diff --git a/Manuscript/si.tex b/Manuscript/si.tex index 717863f..bf65058 100644 --- a/Manuscript/si.tex +++ b/Manuscript/si.tex @@ -90,6 +90,11 @@ \newcommand{\rt}{\rho} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} + % Phantom dash \newcommand{\hd}{\hphantom{-}} @@ -134,5 +139,101 @@ \label{fig:supporting}} \end{figure*} %%% %%% %%% %%% +%%% FIG 2 %%% +\begin{figure*} + \includegraphics[width=\linewidth]{supporting2} + \caption{ + Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal EA of the $GW$50 test set calculated using $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$. + All calculations are performed with the aug-cc-pVTZ basis. + \label{fig:supporting}} +\end{figure*} +%%% %%% %%% %%% + +In this section, the values of the two others SRG-based methods derived in the main manuscript are reported along with their corresponding histogram plot of the errors . +For the sake of completeness, the SRG-regularized quasi-particle equation used for the $G_0W_0$ and ev$GW$ calculations is reported here +\begin{equation} + \epsilon_p^{\text{HF}} + \sum_{i\nu} \frac{W_{pi}^{\nu} W_{pi}^{\nu} }{\omega - \epsilon_i + \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_i + \Omega_{\nu})^2 s} + \sum_{a\nu} \frac{W_{pa}^{\nu} W_{pa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_a - \Omega_{\nu})^2 s} - \omega = 0, +\end{equation} +Therefore, the SRG-$G_0W_0$ values correspond to one-shot solutions of these equations (without linearization). +While the SRG-ev$GW$ results correspond to iterative solutions of these equations where self-consistency on the $\epsilon_p$ has been reached. + +Table \ref{tab:tab1} shows that the SRG-$G_0W_0$ values are the same as the $G_0W_0$ ones for all systems (up to $\num{e-2}$\si{\electronvolt}). +Figure \ref{fig:supporting} shows that ev$GW$ offers a slight improvement over $G_0W_0$ while SRG-ev$GW$ performs similarly to its traditional ev$GW$ counterpart. +One interesting fact to note is that the convergence of SRG-ev$GW$ deteriorates faster than for SRG-qs$GW$. +This is probably due to the absence of the off-diagonal terms. + +%%% TABLE I %%% +\begin{table*} + \caption{Principal IP and EA (in eV) of the $GW$50 test set calculated using $\Delta$CCSD(T) (reference), $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$. + The statistical descriptors associated with the errors with respect to the reference values are also reported. + All calculations are performed with the aug-cc-pVTZ basis.} + \label{tab:tab1} + \begin{ruledtabular} + \begin{tabular}{ldddddddddd} + & \mc{5}{c}{Principal IP} & \mc{5}{c}{Principal EA} \\ + \cline{2-6} \cline{7-11} + & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} \\ + Mol. & \mcc{(Ref.)} & \mcc{($\eta=\num{e-3}$)} & \mcc{($s=\num{e3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} & \mcc{(Ref.)} & & \mcc{($\eta=\num{e-3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} \\ + \hline + \ce{He} & 24.54 & 24.59 & 24.59 & 24.58 & 24.57 & -2.66 & -2.66 & -2.66 & -2.66 & -2.66 \\ + \ce{Ne} & 21.47 & 21.46 & 21.46 & 21.30 & 21.29 & -5.09 & -5.25 & -5.25 & -5.24 & -5.24\\ + \ce{H2} & 16.40 & 16.49 & 16.49 & 16.52 & 16.51 & -1.35 & -1.28 & -1.28 & -1.28 & -1.28 \\ + \ce{Li2} & 5.25 & 5.38 & 5.38 & 5.44 & 5.42 & 0.34 & 0.17 & 0.17 & 0.16 & 0.17 \\ + \ce{LiH} & 8.02 & 8.22 & 8.22 & 8.26 & 8.23 & -0.29 & 0.27 & 0.27 & 0.27 & 0.27 \\ + \ce{HF} & 16.15 & 16.25 & 16.25 & 16.10 & 16.09 & -0.66 & -0.71 & -0.71 & -0.71 & -0.71 \\ + \ce{Ar} & 15.60 & 15.72 & 15.72 & 15.67 & 15.66 & -2.55 & -2.68 & -2.68 & -2.67 & -2.67 \\ + \ce{H2O} & 12.69 & 12.90 & 12.90 & 12.80 & 12.79 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\ + \ce{LiF} & 11.47 & 11.40 & 11.40 & 11.20 & 11.18 & 0.35 & 0.33 & 0.33 & 0.33 & 0.33 \\ + \ce{HCl} & 12.67 & 12.78 & 12.78 & 12.76 & 12.75 & -0.57 & -0.64 & -0.64 & -0.64 & -0.64 \\ + \ce{BeO} & 9.95 & 9.74 & 9.74 & 9.64 & 9.61 & 2.17 & 2.28 & 2.28 & 2.30 & 2.31 \\ + \ce{CO} & 13.99 & 14.80 & 14.80 & 14.77 & 14.76 & -1.57 & -1.66 & -1.66 & -1.65 & -1.65 \\ + \ce{N2} & 15.54 & 17.10 & 17.10 & 17.10 & 17.09 & -2.37 & -2.10 & -2.10 & -2.10 & -2.10 \\ + \ce{CH4} & 14.39 & 14.76 & 14.76 & 14.76 & 14.75 & -0.65 & -0.70 & -0.70 & -0.69 & -0.69 \\ + \ce{BH3} & 13.31 & 13.68 & 13.68 & 13.70 & 13.69 & -0.09 & -0.46 & -0.46 & -0.46 & -0.45 \\ + \ce{NH3} & 10.91 & 11.22 & 11.22 & 11.19 & 11.17 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\ + \ce{BF} & 11.15 & 11.34 & 11.34 & 11.37 & 11.36 & -0.80 & -0.90 & -0.90 & -0.90 & -0.90 \\ + \ce{BN} & 12.05 & 11.76 & 11.76 & 11.78 & 11.76 & 3.02 & 3.90 & 3.90 & 3.95 & 3.95 \\ + \ce{SH2} & 10.39 & 10.51 & 10.51 & 10.51 & 10.50 & -0.52 & -0.60 & -0.60 & -0.60 & -0.60 \\ + \ce{F2} & 15.81 & 16.35 & 16.35 & 16.15 & 16.14 & 0.32 & -0.53 & -0.53 & -0.47 & -0.47 \\ + \ce{MgO} & 7.97 & 8.40 & 8.40 & 8.34 & 8.28 & 1.54 & 1.64 & 1.64 & 1.65 & 1.66 \\ + \ce{O3} & 12.85 & 13.56 & 13.56 & 13.53 & 13.51 & 1.82 & 2.19 & 2.19 & 2.25 & 2.25 \\ + \ce{C2H2} & 11.45 & 11.57 & 11.57 & 11.60 & 11.59 & -0.80 & -0.71 & -0.71 & -0.71 & -0.71 \\ + \ce{HCN} & 13.76 & 13.86 & 13.86 & 13.87 & 13.86 & -0.53 & -0.52 & -0.52 & -0.52 & -0.52 \\ + \ce{B2H6} & 12.27 & 12.81 & 12.81 & 12.81 & 12.80 & -0.52 & -0.56 & -0.56 & -0.56 & -0.56 \\ + \ce{CH2O} & 10.93 & 11.39 & 11.39 & 11.34 & 11.32 & -0.60 & -0.61 & -0.61 & -0.60 & -0.60 \\ + \ce{C2H4} & 10.69 & 10.74 & 10.74 & 10.78 & 10.77 & -1.90 & -0.75 & -0.75 & -0.74 & -0.74 \\ + \ce{SiH4} & 12.79 & 13.22 & 13.22 & 13.23 & 13.22 & -0.53 & -0.59 & -0.59 & -0.59 & -0.59 \\ + \ce{PH3} & 10.60 & 10.79 & 10.79 & 10.82 & 10.81 & -0.51 & -0.58 & -0.58 & -0.58 & -0.58 \\ + \ce{CH4O} & 11.09 & 11.55 & 11.55 & 11.48 & 11.47 & -0.59 & -0.64 & -0.64 & -0.64 & -0.64 \\ + \ce{H2NNH2} & 9.49 & 9.84 & 9.84 & 9.80 & 9.79 & -0.60 & -0.69 & -0.69 & -0.68 & -0.68 \\ + \ce{HOOH} & 11.51 & 11.96 & 11.96 & 11.85 & 11.83 & -0.96 & -0.75 & -0.75 & -0.75 & -0.75 \\ + \ce{KH} & 6.32 & 6.44 & 6.44 & 6.48 & 6.42 & 0.30 & 0.28 & 0.28 & 0.28 & 0.28 \\ + \ce{Na2} & 4.93 & 4.98 & 4.98 & 5.03 & 5.02 & 0.36 & 0.26 & 0.26 & 0.24 & 0.26 \\ + \ce{HN3} & 10.77 & 11.12 & 11.12 & 11.11 & 11.10 & -0.51 & -0.6 & -0.6 & -0.59 & -0.59 \\ + \ce{CO2} & 13.80 & 14.24 & 14.24 & 14.16 & 14.15 & -0.88 & -0.98 & -0.98 & -0.97 & -0.97 \\ + \ce{PN} & 11.90 & 12.33 & 12.33 & 12.34 & 12.33 & -0.02 & -0.03 & -0.03 & 0.01 & 0.01 \\ + \ce{CH2O2} & 11.54 & 12.00 & 12.00 & 11.90 & 11.88 & -0.63 & -0.69 & -0.69 & -0.68 & -0.68 \\ + \ce{C4} & 11.43 & 11.77 & 11.77 & 11.77 & 11.76 & 2.38 & 2.24 & 2.24 & 2.34 & 2.35 \\ + \ce{C3H6} & 10.83 & 11.20 & 11.20 & 11.20 & 11.19 & -0.94 & -0.75 & -0.75 & -0.75 & -0.75 \\ + \ce{C2H3F} & 10.63 & 10.84 & 10.84 & 10.85 & 10.84 & -0.65 & -0.69 & -0.69 & -0.68 & -0.68 \\ + \ce{C2H4O} & 10.29 & 10.84 & 10.84 & 10.76 & 10.75 & -0.54 & -0.56 & -0.56 & -0.56 & -0.56 \\ + \ce{C2H6O} & 10.82 & 11.37 & 11.37 & 11.31 & 11.30 & -0.58 & -0.65 & -0.65 & -0.64 & -0.64 \\ + \ce{C3H8} & 12.13 & 12.61 & 12.61 & 12.60 & 12.59 & -0.63 & -0.70 & -0.70 & -0.70 & -0.70 \\ + \ce{NaCl} & 9.10 & 9.20 & 9.20 & 9.16 & 9.13 & 0.67 & 0.64 & 0.64 & 0.64 & 0.64 \\ + \ce{P2} & 10.72 & 10.49 & 10.49 & 10.52 & 10.51 & 0.43 & 0.47 & 0.47 & 0.53 & 0.54 \\ + \ce{MgF2} & 13.93 & 13.94 & 13.94 & 13.74 & 13.72 & 0.29 & 0.15 & 0.15 & 0.16 & 0.16 \\ + \ce{OCS} & 11.23 & 11.52 & 11.52 & 11.50 & 11.49 & -1.43 & -1.03 & -1.03 & -1.02 & -1.01 \\ + \ce{SO2} & 10.48 & 11.38 & 11.38 & 11.34 & 11.33 & 2.24 & 2.82 & 2.82 & 2.87 & 2.88 \\ + \ce{C2H3Cl} & 10.17 & 10.39 & 10.39 & 10.39 & 10.38 & -0.61 & -0.66 & -0.66 & -0.65 & -0.65 \\ + \hline + MSE & & 0.29 & 0.29 & 0.26 & 0.25 & & 0.02 & 0.02 & 0.03 & 0.00 \\ + MAE & & 0.33 & 0.33 & 0.32 & 0.31 & & 0.16 & 0.16 & 0.16 & 0.17 \\ + RMSE & & 0.43 & 0.43 & 0.41 & 0.40 & & 0.28 & 0.28 & 0.29 & 0.28 \\ + SDE & & 0.31 & 0.31 & 0.31 & 0.32 & & 0.29 & 0.29 & 0.29 & 0.29 \\ + Min & & -0.29 & -0.29 & -0.31 & -0.34 & & -0.85 & -0.85 & -0.79 & -0.82 \\ + Max & & 1.56 & 1.56 & 1.56 & 1.55 & & 1.15 & 1.15 & 1.16 & 1.14 \\ + \end{tabular} + \end{ruledtabular} +\end{table*} \end{document} diff --git a/Manuscript/supporting2.pdf b/Manuscript/supporting2.pdf new file mode 100644 index 0000000..8339580 Binary files /dev/null and b/Manuscript/supporting2.pdf differ