diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 00c3050..c021952 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -652,7 +652,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$ The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively. In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme. However, in order to perform black-box comparisons, these parameters have been fixed to these default values. -The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set. +The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations \titou{where we have not linearized the quasiparticle equation} while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set. The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set. The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively. @@ -737,7 +737,7 @@ However, SRG-qs$GW$ does not suffer from the same problem and improves slightly Finally, we also consider the evolution with respect to $s$ of the principal EA of \ce{F2}, as displayed in Fig.~\ref{fig:fig4}. The HF value is largely underestimating the $\Delta$CCSD(T) reference. Performing a qs$GW$ calculation on top of it brings a quantitative improvement by reducing the error from \SI{-2.03}{\eV} to \SI{-0.24}{\eV}. -The SRG-qs$GW$ EA error is monotonically increasing from the HF value at $s=0$ to an error really close to the qs$GW$ one at $s\to\infty$. +The SRG-qs$GW$ EA (absolute) error is monotonically decreasing from the HF value at $s=0$ to an error close to the qs$GW$ one at $s\to\infty$. % Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations due to the presence of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021} % The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values. @@ -770,17 +770,17 @@ Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in th 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}. -Self-consistency can mitigate the error of the outliers as the maximum absolute error 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} for qs$GW$. +\titou{For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}.} +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}. 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. Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative 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 implemented really easily 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}. -The decrease of the MSE and SDE correspond to a \ant{shift of the maximum} toward zero and a shrink of the distribution width, respectively. +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}. +The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively. \begin{table*} \caption{Principal IP and EA (in eV) of the $GW$50 test set calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$, and SRG-qs$GW$. @@ -869,11 +869,11 @@ Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without a 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 $\eta=1/2s^2$. +This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and \titou{$\eta=1/2s^2$}. The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises 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 $\num{e3}$ seems reasonable. +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 seems to be more erratic. +On the other hand, the qs$GW$ convergence behavior is more erratic. At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be achieved for 13 molecules while 2 molecules were already problematic at $\eta=\num{5e-2}$ ($s=200$). But these convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau before these problem arise (see the orange curve in Fig.~\ref{fig:fig6}). 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}. @@ -893,7 +893,7 @@ 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 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:fig7}. +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 underestimated 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 (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}{\electronvolt}.