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BSEdyn.tex
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@ -733,6 +733,28 @@ All the static and dynamic BSE calculations have been performed with the softwar
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llddddddddd} \begin{tabular}{llddddddddd}
& & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
& \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
& \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
\cline{3-5} \cline{6-8} \cline{9-11}
State & Nature & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
\hline
$^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
$^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
$^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
$^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
$^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
\\
$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
$^3\Pi_g(n \ra \pis)$ & Val. & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
$^3\Delta_u(\pi \ra \pis)$ & Val. & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
\hline
\\
& & \mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)} & & \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-pVTZ ($\Eg^{\GW} = 19.20$ eV)}
& \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = 19.00$ eV)} \\ & \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = 19.00$ eV)} \\
@ -755,44 +777,10 @@ All the static and dynamic BSE calculations have been performed with the softwar
$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 10.71 & -0.81 & -0.68 & 10.89 & -0.82 & -0.30 & 11.00 & -0.83 & -0.53 \\ $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 10.71 & -0.81 & -0.68 & 10.89 & -0.82 & -0.30 & 11.00 & -0.83 & -0.53 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Excitation energy larger than the fundamental gap.}
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
%%%% TABLE I %%%
%\begin{squeezetable}
%\begin{table*}
% \caption{
% Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
% \label{tab:N2}
% }
% \begin{ruledtabular}
% \begin{tabular}{lddddddddd}
% & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
% & \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
% & \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
% \cline{2-4} \cline{5-7} \cline{8-10}
% State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
% \hline
% $^1\Pi_g(n \ra \pis)$ & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
% $^1\Sigma_u^-(\pi \ra \pis)$ & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
% $^1\Delta_u(\pi \ra \pis)$ & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
% $^1\Sigma_g^+$(R) & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
% $^1\Pi_u$(R) & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
% $^1\Sigma_u^+$(R) & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
% $^1\Pi_u$(R) & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
% \\
% $^3\Sigma_u^+(\pi \ra \pis)$ & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
% $^3\Pi_g(n \ra \pis)$ & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
% $^3\Delta_u(\pi \ra \pis)$ & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
% $^3\Sigma_u^-(\pi \ra \pis)$ & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
% \end{tabular}
% \end{ruledtabular}
% \fnt[1]{Excitation energy larger than the fundamental gap.}
%\end{table*}
%\end{squeezetable}
First, we investigate the basis set dependency of the dynamical correction as well as the validity of the dTDA (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian). First, we investigate the basis set dependency of the dynamical correction as well as the validity of the dTDA (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian).
Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA. Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that each corrected transition is well below this gap. The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that each corrected transition is well below this gap.
@ -905,7 +893,7 @@ In accordance with the success of the dTDA, the remaining calculations of the pr
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llldddddddddd} \begin{tabular}{llldddddddddd}
& & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\%& \mc{5}{c}{Density-based methods} \\ & & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\%& \mc{5}{c}{Density-based methods} \\
\cline{4-8} \cline{8-13} %\cline{13-17} \cline{4-8} \cline{9-13} %\cline{13-17}
Mol. & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} Mol. & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$}
& \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{TBE} \\ & \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{TBE} \\
% & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\ % & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\
@ -974,8 +962,8 @@ As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of
Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq.~\eqref{eq:Om1}] yields basically the same results as its (cheaper) renormalized version. Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq.~\eqref{eq:Om1}] yields basically the same results as its (cheaper) renormalized version.
%%% TABLE I %%% %%% TABLE I %%%
%\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table}
\caption{ \caption{
Singlet excitation energies (in eV) for various molecules obtained with the aug-cc-pVDZ basis set computed at various levels of theory. Singlet excitation energies (in eV) for various molecules obtained with the aug-cc-pVDZ basis set computed at various levels of theory.
The dynamical correction is computed in the dTDA. The dynamical correction is computed in the dTDA.
@ -985,52 +973,29 @@ Moreover, we have observed that an iterative, self-consistent resolution [where
\begin{tabular}{llldddddd} \begin{tabular}{llldddddd}
& & & \mc{5}{c}{BSE@{\GOWO}@HF} \\ & & & \mc{5}{c}{BSE@{\GOWO}@HF} \\
\cline{4-8} \cline{4-8}
Mol. & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} & \tabc{TBE} \\ Molecule & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} & \tabc{CC3} \\
\hline \hline
acrolein & $^1A''(n \ra \pis)$ & Val. & \\ acrolein & $^1A''(n \ra \pis)$ & Val. & 11.67 & 4.62 & 4.28 & -0.35 & 1.030 & 3.77 \\
& $^1A'(n \ra \pis)$ & Val. & & 6.86 & 6.70 & -0.16 & 1.023 & 6.67 \\
& $^1A''(n \ra \pis)$ & Val. & & 7.85 & 7.71 & -0.14 & 1.012 & 6.75 \\
& $^1A'(n \ra 3s)$ & Ryd. & & 7.57 & 7.53 & -0.04 & 1.004 & 6.99 \\
\\ \\
butadiene & $^1B_u(\pi \ra \pis)$ & Val. & 9.88 & 6.25 & 6.13 & -0.12 & 1.019 \\ butadiene & $^1B_u(\pi \ra \pis)$ & Val. & 9.88 & 6.25 & 6.13 & -0.12 & 1.019 & 6.25 \\
& $^1A_g(\pi \ra \pis)$ & Val. & & 6.88 & 6.86 & -0.03 & 1.003 \\ & $^1A_g(\pi \ra \pis)$ & Val. & & 6.88 & 6.86 & -0.03 & 1.003 & 6.68 \\
& $^3B_u(\pi \ra \pis)$ & Val. & & 5.09 & 4.61 & -0.48 & 1.054 \\
\\ \\
diacetylene & $^1\Sigma_u^-(\pi \ra \pis)$ & Val. \\ diacetylene & $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 11.01 & 5.62 & 5.35 & -0.28 & 1.025 & 5.44 \\
& $^1\Delta_u(\pi \ra \pis)$ & Val. \\ & $^1\Delta_u(\pi \ra \pis)$ & Val. & & 5.87 & 5.63 & -0.25 & 1.024 & 5.69 \\
& $^3\Sigma_u^+(\pi \ra \pis)$ & Val. \\
& $^3\Delta_u(\pi \ra \pis)$ & Val. \\
\\ \\
glyoxal & $^1A_u(n \ra \pis)$ & Val. & 10.90 & 3.46 & 3.14 & -0.33 & 1.028 \\ glyoxal & $^1A_u(n \ra \pis)$ & Val. & 10.90 & 3.46 & 3.14 & -0.33 & 1.028 & 2.90 \\
& $^1B_g(n \ra \pis)$ & Val. & & 4.96 & 4.55 & -0.41 & 1.034 \\ & $^1B_g(n \ra \pis)$ & Val. & & 4.96 & 4.55 & -0.41 & 1.034 & 4.30 \\
& $^1B_g(n \ra \pis)$ & Val. & & & & & \\ & $^1B_u(n \ra 3p)$ & Ryd. & & 7.90 & 7.86 & -0.04 & 1.004 & 7.55 \\
& $^1B_u(n \ra 3p)$ & Ryd. & & & & & \\
& $^3A_u(n \ra \pis)$ & Val. & & 3.94 & 3.57 & -0.37 & 1.045 \\
& $^3B_g(n \ra \pis)$ & Val. & & 5.70 & 5.30 & -0.40 & 1.051 \\
& $^3B_u(\pi \ra \pis)$ & Val. & & 6.69 & 6.07 & -0.62 & 1.057 \\
\\ \\
streptocyanine & $^1B_2(\pi \ra \pis)$ & Val. & 7.66 & 7.51 & -0.15 & 1.019 & 7.13 \\ streptocyanine & $^1B_2(\pi \ra \pis)$ & Val. & 13.79 & 7.66 & 7.51 & -0.15 & 1.019 & 7.14 \\
& $^3B_2(\pi \ra \pis)$ & Val. & 6.52 & 6.11 & -0.41 & 1.042 & 5.52 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table}
%\end{squeezetable} \end{squeezetable}
%%% 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} \section{Conclusion}

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