diff --git a/Manuscript/kappa_1.pdf b/Manuscript/kappa_1.pdf new file mode 100644 index 0000000..d27de77 Binary files /dev/null and b/Manuscript/kappa_1.pdf differ diff --git a/Manuscript/ufGW-SI.tex b/Manuscript/ufGW-SI.tex index 20496a8..dc88849 100644 --- a/Manuscript/ufGW-SI.tex +++ b/Manuscript/ufGW-SI.tex @@ -59,57 +59,35 @@ \maketitle -%%%%%%%%%%%%%%%%%%%%%%%% -\section{Energy differences} -%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection{$\eta$ shift} +\begin{figure} + \includegraphics[width=0.33\linewidth]{eta_0_1} + \includegraphics[width=0.33\linewidth]{eta_1} + \includegraphics[width=0.33\linewidth]{eta_10} + \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = 0.1$ (left), $\eta = 1$ (center), and $\eta = 10$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } +\end{figure} \begin{figure} - \includegraphics[width=0.6\linewidth]{eta_0_1} - \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = 0.1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } + \includegraphics[width=0.33\linewidth]{kappa_0_1} + \includegraphics[width=0.33\linewidth]{kappa_1} + \includegraphics[width=0.33\linewidth]{kappa_10} + \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with computed with $\kappa = 0.1$ (left), $\kappa = 1$ (center), and $\kappa = 10$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } \end{figure} \begin{figure} - \includegraphics[width=0.6\linewidth]{eta_1} - \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } + \includegraphics[width=0.45\linewidth]{f2_eta_1} + \hspace{0.05\textwidth} +% \includegraphics[width=0.45\linewidth]{f2_eta_1} + \caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\eta = 1$.} \end{figure} \begin{figure} - \includegraphics[width=0.6\linewidth]{eta_10} - \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = 10$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } + \includegraphics[width=0.45\linewidth]{f2_kappa_1} + \hspace{0.05\textwidth} + \includegraphics[width=0.45\linewidth]{f2_kappa_10} + \caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = 1$ (left) and $\kappa = 10$ (right). + For $\kappa = 10$, the black and gray curves are superposed.} \end{figure} -\subsection{$\kappa$ shift} - -\begin{figure} - \includegraphics[width=0.6\linewidth]{kappa_0_1} - \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\kappa = 0.1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } -\end{figure} - -\begin{figure} - \includegraphics[width=0.6\linewidth]{kappa_10} - \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\kappa = 10$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } -\end{figure} - -%%%%%%%%%%%%%%%%%%%%%%%% -\section{\ce{F2} ground state} -%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{figure} - \includegraphics[width=0.6\linewidth]{f2_eta_1} - \caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.} -\end{figure} - -\begin{figure} - \includegraphics[width=0.6\linewidth]{f2_kappa_1} - \caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.} -\end{figure} - -\begin{figure} - \includegraphics[width=0.6\linewidth]{f2_kappa_10} - \caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.} -\end{figure} %%%%%%%%%%%%%%%%%%%%%%%% \bibliography{ufGW} diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index ea87631..9fb11c3 100644 --- a/Manuscript/ufGW.tex +++ b/Manuscript/ufGW.tex @@ -373,7 +373,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str \includegraphics[width=\linewidth]{fig4} \caption{ \label{fig:H2reg} - Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\eta = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. + Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\alert{\kappa} = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -430,7 +430,7 @@ Our investigations have shown that the following energy-dependent regularizer f_\kappa(\Delta) = \frac{1-e^{-2\Delta^2/\kappa^2}}{\Delta} \end{equation} derived from the (second-order) perturbative analysis of the similarity renormalization group (SRG) equations \cite{Wegner_1994,Glazek_1994,White_2002} by Evangelista \cite{Evangelista_2014} is particularly convenient and effective for our purposes. -Increasing $\kappa$ gradually integrates out states with denominators $\Delta$ larger than $\kappa$ while the states with $\Delta \ll \kappa$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a} +Increasing $\alert{\kappa}$ gradually integrates out states with denominators $\Delta$ larger than $\alert{\kappa}$ while the states with $\Delta \ll \alert{\kappa}$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a} Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest for various $\eta$ values.