From 07ed3bd11eb925fcb8b01bb9646df960ffd2d505 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 26 Apr 2022 13:18:28 +0200 Subject: [PATCH] adding units --- Manuscript/ufGW-SI.tex | 34 +++++++++++----------------------- Manuscript/ufGW.tex | 12 ++++++------ 2 files changed, 17 insertions(+), 29 deletions(-) diff --git a/Manuscript/ufGW-SI.tex b/Manuscript/ufGW-SI.tex index c5352e2..72800c4 100644 --- a/Manuscript/ufGW-SI.tex +++ b/Manuscript/ufGW-SI.tex @@ -60,23 +60,11 @@ \maketitle \begin{figure} - \includegraphics[width=0.8\linewidth]{h2_kappa_1} - \caption{ - Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\kappa = 1$. -} -\end{figure} - -\begin{figure} - \includegraphics[width=0.8\linewidth]{h2_eta_1} - \caption{ - Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = 1$. -} -\end{figure} - -\begin{figure} - \includegraphics[width=0.8\linewidth]{h2_eta_100_meV} - \caption{ - Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = \SI{100}{\milli\eV}$. + \includegraphics[width=0.6\linewidth]{h2_eta_100_meV} + \includegraphics[width=0.6\linewidth]{h2_kappa_1} + \includegraphics[width=0.6\linewidth]{h2_eta_1} + \caption{ + Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = \SI{100}{\milli\eV}$ (top), $\eta = \SI{1}{\hartree}$ (center), and $\kappa = \SI{1}{\hartree}$ (bottom). } \end{figure} @@ -84,14 +72,14 @@ \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. } + \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and $\eta = \SI{10}{\hartree}$ (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=0.33\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. } + \caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with computed with $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and $\kappa = \SI{10}{\hartree}$ (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} @@ -100,7 +88,7 @@ \includegraphics[width=0.3\linewidth]{f2_eta_1} \hspace{0.03\textwidth} \includegraphics[width=0.3\linewidth]{f2_eta_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 $\kappa = 0.1$ (left), $\eta = 1$ (center), and $\eta = 10$ (right).} + \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 $\kappa = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and $\eta = \SI{10}{\hartree}$ (right).} \end{figure} \begin{figure} @@ -109,9 +97,9 @@ \includegraphics[width=0.3\linewidth]{f2_kappa_1} \hspace{0.03\textwidth} \includegraphics[width=0.3\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 $\kappa = 0.1$ (left), $\kappa = 1$ (center), and $\kappa = 10$ (right). - For $\kappa = 0.1$, the BSE@ev$GW$@HF calculations do not converge for numerous values of $R_{\ce{F-F}}$ and are not shown in this figure. - For $\kappa = 10$, the black and gray curves are superposed.} + \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 $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and $\kappa = \SI{10}{\hartree}$ (right). + For $\kappa = \SI{0.1}{\hartree}$, the BSE@ev$GW$@HF calculations do not converge for numerous values of $R_{\ce{F-F}}$ and are not shown in this figure. + For $\kappa = \SI{10}{\hartree}$, the black and gray curves are superposed.} \end{figure} diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index ced4cb1..bbeed88 100644 --- a/Manuscript/ufGW.tex +++ b/Manuscript/ufGW.tex @@ -374,8 +374,8 @@ 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 $\alert{\kappa} = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. - \alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.} + Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\alert{\kappa} = \SI{1}{\hartree}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. + \alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ are reported as {\SupMat}.} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -388,7 +388,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str \caption{ \label{fig:F2} Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set. - \alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.} + \alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ are reported as {\SupMat}.} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -457,14 +457,14 @@ However, it can be certainly refined for specific applications. To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized (computed at $\alert{\kappa} = \SI{1.0}{\hartree}$ with the SRG-based regularizer) and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital. The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} for the HOMO ($p = 1$) and LUMO ($p = 2$), which is practically viable. Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger (as it should) but it has the undeniable advantage to provide smooth curves. -\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.} +\alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.} As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis. In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol as detailed in Ref.~\onlinecite{Loos_2020e}. These results are compared to high-level coupled-cluster \alert{(CC)} calculations extracted from the same work: \alert{CC with singles and doubles (CCSD) \cite{Purvis_1982} and the non-perturbative third-order approximate CC method (CC3). \cite{Christiansen_1995b}} -As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\alert{\kappa} = 1$) allows to smooth it out without significantly altering the overall accuracy. +As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\alert{\kappa} = \SI{1}{\hartree}$) allows to smooth it out without significantly altering the overall accuracy. Moreover, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface (gray curve). -\alert{For the sake of completeness, a similar graph for $\kappa = 10$ is reported as {\SupMat}. +\alert{For the sake of completeness, a similar graph for $\kappa = \SI{10}{\hartree}$ is reported as {\SupMat}. Interestingly, for this rather large value of $\kappa$, the smooth BSE@{\GOWO}@HF and BSE@ev$GW$@HF curves are superposed, and of very similar quality as CCSD. It is therefore clear that a smaller value of $\kappa$ is more suitable.}