diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index da24648..f62ac12 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -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.