PTEROSOR/BSEdyn
10
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

discussion

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-07-23 09:50:47 +02:00
parent 8d72ac0567
commit 15a69329c9
1 changed files with 93 additions and 82 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

175
BSEdyn.tex
View File

@ -816,11 +816,13 @@ All the static and dynamic BSE calculations have been performed with the softwar
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
First, we investigate the basis set dependency of the dynamical correction. First, we investigate the basis set dependence of the dynamical correction.
The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the cc-pVXZ and aug-cc-pVXZ families of basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that corrected transitions are usually 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 cc-pVXZ and aug-cc-pVXZ families of basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that corrected transitions are usually well below this gap.
The \ce{N2} molecule is a very convenient example for this kind of study as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions. The \ce{N2} molecule is a very convenient example for this kind of study as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions.
As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions. As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions.
One key result of the present investigation is that the dynamical correction is quite basis set insensitive with a maximum variation of $0.03$ eV between in smallest (aug-cc-pVDZ) and largest (aug-cc-pVQZ) basis sets. One key result of the present investigation is that the dynamical correction is quite basis set insensitive with a maximum variation of $0.03$ eV between aug-cc-pVDZ and aug-cc-pVQZ.
It is only for the smallest basis set (cc-pVDZ) that one can observe significant differences.
We can then safely conclude that the dynamical correction converges rapidly with respect to the size of the one-electron basis set, a triple-$\zeta$ or an augmented double-$\zeta$ basis set being enough to obtain near complete basis set limit values.
This is quite a nice feature as it means that one does not need to compute the dynamical correction in a very large basis to get a meaningful estimate of its magnitude. This is quite a nice feature as it means that one does not need to compute the dynamical correction in a very large basis to get a meaningful estimate of its magnitude.
%The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies. %The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies.
%The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate. %The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate.
@ -840,88 +842,88 @@ This is quite a nice feature as it means that one does not need to compute the d
& & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\ & & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\
\cline{4-8} \cline{9-13} \cline{4-8} \cline{9-13}
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{CC2} & \tabc{CCSD} & \tabc{TBE} \\
\hline \hline
\ce{HCl} & $^1\Pi$ & CT & 13.43 & 8.30 & 8.19 & -0.11 & 1.009 \ce{HCl} & $^1\Pi$ & CT & 13.43 & 8.30 & 8.19 & -0.11 & 1.009
& 6.07 & 7.97 & 7.91 & 7.96 & 7.84 \\ & 6.07 & 7.97 & 7.96 & 7.91 & 7.84 \\
\\ \\
\ce{H2O} & $^1B_1(n \ra 3s)$ & Ryd. & 13.58 & 8.09 & 8.00 & -0.09 & 1.007 \ce{H2O} & $^1B_1(n \ra 3s)$ & Ryd. & 13.58 & 8.09 & 8.00 & -0.09 & 1.007
& 7.62 & 7.18 & 7.60 & 7.23 & 7.17 \\ & 7.62 & 7.18 & 7.23 & 7.60 & 7.17 \\
& $^1A_2(n \ra 3p)$ & Ryd. & & 9.79 & 9.72 & -0.07 & 1.005 & $^1A_2(n \ra 3p)$ & Ryd. & & 9.79 & 9.72 & -0.07 & 1.005
& 9.41 & 8.84 & 9.36 & 8.89 & 8.92 \\ & 9.41 & 8.84 & 8.89 & 9.36 & 8.92 \\
& $^1A_1(n \ra 3s)$ & Ryd. & & 10.42 & 10.35 & -0.07 & 1.006 & $^1A_1(n \ra 3s)$ & Ryd. & & 10.42 & 10.35 & -0.07 & 1.006
& 9.99 & 9.52 & 9.96 & 9.58 & 9.52 \\ & 9.99 & 9.52 & 9.58 & 9.96 & 9.52 \\
\\ \\
\ce{N2} & $^1\Pi_g(n \ra \pis)$ & Val. & 19.20 & 10.42 & 9.99 & -0.42 & 1.031 \ce{N2} & $^1\Pi_g(n \ra \pis)$ & Val. & 19.20 & 10.42 & 9.99 & -0.42 & 1.031
& 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\ & 9.66 & 9.48 & 9.44 & 9.41 & 9.34 \\
& $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.11 & 9.66 & -0.45 & 1.029 & $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.11 & 9.66 & -0.45 & 1.029
& 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\ & 10.31 & 10.26 & 10.32 & 10.00 & 9.88 \\
& $^1\Delta_u(\pi \ra \pis)$ & Val. & & 10.75 & 10.33 & -0.42 & 1.030 & $^1\Delta_u(\pi \ra \pis)$ & Val. & & 10.75 & 10.33 & -0.42 & 1.030
& 10.85 & 10.79 & 10.44 & 10.86 & 10.29 \\ & 10.85 & 10.79 & 10.86 & 10.44 & 10.29 \\
& $^1\Sigma_g^+$ & Ryd. & & 13.60 & 13.57 & -0.03 & 1.003 & $^1\Sigma_g^+$ & Ryd. & & 13.60 & 13.57 & -0.03 & 1.003
& 13.67 & 12.99 & 13.15 & 12.83 & 12.98 \\ & 13.67 & 12.99 & 12.83 & 13.15 & 12.98 \\
& $^1\Pi_u$ & Ryd. & & 13.98 & 13.94 & -0.04 & 1.004 & $^1\Pi_u$ & Ryd. & & 13.98 & 13.94 & -0.04 & 1.004
& 13.64 & 13.32 & 13.43 & 13.15 & 13.03 \\ & 13.64 & 13.32 & 13.15 & 13.43 & 13.03 \\
& $^1\Sigma_u^+$ & Ryd. & & 13.98 & 13.91 & -0.07 & 1.008 & $^1\Sigma_u^+$ & Ryd. & & 13.98 & 13.91 & -0.07 & 1.008
& 13.75 & 13.07 & 13.26 & 12.89 & 13.09 \\ & 13.75 & 13.07 & 12.89 & 13.26 & 13.09 \\
& $^1\Pi_u$ & Ryd. & & 14.24 & 14.21 & -0.03 & 1.002 & $^1\Pi_u$ & Ryd. & & 14.24 & 14.21 & -0.03 & 1.002
& 14.52 & 14.00 & 13.67 & 13.96 & 13.46 \\ & 14.52 & 14.00 & 13.96 & 13.67 & 13.46 \\
\\ \\
\ce{CO} & $^1\Pi(n \ra \pis)$ & Val. & 16.46 & 9.54 & 9.19 & -0.34 & 1.029 \ce{CO} & $^1\Pi(n \ra \pis)$ & Val. & 16.46 & 9.54 & 9.19 & -0.34 & 1.029
& 8.78 & 8.69 & 8.59 & 8.64 & 8.49 \\ & 8.78 & 8.69 & 8.64 & 8.59 & 8.49 \\
& $^1\Sigma^-(\pi \ra \pis)$ & Val. & & 10.25 & 9.90 & -0.35 & 1.023 & $^1\Sigma^-(\pi \ra \pis)$ & Val. & & 10.25 & 9.90 & -0.35 & 1.023
& 10.13 & 10.03 & 9.99 & 10.30 & 9.92 \\ & 10.13 & 10.03 & 10.30 & 9.99 & 9.92 \\
& $^1\Delta(\pi \ra \pis)$ & Val. & & 10.71 & 10.39 & -0.32 & 1.023 & $^1\Delta(\pi \ra \pis)$ & Val. & & 10.71 & 10.39 & -0.32 & 1.023
& 10.41 & 10.30 & 10.12 & 10.60 & 10.06 \\ & 10.41 & 10.30 & 10.60 & 10.12 & 10.06 \\
& $^1\Sigma^+$ & Ryd. & & 11.88 & 11.85 & -0.03 & 1.005 & $^1\Sigma^+$ & Ryd. & & 11.88 & 11.85 & -0.03 & 1.005
& 11.48 & 11.32 & 11.22 & 11.11 & 10.95 \\ & 11.48 & 11.32 & 11.11 & 11.22 & 10.95 \\
& $^1\Sigma^+$ & Ryd. & & 12.39 & 12.37 & -0.02 & 1.003 & $^1\Sigma^+$ & Ryd. & & 12.39 & 12.37 & -0.02 & 1.003
& 11.71 & 11.83 & 11.75 & 11.63 & 11.52 \\ & 11.71 & 11.83 & 11.63 & 11.75 & 11.52 \\
& $^1\Pi$ & Ryd. & & 12.37 & 12.32 & -0.05 & 1.004 & $^1\Pi$ & Ryd. & & 12.37 & 12.32 & -0.05 & 1.004
& 12.06 & 12.03 & 11.96 & 11.83 & 11.72 \\ & 12.06 & 12.03 & 11.83 & 11.96 & 11.72 \\
\\ \\
\ce{HNO} & $^1A''(n \ra \pis)$ & Val. & 11.71 & 2.46 & 1.98 & -0.48 & 1.035 \ce{HNO} & $^1A''(n \ra \pis)$ & Val. & 11.71 & 2.46 & 1.98 & -0.48 & 1.035
& 1.80 & 1.68 & 1.76 & 1.74 & 1.74 \\ & 1.80 & 1.68 & 1.74 & 1.76 & 1.74 \\
& $^1A'$ & Ryd. & & 7.05 & 7.01 & -0.04 & 1.003 & $^1A'$ & Ryd. & & 7.05 & 7.01 & -0.04 & 1.003
& 5.81 & 5.73 & 6.30 & 5.72 & 6.27 \\ & 5.81 & 5.73 & 5.72 & 6.30 & 6.27 \\
\\ \\
\ce{C2H2} & $^1\Sigma_{u}^-(\pi \ra \pis)$ & Val. & 12.28 & 7.37 & 7.05 & -0.32 & 1.026 \ce{C2H2} & $^1\Sigma_{u}^-(\pi \ra \pis)$ & Val. & 12.28 & 7.37 & 7.05 & -0.32 & 1.026
& 7.28 & 7.24 & 7.15 & 7.26 & 7.10 \\ & 7.28 & 7.24 & 7.26 & 7.15 & 7.10 \\
& $^1\Delta_{u}(\pi \ra \pis)$ & Val. & & 7.74 & 7.46 & -0.29 & 1.025 & $^1\Delta_{u}(\pi \ra \pis)$ & Val. & & 7.74 & 7.46 & -0.29 & 1.025
& 7.62 & 7.56 & 7.48 & 7.59 & 7.44\\ & 7.62 & 7.56 & 7.59 & 7.48 & 7.44 \\
\\ \\
\ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ & Ryd. & 11.49 & 7.64 & 7.62 & -0.03 & 1.004 \ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ & Ryd. & 11.49 & 7.64 & 7.62 & -0.03 & 1.004
& 7.35 & 7.34 & 7.42 & 7.29 & 7.39 \\ & 7.35 & 7.34 & 7.29 & 7.42 & 7.39 \\
& $^1B_{1u}(\pi \ra \pis)$ & Val. & & 8.18 & 8.03 & -0.15 & 1.022 & $^1B_{1u}(\pi \ra \pis)$ & Val. & & 8.18 & 8.03 & -0.15 & 1.022
& 7.95 & 7.91 & 8.02 & 7.92 & 7.93 \\ & 7.95 & 7.91 & 7.92 & 8.02 & 7.93 \\
& $^1B_{1g}(\pi \ra 3p)$ & Ryd. & & 8.29 & 8.26 & -0.03 & 1.003 & $^1B_{1g}(\pi \ra 3p)$ & Ryd. & & 8.29 & 8.26 & -0.03 & 1.003
& 8.01 & 7.99 & 8.08 & 7.95 & 8.08 \\ & 8.01 & 7.99 & 7.95 & 8.08 & 8.08 \\
\\ \\
\ce{CH2O} & $^1A_2(n \ra \pis)$ & Val. & 12.00 & 5.03 & 4.68 & -0.35 & 1.027 \ce{CH2O} & $^1A_2(n \ra \pis)$ & Val. & 12.00 & 5.03 & 4.68 & -0.35 & 1.027
& 4.04 & 3.92 & 4.01 & 4.07 & 3.98 \\ & 4.04 & 3.92 & 4.07 & 4.01 & 3.98 \\
& $^1B_2(n \ra 3s)$ & Ryd. & & 7.87 & 7.85 & -0.02 & 1.001 & $^1B_2(n \ra 3s)$ & Ryd. & & 7.87 & 7.85 & -0.02 & 1.001
& 6.64 & 6.50 & 7.23 & 6.56 & 7.23 \\ & 6.64 & 6.50 & 6.56 & 7.23 & 7.23 \\
& $^1B_2(n \ra 3p)$ & Ryd. & & 8.76 & 8.72 & -0.04 & 1.003 & $^1B_2(n \ra 3p)$ & Ryd. & & 8.76 & 8.72 & -0.04 & 1.003
& 7.56 & 7.53 & 8.12 & 7.57 & 8.13 \\ & 7.56 & 7.53 & 7.57 & 8.12 & 8.13 \\
& $^1A_1(n \ra 3p)$ & Ryd. & & 8.85 & 8.84 & -0.01 & 1.000 & $^1A_1(n \ra 3p)$ & Ryd. & & 8.85 & 8.84 & -0.01 & 1.000
& 8.16 & 7.47 & 8.21 & 7.52 & 8.23 \\ & 8.16 & 7.47 & 7.52 & 8.21 & 8.23 \\
& $^1A_2(n \ra 3p)$ & Ryd. & & 8.87 & 8.85 & -0.02 & 1.002 & $^1A_2(n \ra 3p)$ & Ryd. & & 8.87 & 8.85 & -0.02 & 1.002
& 8.04 & 7.99 & 8.65 & 8.04 & 8.67 \\ & 8.04 & 7.99 & 8.04 & 8.65 & 8.67 \\
& $^1B_1(\si \ra \pis)$ & Val. & & 10.18 & 9.77 & -0.42 & 1.032 & $^1B_1(\si \ra \pis)$ & Val. & & 10.18 & 9.77 & -0.42 & 1.032
& 9.38 & 9.17 & 9.28 & 9.32 & 9.22 \\ & 9.38 & 9.17 & 9.32 & 9.28 & 9.22 \\
& $^1A_1(\pi \ra \pis)$ & Val. & & 10.05 & 9.81 & -0.24 & 1.026 & $^1A_1(\pi \ra \pis)$ & Val. & & 10.05 & 9.81 & -0.24 & 1.026
& 9.08 & 9.46 & 9.67 & 9.54 & 9.43 \\ & 9.08 & 9.46 & 9.54 & 9.67 & 9.43 \\
\hline \hline
MAE & & & & 0.65 & 0.50 & & MAE & & & & 0.65 & 0.50 & &
& 0.41 & 0.24 & 0.14 & 0.25 & 0.00 \\ & 0.41 & 0.24 & 0.25 & 0.14 & 0.00 \\
MSE & & & & 0.65 & 0.48 & & MSE & & & & 0.65 & 0.48 & &
& 0.12 & 0.00 & 0.13 & 0.00 & 0.00 \\ & 0.12 & 0.00 & 0.00 & 0.13 & 0.00 \\
RMSE & & & & 0.71 & 0.58 & & RMSE & & & & 0.71 & 0.58 & &
& 0.54 & 0.34 & 0.19 & 0.33 & 0.00 \\ & 0.54 & 0.34 & 0.33 & 0.19 & 0.00 \\
Max($+$) & & & & 1.08 & 0.91 & & Max($+$) & & & & 1.08 & 0.91 & &
& 1.06 & 0.54 & 0.44 & 0.57 & 0.00 \\ & 1.06 & 0.54 & 0.57 & 0.44 & 0.00 \\
Max($-$) & & & & 0.20 & -0.22 & & Max($-$) & & & & 0.20 & -0.22 & &
& -1.77 & -0.76 & -0.02 & -0.71 & 0.00 \\ & -1.77 & -0.76 & -0.71 & -0.02 & 0.00 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
@ -939,79 +941,79 @@ This is quite a nice feature as it means that one does not need to compute the d
& & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\ & & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\
\cline{4-8} \cline{9-13} \cline{4-8} \cline{9-13}
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{CC2} & \tabc{CCSD} & \tabc{TBE} \\
\hline \hline
\ce{H2O} & $^3B_1(n \ra 3s)$ & Ryd. & 13.58 & 7.62 & 7.48 & -0.14 & 1.009 \ce{H2O} & $^3B_1(n \ra 3s)$ & Ryd. & 13.58 & 7.62 & 7.48 & -0.14 & 1.009
& 7.25 & 6.86 & 7.20 & 6.91 & 6.92 \\ & 7.25 & 6.86 & 6.91 & 7.20 & 6.92 \\
& $^3A_2(n \ra 3p)$ & Ryd. & & 9.61 & 9.50 & -0.11 & 1.007 & $^3A_2(n \ra 3p)$ & Ryd. & & 9.61 & 9.50 & -0.11 & 1.007
& 9.24 & 8.72 & 9.20 & 8.77 & 8.91 \\ & 9.24 & 8.72 & 8.77 & 9.20 & 8.91 \\
& $^3A_1(n \ra 3s)$ & Ryd. & & 9.80 & 9.66 & -0.14 & 1.008 & $^3A_1(n \ra 3s)$ & Ryd. & & 9.80 & 9.66 & -0.14 & 1.008
& 9.54 & 9.15 & 9.49 & 9.20 & 9.30 \\ & 9.54 & 9.15 & 9.20 & 9.49 & 9.30 \\
\\ \\
\ce{N2} & $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 19.20 & 8.02 & 7.38 & -0.64 & 1.032 \ce{N2} & $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 19.20 & 8.02 & 7.38 & -0.64 & 1.032
& 8.20 & 8.15 & 7.66 & 8.19 & 7.70 \\ & 8.20 & 8.15 & 8.19 & 7.66 & 7.70 \\
& $^3\Pi_g(n \ra \pis)$ & Val. & & 8.66 & 8.10 & -0.56 & 1.031 & $^3\Pi_g(n \ra \pis)$ & Val. & & 8.66 & 8.10 & -0.56 & 1.031
& 8.33 & 8.20 & 8.09 & 8.19 & 8.01 \\ & 8.33 & 8.20 & 8.19 & 8.09 & 8.01 \\
& $^3\Delta_u(\pi \ra \pis)$ & Val. & & 9.04 & 8.48 & -0.56 & 1.031 & $^3\Delta_u(\pi \ra \pis)$ & Val. & & 9.04 & 8.48 & -0.56 & 1.031
& 9.30 & 9.25 & 8.91 & 9.30 & 8.87 \\ & 9.30 & 9.25 & 9.30 & 8.91 & 8.87 \\
& $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.11 & 9.66 & -0.45 & 1.029 & $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.11 & 9.66 & -0.45 & 1.029
& 10.29 & 10.23 & 9.83 & 10.29 & 9.66 \\ & 10.29 & 10.23 & 10.29 & 9.83 & 9.66 \\
\\ \\
\ce{CO} & $^3\Pi(n \ra \pis)$ & Val. & 16.46 & 6.80 & 6.25 & -0.55 & 1.031 \ce{CO} & $^3\Pi(n \ra \pis)$ & Val. & 16.46 & 6.80 & 6.25 & -0.55 & 1.031
& 6.51 & 6.45 & 6.36 & 6.42 & 6.28 \\ & 6.51 & 6.45 & 6.42 & 6.36 & 6.28 \\
& $^3\Sigma^+(\pi \ra \pis)$ & Val. & & 8.56 & 8.06 & -0.50 & 1.025 & $^3\Sigma^+(\pi \ra \pis)$ & Val. & & 8.56 & 8.06 & -0.50 & 1.025
& 8.63 & 8.54 & 8.34 & 8.72 & 8.45 \\ & 8.63 & 8.54 & 8.72 & 8.34 & 8.45 \\
& $^3\Delta(\pi \ra \pis)$ & Val. & & 9.39 & 8.96 & -0.43 & 1.024 & $^3\Delta(\pi \ra \pis)$ & Val. & & 9.39 & 8.96 & -0.43 & 1.024
& 9.44 & 9.33 & 9.23 & 9.56 & 9.27 \\ & 9.44 & 9.33 & 9.56 & 9.23 & 9.27 \\
& $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.25 & 9.90 & -0.35 & 1.023 & $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & & 10.25 & 9.90 & -0.35 & 1.023
& 10.10 & 10.01 & 9.81 & 10.27 & 9.80 \\ & 10.10 & 10.01 & 10.27 & 9.81 & 9.80 \\
& $^3\Sigma_u^+$ & Ryd. & & 11.17 & 11.07 & -0.10 & 1.008 & $^3\Sigma_u^+$ & Ryd. & & 11.17 & 11.07 & -0.10 & 1.008
& 10.98 & 10.83 & 10.71 & 10.60 & 10.47 \\ & 10.98 & 10.83 & 10.60 & 10.71 & 10.47 \\
\\ \\
\ce{HNO} & $^3A''(n \ra \pis)$ & Val. & 11.71 & 1.27 & 0.67 & -0.60 & 1.036 \ce{HNO} & $^3A''(n \ra \pis)$ & Val. & 11.71 & 1.27 & 0.67 & -0.60 & 1.036
& 0.91 & 0.78 & 0.85 & 0.84 & 0.88 \\ & 0.91 & 0.78 & 0.84 & 0.85 & 0.88 \\
& $^3A'(\pi \ra \pis)$ & Val. & & 5.55 & 4.87 & -0.69 & 1.037 & $^3A'(\pi \ra \pis)$ & Val. & & 5.55 & 4.87 & -0.69 & 1.037
& 5.72 & 5.46 & 5.49 & 5.44 & 5.61 \\ & 5.72 & 5.46 & 5.44 & 5.49 & 5.61 \\
\\ \\
\ce{C2H2} & $^3\Sigma_{u}^+(\pi \ra \pis)$ & Val. & 12.28 & 5.83 & 5.32 & -0.51 & 1.031 \ce{C2H2} & $^3\Sigma_{u}^+(\pi \ra \pis)$ & Val. & 12.28 & 5.83 & 5.32 & -0.51 & 1.031
& 5.79 & 5.75 & 5.45 & 5.76 & 5.53 \\ & 5.79 & 5.75 & 5.76 & 5.45 & 5.53 \\
& $^3\Delta_{u}(\pi \ra \pis)$ & Val. & & 6.64 & 6.23 & -0.41 & 1.028 & $^3\Delta_{u}(\pi \ra \pis)$ & Val. & & 6.64 & 6.23 & -0.41 & 1.028
& 6.62 & 6.57 & 6.41 & 6.60 & 6.40 \\ & 6.62 & 6.57 & 6.60 & 6.41 & 6.40 \\
& $^3\Sigma_{u}^-(\pi \ra \pis)$ & Val. & & 7.37 & 7.05 & -0.32 & 1.026 & $^3\Sigma_{u}^-(\pi \ra \pis)$ & Val. & & 7.37 & 7.05 & -0.32 & 1.026
& 7.31 & 7.27 & 7.12 & 7.29 & 7.08 \\ & 7.31 & 7.27 & 7.29 & 7.12 & 7.08 \\
\\ \\
\ce{C2H4} & $^3B_{1u}(\pi \ra \pis)$ & Val. & 11.49 & 4.95 & 4.49 & -0.46 & 1.032 \ce{C2H4} & $^3B_{1u}(\pi \ra \pis)$ & Val. & 11.49 & 4.95 & 4.49 & -0.46 & 1.032
& 4.62 & 4.59 & 4.46 & 4.59 & 4.54 \\ & 4.62 & 4.59 & 4.59 & 4.46 & 4.54 \\
& $^3B_{3u}(\pi \ra 3s)$ & Ryd. & & 7.46 & 7.42 & -0.04 & 1.004 & $^3B_{3u}(\pi \ra 3s)$ & Ryd. & & 7.46 & 7.42 & -0.04 & 1.004
& 7.26 & 7.23 & 7.29 & 7.19 & 7.23 \\ & 7.26 & 7.23 & 7.19 & 7.29 & 7.23 \\
& $^3B_{1g}(\pi \ra 3p)$ & Ryd. & & 8.23 & 8.19 & -0.04 & 1.004 & $^3B_{1g}(\pi \ra 3p)$ & Ryd. & & 8.23 & 8.19 & -0.04 & 1.004
& 7.97 & 7.95 & 8.03 & 7.91 & 7.98 \\ & 7.97 & 7.95 & 7.91 & 8.03 & 7.98 \\
\\ \\
\ce{CH2O} & $^3A_2(n \ra \pis)$ & Val. & 12.00 & 4.28 & 3.87 & -0.40 & 1.027 \ce{CH2O} & $^3A_2(n \ra \pis)$ & Val. & 12.00 & 4.28 & 3.87 & -0.40 & 1.027
& 3.58 & 3.46 & 3.56 & 3.59 & 3.58 \\ & 3.58 & 3.46 & 3.59 & 3.56 & 3.58 \\
& $^3A_1(\pi \ra \pis)$ & Val. & & 6.31 & 5.75 & -0.56 & 1.033 & $^3A_1(\pi \ra \pis)$ & Val. & & 6.31 & 5.75 & -0.56 & 1.033
& 6.27 & 6.20 & 5.97 & 6.30 & 6.06 \\ & 6.27 & 6.20 & 6.30 & 5.97 & 6.06 \\
& $^3B_2(n \ra 3s)$ & Ryd. & & 7.60 & 7.56 & -0.05 & 1.002 & $^3B_2(n \ra 3s)$ & Ryd. & & 7.60 & 7.56 & -0.05 & 1.002
& 6.66 & 6.39 & 7.08 & 6.44 & 7.06 \\ & 6.66 & 6.39 & 6.44 & 7.08 & 7.06 \\
\hline \hline
MAE & & & & 0.39 & 0.29 & & MAE & & & & 0.39 & 0.29 & &
& 0.25 & 0.21 & 0.09 & 0.22 & 0.00 \\ & 0.25 & 0.21 & 0.22 & 0.09 & 0.00 \\
MSE & & & & 0.39 & 0.01 & & MSE & & & & 0.39 & 0.01 & &
& 0.21 & 0.08 & 0.04 & 0.12 & 0.00 \\ & 0.21 & 0.08 & 0.12 & 0.04 & 0.00 \\
RMSE & & & & 0.44 & 0.35 & & RMSE & & & & 0.44 & 0.35 & &
& 0.30 & 0.27 & 0.13 & 0.29 & 0.00 \\ & 0.30 & 0.27 & 0.29 & 0.13 & 0.00 \\
Max($+$) & & & & 0.70 & 0.60 & & Max($+$) & & & & 0.70 & 0.60 & &
& 0.63 & 0.57 & 0.29 & 0.63 & 0.00 \\ & 0.63 & 0.57 & 0.63 & 0.29 & 0.00 \\
Max($-$) & & & & -0.06 & -0.74 & & Max($-$) & & & & -0.06 & -0.74 & &
& -0.40 & -0.67 & -0.12 & -0.62 & 0.00 \\ & -0.40 & -0.67 & -0.62 & -0.12 & 0.00 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set. Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported. For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CC2, and CCSD] are also reported.
The highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference, and statistical quantities [MAE, MSE, RMSE, Max($+$), and Max($-$)] are computed with respect to these references. The highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} (computed in the same basis) will serve us as reference, and statistical quantities [MAE, MSE, RMSE, Max($+$), and Max($-$)] are defined with respect to these references.
For each excitation, we report the static and dynamic excitation energies, $\Om{S}{\stat}$ and $\Om{S}{\dyn}$, as well as the value of the renormalization factor $Z_S$ defined in Eq.~\eqref{eq:Z}. For each excitation, we report the static and dynamic excitation energies, $\Om{S}{\stat}$ and $\Om{S}{\dyn}$, as well as the value of the renormalization factor $Z_S$ defined in Eq.~\eqref{eq:Z}.
As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result. As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
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.
@ -1027,14 +1029,16 @@ A clear general trend is the consistent red shift of the static BSE excitation e
\label{fig:SiTr-SmallMol}} \label{fig:SiTr-SmallMol}}
\end{figure*} \end{figure*}
The results gathered in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} are depicted in Fig.~\ref{fig:SiTr-SmallMol} where we report the error (with respect to the TBEs) for the singlet and triplet excitation energies computed within the static and dynamic BSE formalism. The results gathered in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} are depicted in Fig.~\ref{fig:SiTr-SmallMol}, where we report the error (with respect to the TBEs) for the singlet and triplet excitation energies computed within the static and dynamic BSE formalism.
From this figure, it is quite clear that the dynamical correction systematically improves upon its static analog, except in a very few cases. From this figure, it is quite clear that the dynamically-corrected excitation energies are systematically improved upon their static analogs, especially for singlet states.
(In the case of triplets, one would notice a few cases where the excitation energies is underestimated.)
In particular, the MAE is reduced from $0.65$ to $0.50$ eV for singlets, and from $0.39$ to $0.29$ eV for triplets. In particular, the MAE is reduced from $0.65$ to $0.50$ eV for singlets, and from $0.39$ to $0.29$ eV for triplets.
The MSE and RMSE are also systematically improved when one takes into account dynamical effects. The MSE and RMSE are also systematically improved when one takes into account dynamical effects.
The second important observation extracted from Fig.~\ref{fig:SiTr-SmallMol} is that the (singlet and triplet) Rydberg states are rather unaltered by the dynamical effects with a correction of few hundredths of eV in most cases. The second important observation extracted from Fig.~\ref{fig:SiTr-SmallMol} is that the (singlet and triplet) Rydberg states are rather unaltered by the dynamical effects with a correction of few hundredths of eV in most cases.
The magnitude of the dynamical correction for $n \ra \pis$ and $\pi \ra \pis$ transitions is much more important: $0.3$--$0.5$ eV for singlets and $0.3$--$0.7$ eV for triplets. The magnitude of the dynamical correction for $n \ra \pis$ and $\pi \ra \pis$ transitions is much more important: $0.3$--$0.5$ eV for singlets and $0.3$--$0.7$ eV for triplets.
\titou{Comparison with second-order methods comes here.} Dynamical BSE does not quite reach the accuracy of second-order methods [CIS(D), ADC(2), CC2, and CCSD] for the singlet and triplet optical excitations of these small molecules.
However, it is definitely an improvement in terms of performances as compared to static BSE, especially for triplet states, where dynamical BSE reaches an accuracy close to CIS(D), ADC(2), and CC2.
%%% TABLE III %%% %%% TABLE III %%%
\begin{squeezetable} \begin{squeezetable}
@ -1098,12 +1102,19 @@ The magnitude of the dynamical correction for $n \ra \pis$ and $\pi \ra \pis$ tr
\label{fig:SiTr-BigMol}} \label{fig:SiTr-BigMol}}
\end{figure*} \end{figure*}
Table \ref{tab:BigMol} reports singlet and triplet excitation energies for larger molecules at the static and dynamic BSE levels with the aug-cc-pVDZ basis set. Table \ref{tab:BigMol} reports singlet and triplet excitation energies for larger molecules (acrolein \ce{H2C=CH-CH=O}, butadiene \ce{H2C=CH-CH=CH2}, diacetylene \ce{HC#C-C#CH}, glyoxal \ce{O=CH-CH=O}, and streptocyanine-C1 \ce{H2N-CH=NH2+}) at the static and dynamic BSE levels with the aug-cc-pVDZ basis set.
We also report the CC3 excitation energies computed in Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} with the same basis set. We also report the CC3 excitation energies computed in Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} with the same basis set.
These will be our reference as they are known to be extremely accurate ($0.03$--$0.04$ eV from the TBEs). \cite{Loos_2020g} These will be our reference as they are known to be extremely accurate ($0.03$--$0.04$ eV from the TBEs). \cite{Loos_2020g}
Error (in eV) with respect to CC3 are represented in Fig.~\ref{fig:SiTr-BigMol}. Errors associated with these excitation energies (with respect to CC3) are represented in Fig.~\ref{fig:SiTr-BigMol}.
As expected the static BSE excitation energies are much more accurate for these larger molecules with a MAE of $0.32$ eV and a MSE of $0.30$ eV. As expected the static BSE excitation energies are much more accurate for these larger molecules with a MAE of $0.32$ eV, a MSE of $0.30$ eV, and a RMSE of $0.38$ eV.
Here again, the dynamical correction improves the accuracy of BSE by lowering the MAE and MSE to $0.23$ and $0.00$ eV, respectively. Here again, the dynamical correction improves the accuracy of BSE by lowering the MAE, MSE, and RMSE to $0.23$, $0.00$, and $0.29$ eV, respectively.
For these larger systems, Rydberg states are again very slightly affected by dynamical effects, while the dynamical corrections associated with the $n \ra \pis$ and $\pi \ra \pis$ transitions are much larger and of the same magnitude ($0.3$--$0.6$ eV) for both types of transitions.
This latter observation is quite different from the outcomes reached by Rohlfing and coworkers in previous works \cite{Ma_2009a,Ma_2009b} (see Sec.~\ref{sec:intro}) where they observed i) smaller corrections (maybe due to the plasmon-pole approximation), and ii) that $n \ra \pis$ transitions are more affected by the dynamical screening than $\pi \ra \pis$ transitions.
As a final comment, let us discuss the two singlet states of butadiene reported in Table \ref{tab:BigMol}.\cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
As discussed in Sec.~\ref{sec:intro}, these corresponds to a bright $^1B_u$ state with a clear single-excitation character, and a dark $^1A_g$ state including a substantial fraction of double excitation character (roughly $30\%$).
Although they are both of $\pi \ra \pis$ nature, they are very slightly altered by dynamical screening with corrections of $-0.12$ and $-0.03$ eV for the $^1B_u$ and $^1A_g$ states, respectively.
The small correction on the $^1A_g$ state might be explained by its rather diffuse nature (similar to a Rydberg states). \cite{Boggio-Pasqua_2004}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}