From 158253f3153aee77799b6fff55896fc25ad675b6 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 20 May 2020 17:17:34 +0200 Subject: [PATCH] basis effects and TDA --- BSEdyn.tex | 64 +++++++++++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 58 insertions(+), 6 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 5bba572..c3c3e7b 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -69,6 +69,7 @@ \newcommand{\GW}{GW} \newcommand{\stat}{\text{stat}} \newcommand{\dyn}{\text{dyn}} +\newcommand{\TDA}{\text{TDA}} % energies \newcommand{\Enuc}{E^\text{nuc}} @@ -586,12 +587,12 @@ From a practical point of view, if one enforces the TDA, we obtain the very simp \label{eq:Om1-TDA} \Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}. \end{equation} -This correction can be renormalized by computing, at basically no extra cost, the renormalization factor +This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the TDA, \begin{equation} \label{eq:Z} - Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}, + Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}. \end{equation} -which finally yields +This finally yields \begin{equation} \Om{m}{\text{dyn}} = \Om{m}{\text{stat}} + \Delta\Om{m}{\text{dyn}} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}. \end{equation} @@ -637,6 +638,38 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \label{sec:resdis} %%%%%%%%%%%%%%%%%%%%%%%% +%%% TABLE I %%% +\begin{table*} + \caption{ + Excitation energies (in eV) of \ce{N2} at the BSE@{\GOWO}@HF level for various basis sets. + \label{tab:N2} + } + \begin{ruledtabular} + \begin{tabular}{lddddddddd} + & \mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)} + & \mc{3}{c}{aug-cc-pVTZ ($\Eg^{\GW} = 19.20$ eV)} + & \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = $ eV)} \\ + \cline{2-4} \cline{5-7} \cline{8-10} + State & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$} + & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$} + & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$} \\ + \hline + $^1\Pi_g(n \ra \pis)$ & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.40 & & & \\ + $^1\Sigma_u^-(\pi \ra \pis)$ & 9.95 & -0.44 & -0.44 & 10.11 & -0.45 & -0.45 & & & \\ + $^1\Delta_u(\pi \ra \pis)$ & 10.57 & -0.41 & -0.40 & 10.75 & -0.42 & -0.41 & & & \\ + $^1\Sigma_g^+$(R) & 13.72 & -0.04 & -0.04 & 13.60 & -0.03 & -0.03 & & & \\ + $^1\Pi_u$(R) & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & & & \\ + $^1\Sigma_u^+$(R) & 13.80 & -0.08 & -0.08 & 13.98 & -0.07 & -0.08 & & & \\ + $^1\Pi_u$(R) & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & & & \\ + $^3\Sigma_u^+(\pi \ra \pis)$ & 9.21 & -1.01 & -0.71 & 9.50 & -1.04 & -0.73 & & & \\ + $^3\Pi_g(n \ra \pis)$ & 9.58 & -0.57 & -0.34 & 9.85 & -0.58 & -0.33 & & & \\ + $^3\Delta_u(\pi \ra \pis)$ & 9.97 & -0.92 & -0.58 & 10.19 & -0.95 & -0.36 & & & \\ + $^3\Sigma_u^-(\pi \ra \pis)$ & 10.71 & -0.81 & -0.68 & 10.89 & -0.82 & -0.30 & & & \\ + \end{tabular} + \end{ruledtabular} +\end{table*} + + %%% TABLE I %%% \begin{table*} \caption{ @@ -656,8 +689,8 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \ce{H2O} & $^1B_1(n \ra 3s)$ & 13.58 & 8.09 & 8.00 & -0.09 & 1.007 & & & & & & 7.62 & 7.18 & 7.60 & 7.23 & 7.65 \\ & $^1A_2(n \ra 3p)$ & & 9.79 & 9.72 & -0.07 & 1.005 & & & & & & 9.41 & 8.84 & 9.36 & 8.89 & 9.43 \\ & $^1A_1(n \ra 3s)$ & & 10.42 & 10.35 & -0.07 & 1.006 & & & & & & 9.99 & 9.52 & 9.96 & 9.58 & 10.00 \\ - \ce{N2} & $^1\Pi_g(n \ra \pis)$ & 19.20 & 10.11 & 9.66 & -0.45 & 1.029 & & & & & & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\ - & $^1\Sigma_u^-(\pi \ra \pis)$ & & 10.42 & 9.99 & -0.42 & 1.031 & & & & & & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\ + \ce{N2} & $^1\Pi_g(n \ra \pis)$ & 19.20 & 10.42 & 9.99 & -0.42 & 1.031 & & & & & & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\ + & $^1\Sigma_u^-(\pi \ra \pis)$ & & 10.11 & 9.66 & -0.45 & 1.029 & & & & & & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\ & $^1\Delta_u(\pi \ra \pis)$ & & 10.75 & 10.33 & -0.42 & 1.030 & & & & & & 10.85 & 10.79 & 10.44 & 10.86 & 10.29 \\ & $^1\Sigma_g^+$(R) & & 13.60 & 13.57 & -0.03 & 1.003 & & & & & & 13.67 & 12.99 & 13.15 & 12.83 & 13.01 \\ & $^1\Pi_u$(R) & & 13.98 & 13.94 & -0.04 & 1.004 & & & & & & 13.64 & 13.32 & 13.43 & 13.15 & 13.22 \\ @@ -726,6 +759,25 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \end{ruledtabular} \end{table*} +%%% TABLE III %%% +\begin{table} + \caption{ + Excitation energies (in eV) of CN3 obtained with the aug-cc-pVDZ basis set at various levels of theory. + %$\Eg^{\GW} = 13.79$ eV. + \label{tab:CN3} + } + \begin{ruledtabular} + \begin{tabular}{lcc} + & \mc{2}{c}{Excitation} \\ + Method & $^1B_2(\pi \ra \pis)$ & $^3B_2(\pi \ra \pis)$ \\ + \hline + BSE@{\GOWO}@HF & 7.66 & 6.52 \\ + dBSE(TDA)@{\GOWO}@HF & 7.51 & 6.11 \\ + FCI & 7.14 & 5.47 \\ + \end{tabular} + \end{ruledtabular} +\end{table} + %%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} \label{sec:conclusion} @@ -735,7 +787,7 @@ This is the conclusion %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ %%%%%%%%%%%%%%%%%%%%%%%% -%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support. +PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support. This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005. Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}