From 15a69329c9863da7242261bb97226ce28f7f2373 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 23 Jul 2020 09:50:47 +0200 Subject: [PATCH] discussion --- BSEdyn.tex | 175 ++++++++++++++++++++++++++++------------------------- 1 file changed, 93 insertions(+), 82 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 1f090ab..747db5c 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -816,11 +816,13 @@ All the static and dynamic BSE calculations have been performed with the softwar \end{table*} \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 \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. -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. %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. @@ -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} \\ \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{$Z_{S}$} - & \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 \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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 9.08 & 9.46 & 9.67 & 9.54 & 9.43 \\ + & 9.08 & 9.46 & 9.54 & 9.67 & 9.43 \\ \hline - MAE & & & & 0.65 & 0.50 & & - & 0.41 & 0.24 & 0.14 & 0.25 & 0.00 \\ - MSE & & & & 0.65 & 0.48 & & - & 0.12 & 0.00 & 0.13 & 0.00 & 0.00 \\ - RMSE & & & & 0.71 & 0.58 & & - & 0.54 & 0.34 & 0.19 & 0.33 & 0.00 \\ - Max($+$) & & & & 1.08 & 0.91 & & - & 1.06 & 0.54 & 0.44 & 0.57 & 0.00 \\ - Max($-$) & & & & 0.20 & -0.22 & & - & -1.77 & -0.76 & -0.02 & -0.71 & 0.00 \\ + MAE & & & & 0.65 & 0.50 & & + & 0.41 & 0.24 & 0.25 & 0.14 & 0.00 \\ + MSE & & & & 0.65 & 0.48 & & + & 0.12 & 0.00 & 0.00 & 0.13 & 0.00 \\ + RMSE & & & & 0.71 & 0.58 & & + & 0.54 & 0.34 & 0.33 & 0.19 & 0.00 \\ + Max($+$) & & & & 1.08 & 0.91 & & + & 1.06 & 0.54 & 0.57 & 0.44 & 0.00 \\ + Max($-$) & & & & 0.20 & -0.22 & & + & -1.77 & -0.76 & -0.71 & -0.02 & 0.00 \\ \end{tabular} \end{ruledtabular} \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} \\ \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{$Z_{S}$} - & \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 \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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 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 - & 6.66 & 6.39 & 7.08 & 6.44 & 7.06 \\ + & 6.66 & 6.39 & 6.44 & 7.08 & 7.06 \\ \hline 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 & & - & 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 & & - & 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 & & - & 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 & & - & -0.40 & -0.67 & -0.12 & -0.62 & 0.00 \\ + & -0.40 & -0.67 & -0.62 & -0.12 & 0.00 \\ \end{tabular} \end{ruledtabular} \end{table*} \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. -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. -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. +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} (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}. 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. @@ -1027,14 +1029,16 @@ A clear general trend is the consistent red shift of the static BSE excitation e \label{fig:SiTr-SmallMol}} \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. -From this figure, it is quite clear that the dynamical correction systematically improves upon its static analog, except in a very few cases. +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 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. 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 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 %%% \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}} \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. 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}. -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. -Here again, the dynamical correction improves the accuracy of BSE by lowering the MAE and MSE to $0.23$ and $0.00$ eV, respectively. +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, 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, 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}