new results for HeH+ and H2
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dynker.tex
191
dynker.tex
@ -17,12 +17,11 @@
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\title{Dynamical Kernels for Optical Excitations}
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\author{Juliette \surname{Authier}}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Juliette \surname{Authier}}
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\affiliation{\LCPQ}
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%\author{Friends}
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\begin{abstract}
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We discuss the physical properties and accuracy of three distinct dynamical (\ie, frequency-dependent) kernels for the computation of optical excitations within linear response theory:
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@ -307,19 +306,20 @@ Because, there is nothing to dress for the triplet state, only the static TDHF e
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%%% TABLE I %%%
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\begin{table}
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\caption{Singlet and triplet excitation energies (in hartree) at various levels of theory.
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\caption{Singlet and triplet excitation energies (in eV) at various levels of theory for \ce{He} at the HF/6-31G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:Maitra}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|cccc|c|}
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Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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\hline
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$\omega_1^{\updw}$ & 1.91119 & 1.89758 & 1.90636 & 1.89314 & 1.92145 \\
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$\omega_3^{\updw}$ & & & 3.44888 & 3.44865 & 3.47880 \\
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$\omega_1^{\updw}$ & 52.01 & 51.64 & 51.87[-0.14]& 51.52[-0.12]& 52.29 \\
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$\omega_3^{\updw}$ & & & 93.85 & 93.84 & 94.66 \\
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\hline
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Triplets & & & & & Exact \\
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Triplets & & & & & Exact \\
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\hline
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$\omega_1^{\upup}$ & 1.45585 & 1.43794 & 1.45585 & 1.43794 & 1.47085 \\
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$\omega_1^{\upup}$ & 39.62 & 39.13 & 39.62[+0.00]& 39.13[+0.00]& 40.18 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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@ -440,24 +440,25 @@ Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^
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It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
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This makes these two latter solutions quite hard to locate with the Newton-Raphson method.
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%%% TABLE I %%%
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%%% TABLE II %%%
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\begin{table*}
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\caption{BSE singlet and triplet excitation energies (in hartree) at various levels of theory.
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\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory for \ce{He} at the HF/6-31G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
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$\omega_2^{\updw}$ & & & & 2.78377 & & & & \\
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$\omega_3^{\updw}$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\
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$\omega_1^{\updw}$ & 52.46 & 51.71[-0.75]& 52.12[-0.33]& 51.85[-0.61]& 53.10 & 52.79[-0.31]& 52.79[-0.31]& 52.29 \\
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$\omega_2^{\updw}$ & & & & 75.75 & & & & \\
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$\omega_3^{\updw}$ & & & & 133.37 & & & 133.37 & 94.66 \\
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\hline
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Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
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$\omega_2^{\upup}$ & & & & 2.76178 & & & & \\
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$\omega_3^{\upup}$ & & & & 4.91545 & & & 4.91517 & \\
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$\omega_1^{\upup}$ & 40.50 & 39.96[-0.53]& 39.80[-0.70]& 39.90[-0.60]& 40.71 & 40.02[-0.69]& 40.02[-0.69]& 40.18 \\
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$\omega_2^{\upup}$ & & & & 75.15 & & & & \\
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$\omega_3^{\upup}$ & & & & 133.76 & & & 133.75 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -635,24 +636,23 @@ In the case of BSE2, the perturbative partitioning is simply
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+ \underbrace{\qty[ \bH_{\dBSE2}^{\sigma}(\omega) - \bH_{\BSE2}^{\sigma} ]}_{\bH_{\pBSE2}^{(1)}(\omega)}
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\end{equation}
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%%% TABLE II %%%
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%%% TABLE III %%%
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\begin{table*}
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\caption{BSE2 singlet and triplet excitation energies (in hartree) at various levels of theory.
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\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{He} at the HF/6-31G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE2}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 1.84903 & 1.90940 & 1.90950 & 1.91454 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
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$\omega_2^{\updw}$ & & & & & & & & \\
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$\omega_3^{\updw}$ & & & & 4.47109 & & & 4.47097 & 3.47880 \\
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$\omega_1^{\updw}$ & 50.31 & 51.96[+1.64]& 51.96[+1.65]& 52.10[+1.79]& 50.69 & 52.34[+1.65]& 52.34[+1.65]& 52.29 \\
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$\omega_3^{\updw}$ & & & & 121.67 & & & 121.66 & 94.66 \\
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\hline
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Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 1.38912 & 1.44285 & 1.44304 & 1.45489 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
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$\omega_2^{\upup}$ & & & & & & & & \\
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$\omega_3^{\upup}$ & & & & 4.47773 & & & 4.47767 & \\
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$\omega_1^{\upup}$ & 37.80 & 39.26[+1.46]& 39.27[+1.47]& 39.59[+1.79]& 38.30 & 39.77[+1.47]& 39.77[+1.47]& 40.18 \\
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$\omega_3^{\upup}$ & & & & 121.85 & & & 121.84 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -741,6 +741,139 @@ PFL would like to thank Xavier Blase, Elisa Rebolini, Pina Romaniello, Arjan Ber
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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.}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%% TABLE I %%%
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\begin{table}
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\caption{Singlet and triplet excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:Maitra}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|cccc|c|}
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Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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\hline
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$\omega_1^{\updw}$ & 29.68 & 29.42 & 27.75[-1.93] & 27.64[-1.78] & 28.05 \\
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$\omega_3^{\updw}$ & & & 63.59 & 63.52 & 64.09 \\
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\hline
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Triplets & & & & & Exact \\
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\hline
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$\omega_1^{\upup}$ & 21.77 & 21.41 & 21.77[+0.00] & 21.41[+0.00] & 22.03 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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%%% TABLE II %%%
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\begin{table*}
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\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 28.56 & 28.41[-0.15] & 28.63[+0.07]& 28.52[-0.04]& 29.04 & 29.11[+0.07]& 29.11[+0.07]& 28.05 \\
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$\omega_2^{\updw}$ & & & & 47.85 & & & & \\
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$\omega_3^{\updw}$ & & & & 87.47 & & & 87.47 & 64.09 \\
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\hline
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Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 20.96 & 21.16[+0.20] & 21.07[+0.11]& 21.12[+0.16]& 21.13 & 21.24[+0.11]& 21.24[+0.11]& 22.03 \\
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$\omega_2^{\upup}$ & & & & 47.54 & & & & \\
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$\omega_3^{\upup}$ & & & & 87.43 & & & 87.43 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% %%% %%% %%%
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%%% TABLE III %%%
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\begin{table*}
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\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE2}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 29.23 & 28.40[-0.83]& 28.40[-0.83]& 28.56[-0.67]& 29.50 & 28.66[-0.84]& 28.66[-0.84]& 28.05 \\
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$\omega_3^{\updw}$ & & & & 79.94 & & & 79.94 & 64.09 \\
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\hline
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Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 21.22 & 21.63[+0.41]& 21.63[+0.41]& 21.93[+0.71]& 21.59 & 21.99[+0.40]& 21.99[+0.40]& 22.03 \\
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$\omega_3^{\upup}$ & & & & 78.70 & & & 78.70 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% %%% %%% %%%
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%%% TABLE I %%%
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\begin{table}
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\caption{Singlet and triplet excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:Maitra}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|cccc|c|}
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Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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\hline
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$\omega_1^{\updw}$ & 25.78 & 25.30 & 25.78[+0.00] & 25.30[+0.00] & 26.34 \\
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$\omega_3^{\updw}$ & & & 63.59 & 63.52 & 44.04 \\
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\hline
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Triplets & & & & & Exact \\
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\hline
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$\omega_1^{\upup}$ & 15.92 & 15.13 & 15.92[+0.00] & 15.13[+0.00] & 16.48 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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%%% TABLE II %%%
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\begin{table*}
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\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 26.06 & 25.52[-0.54] & 26.06[+0.00]& 25.78[-0.04]& 27.02 & 27.02[+0.00]& 27.02[+0.00]& 26.34 \\
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$\omega_3^{\updw}$ & & & & 44.30 & & & & 44.04 \\
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\hline
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Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 16.94 & 17.10[+0.16] & 16.94[+0.00]& 17.03[+0.16]& 17.16 & 17.16[+0.00]& 17.16[+0.00]& 16.48 \\
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$\omega_3^{\upup}$ & & & & 43.61 & & & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% %%% %%% %%%
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%%% TABLE III %%%
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\begin{table*}
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\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\label{tab:BSE2}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 26.03 & 26.03[+0.00]& 26.03[+0.00]& 26.24[+0.21]& 26.49 & 26.49[+0.00]& 26.49[+0.00]& 26.34 \\
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\hline
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Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 15.88 & 15.88[+0.00]& 15.88[+0.00]& 16.47[+0.59]& 16.63 & 16.63[+0.00]& 16.63[+0.00]& 16.48 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% %%% %%% %%%
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% BIBLIOGRAPHY
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\bibliography{dynker}
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