64 \\
\ce{P2} & 10.72 & 10.05 & 10.49 & 10.43 & 10.40 & 0.43 & -0.35 & 0.47 & 0.48 & 0.47 \\
\ce{F2Mg} & 13.93 & 15.46 & 13.94 & 14.23 & 14.07 & 0.29 & -0.03 & 0.15 & 0.21 & 0.21 \\
\ce{MgF2} & 13.93 & 15.46 & 13.94 & 14.23 & 14.07 & 0.29 & -0.03 & 0.15 & 0.21 & 0.21 \\
\ce{OCS} & 11.23 & 11.44 & 11.52 & 11.37 & 11.32 & -1.43 & -1.27 & -1.03 & -0.97 & -0.98 \\
\ce{SO2} & 10.48 & 11.47 & 11.38 & 10.85 & 10.82 & 2.24 & 1.84 & 2.82 & 2.74 & 2.68 \\
\ce{C2H3Cl} & 10.17 & 10.13 & 10.39 & 10.27 & 10.24 & -0.61 & -0.79 & -0.66 & -0.65 & -0.65 \\
@ -896,17 +896,40 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
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.
The raw results are given in Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
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.
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.
The HF EAs are understimated in average with a MAE of \SI{0.31}{\eV} and some large outliers, for \SI{-2.03}{\eV} for \ce{F2} and \SI{1.04}{\eV} for \ce{CH2O} for example.
$G_0W_0$@HF mitigates the average error (\SI{0.16}{\eV} MAE) but the minimum and maximum values are not yet satisfactory.
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt}.
The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavours of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
\ANT{Maybe we should mention that some EA are not chemically meaningful.}
Note that a positive EA means that the anion state is bound and therefore the methods considered here are well-suited to describe the EAs.
On the other hand, a positive EA means that this is a resonance state and the methods considered in this study, even the reference, are not able to describe the physics of resonance states.
Yet, one can still compare the $GW$ values with their CCSD(T) counterparts in these cases.
%=================================================================%
\section{Conclusion}
\label{sec:conclusion}
%=================================================================%
Here comes the conclusion.
In this work, we have investigated the application of the similarity renormalization group to many-body perturbation theory in its $GW$ form.
The latter one is known to be plagued by intruder states while the first one is designed to avoid them.
The problems caused by intruder states in many-body perturbatin theory are multiple but here the focus was on convergence problems caused by such states.
The central equation of the SRG formalism, the flow equation, can be solved analytically for low perturbation order.
These analytical expressions for the Fock matrices elements and two-electrons screened integrals lead to a renormalized $GW$ quasiparticle equation.
Isolating the static part of the equation yields an alternative Hermitian static and intruder-state-free self-energy that can be used for qs$GW$ calculation.
In addition, to this new static form we also explained how to use the SRG formalism to cure discontinuity problems.
This gave a first-principle rationale for the SRG-inspired regularizer introduced in Ref.~\onlinecite{Monino_2022}.
The flow parameter dependence of SRG-qs$GW$ has been studied for a few test cases.
In particular it has been shown that the flow parameter gradually introduce correction in the static self-energy and therefore the IP gradually evolves from the HF one to a plateau value for $s\to\infty$.
For small values of the flow parameter the SRG-qs$GW$ IPs are actually worst than their starting point so one should always use a value of $s$ as large as possible.
As a second stage to this study, the SRG-qs$GW$ performance has been statistically gauged for a test set of 50 atoms and molecules (referred to as $GW$50).
It has been shown that in averaged SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
Note that while the accuracy improvement is quite small, it comes with no additional computational cost and its really fast to implement as one just need to change the static self-energy expression.
In addition, the SRG-qs$GW$ can be converged in a much more black-box fashion than traditional qs$GW$ thanks to its intruder-state free nature.
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{