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@ -81,7 +81,7 @@ The family of Green's function methods based on the $GW$ approximation has gaine
Despite this, self-consistent versions still pose challenges in terms of convergence.
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 SRG formalism enables us to derive, from first principles, the expression of a naturally static and Hermitian form of the 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, slightly improves the overall accuracy, and is straightforward to implement in existing code.
\bigskip
\begin{center}
@ -114,7 +114,7 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
\label{eq:gw_selfenergy}
\Sigma(1,2) = \ii G(1,2) W(1,2).
\end{equation}
Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
Diagrammatically, $GW$ involves a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
For example, modeling core-electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{vanSetten_2018,Golze_2018,Golze_2020,Li_2022}
@ -122,7 +122,7 @@ Many-body perturbation theory can also be used to access optical excitation ener
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require higher precision.
Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of vertex corrections has been demonstrated to be a tricky task. \cite{Baym_1961,Baym_1962,DeDominicis_1964a,DeDominicis_1964b,Bickers_1989a,Bickers_1989b,Bickers_1991,Hedin_1999,Bickers_2004,Shirley_1996,DelSol_1994,Schindlmayr_1998,Morris_2007,Shishkin_2007b,Romaniello_2009a,Romaniello_2012,Gruneis_2014,Hung_2017,Maggio_2017b,Mejuto-Zaera_2022}
For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in Green's function methods.
We refer the reader to the recent review by Golze and co-workers \cite{Golze_2019} for an extensive list of current challenges in Green's function methods.
Many-body perturbation theory also suffers from the infamous intruder-state problem,\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} where they manifest themselves as solutions of the quasiparticle equation with non-negligible spectral weights.
In some cases, this transfer of spectral weight makes it difficult to distinguish between a quasiparticle and a satellite.
@ -130,7 +130,7 @@ These multiple solutions hinder the convergence of partially self-consistent sch
The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007a} is also impacted by these intruder states, leading to discontinuities and/or irregularities in a variety of physical quantities including charged and neutral excitation energies, correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction, in the quasiparticle equation, of a regularizer inspired by the similarity renormalization group (SRG). \cite{Monino_2022}
Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022,Coveney_2023} the present work investigates the application of the SRG formalism in $GW$-based methods.
In particular, we focus here on the possibility of curing the qs$GW$ convergence issues using the SRG.
@ -768,19 +768,19 @@ The SRG-qs$GW$ EA (absolute) error is monotonically decreasing from the HF value
%%% %%% %%% %%%
Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in this work computed at various levels of theory.
As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
As mentioned previously, the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
However, there are still outliers with large errors.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}, a large error that is due to the HF starting point.
Self-consistency can mitigate the error of the outliers as the MAE at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$ to \SI{0.18}{\eV}.
Self-consistency mitigates the error of the outliers as the MAE at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$@HF to \SI{0.18}{\eV} for qs$GW$.
In addition, the MSE and MAE (\SI{0.23}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
Let us now turn to the new method of this manuscript, the SRG-qs$GW$ self-consistent scheme.
Let us now turn to our new method, the SRG-qs$GW$ self-consistent scheme.
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=\num{e3}$.
The statistical descriptors corresponding to this alternative static self-energy are all improved with respect to qs$GW$.
In particular, the MSE and MAE are decreased by \SI{0.06}{\eV}.
Of course, these are slight improvements but this is done with no additional computational cost and can be easily implemented by changing the form of the static approximation.
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}.
Of course, these are small improvements but this is done with no additional computational cost and it can be easily implemented in existing code by changing the form of the static self-energy.
The evolution of the statistical descriptors with respect to the various methods considered in Table \ref{tab:tab1} is graphically illustrated in Fig.~\ref{fig:fig4}.
The decrease of the MSE and SDE correspond to a shift of the maximum of the distribution toward zero and a contraction of the distribution width, respectively.
%%% TABLE I %%%
@ -866,23 +866,23 @@ The decrease of the MSE and SDE correspond to a shift of the maximum of the dist
\end{figure}
%%% %%% %%% %%%
In addition to this improvement in accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=\num{e3}$ SRG-qs$GW$, it is straightforward to reach self-consistency for the 50 compounds.
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50.
This means that some intruder states were included in the static correction for this value of $s$.
However, this is not a problem as the MAE of the test set is already well converged at $s=\num{e3}$.
This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and $s=1/2\eta^2$.
The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises for $s>\num{e3}$.
In addition to this improvement in terms of accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ parent.
Indeed, up to $s=\num{e3}$, it is straightforward to reach self-consistency for the 50 compounds at the SRG-qs$GW$ level.
For $s=\num{5e3}$, convergence could not be attained for 11 systems out of 50.
%This means that some intruder states were included in the static correction for this value of $s$.
However, this is not a serious issue as the MAE of the test set is already well converged at $s=\num{e3}$.
This is illustrated by the green curve of Fig.~\ref{fig:fig6} which shows the evolution of the SRG-qs$GW$ MAE with respect to $s$.
The convergence plateau of the MAE is reached around $s=50$ while the convergence problems arise for $s>\num{e3}$.
Therefore, for future studies using the SRG-qs$GW$ method, a default value of the flow parameter equal to $\num{5e2}$ or $\num{e3}$ is recommended.
On the other hand, the qs$GW$ convergence behavior is more erratic.
On the other hand, the qs$GW$ convergence behavior is more erratic as shown by the blue curve of Fig.~\ref{fig:fig6} where we report the variation of the qs$GW$ MAE as a function of $\eta=\sqrt{1/(2s)}$.
At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be reached for 13 molecules while 2 systems were already problematic at $\eta=\num{5e-2}$ ($s=200$).
These convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE has not reached a convergence plateau before these problems arise (see the orange curve in Fig.~\ref{fig:fig6}).
These convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE has not reached its limiting value before these issues arise.
For example, out of the 37 molecules that could be converged for $\eta=\num{e-2}$, the variation of the IP with respect to $\eta=\num{5e-2}$ can go up to \SI{0.1}{\eV}.
This difference in behavior is due to the energy (in)dependence of the regularizers.
Indeed, the SRG regularizer first includes the terms that are important for the energy and finally adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on intruder states and terms that contribute to the energy.
Indeed, the SRG regularizer first includes the terms \titou{that are important for the energy} and finally adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on all terms.
%%% FIG 7 %%%
\begin{figure*}
@ -894,16 +894,16 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
\end{figure*}
%%% %%% %%% %%%
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal EA energies.
The raw results are given in Table \ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig7}.
The HF EAs are, on average, underestimated with a MAE of \SI{0.31}{\eV} and some clear 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 (MAE equal to \SI{0.16}{\eV}) but the minimum and maximum error 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}{\eV}.
The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavors of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$, and SRG-qs$GW$ again but for the principal EAs of $GW$50.
The raw data are reported in Table \ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig7}.
The HF EAs are, on average, underestimated with a MAE of \SI{0.31}{\eV} and some clear outliers: \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 (MAE equals to \SI{0.16}{\eV}) but the minimum and maximum error values are not satisfactory.
The performance of the two qs$GW$ schemes are quite similar for EAs with MAEs of the order of \SI{0.1}{\eV}.
These two partially self-consistent methods reduce also the minimum errors but, interestingly, they do not decrease the maximum error compared to HF.
Note that a positive EA means that the anion state is bound and therefore the methods considered here are well-suited to describe these EAs.
On the other hand, a negative EA means that this is a resonance state.
The methods considered in this study, even the $\Delta$CCSD(T) reference, are not able to describe the physics of resonance states, therefore one should not try to give a physical interpretation to these values.
Note that a positive EA means that the anion state is bound and, therefore, the methods that we consider here are well-suited to describe these states.
On the other hand, a negative EA means that we are potentially dealing with a resonance state.
The methods considered in this study, even the $\Delta$CCSD(T) reference, are not able to describe the physics of resonance states. Therefore, one should not try to give a physical interpretation to these values.
Yet, one can still compare the $GW$ values with their $\Delta$CCSD(T) counterparts within a given basis set in these cases.
%=================================================================%
@ -937,7 +937,7 @@ Note that while the accuracy improvements are quite small, it comes with no addi
In addition, it has been shown that the SRG-qs$GW$ can be converged in a much more black-box fashion than the traditional qs$GW$ thanks to its intruder-state-free nature.
Finally, the EAs have been investigated as well.
It has been found that the performances of the two qs$GW$ flavours for the EAs of the $GW$50 set are quite similar.
It has been found that the performances of qs$GW$ and SRG-qs$GW$ are quite similar for the EAs of the $GW$50 set.
However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physics can not be accurately described by the methods considered in this study.
A test set of molecules with bound anions with an accompanying benchmark of accurate reference values would certainly be valuable to the many-body perturbation theory community.