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Pierre-Francois Loos 2020-05-20 00:28:03 +02:00
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BSEdyn.tex
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@ -174,6 +174,8 @@
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
\newcommand{\si}{\sigma}
\newcommand{\sis}{\sigma^*}
\newcommand{\pis}{\pi^*}
\newcommand{\ra}{\rightarrow}
@ -608,8 +610,8 @@ These are then used to compute the first-order correction from Eq.~\eqref{eq:Om1
The static BSE Hamiltonian is computed once during the static BSE calculation and does not dependent on the targeted excitation.
Searching iteratively for the lowest eigenstates, via the Davidson algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is, by far, the computational bottleneck in our current implementation.
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$ computational cost.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in our current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -635,40 +637,65 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
\begin{table*}
\caption{
BSE excitation energies for various molecules obtained with the aug-cc-pVTZ basis set.
Excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set at various levels of theory.
\label{tab:BigTab}
}
\begin{ruledtabular}
\begin{tabular}{lccccccccccc}
& & \mc{4}{c}{BSE@{\GOWO}@HF} & \mc{4}{c}{BSE@{\evGW}@HF} \\
\cline{4-7} \cline{8-11}
Mol. & State & $\Om{m}{\stat}$ & $\Om{m}{\dyn}$ & $\Delta\Om{m}{\dyn}$ & $Z_{m}$
& $\Om{m}{\stat}$ & $\Om{m}{\dyn}$ & $\Delta\Om{m}{\dyn}$ & $Z_{m}$ & CC2 & CC3 \\
\begin{tabular}{llrrrrrrrrrrrrrrr}
& & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{BSE@{\evGW}@HF} \\
\cline{3-7} \cline{8-12}
Mol. & State & $\Eg^{\GW}$ & $\Om{m}{\stat}$ & $\Om{m}{\dyn}$ & $\Delta\Om{m}{\dyn}$ & $Z_{m}$
& $\Eg^{\GW}$ & $\Om{m}{\stat}$ & $\Om{m}{\dyn}$ & $\Delta\Om{m}{\dyn}$ & $Z_{m}$
& CISD(D) & ADC(2) & CCSD & CC2 & CC3 \\
\hline
\ce{HCl} & $^1\Pi$(CT)\\
\ce{H2O} & \\
\ce{N2} & $^1\Pi_g(n \ra \pis)$ \\
& $^1\Sigma_u^-(\pi \ra \pis)$ \\
& $^1\Delta_u(\pi \ra \pis)$ \\
& $^3\Sigma_u^+(\pi \ra \pis)$\\
& $^3\Pi_g(n \ra \pis)$ \\
& $^3\Delta_u(\pi \ra \pis)$ \\
& $^3\Sigma_u^-(\pi \ra \pis)$ \\
\ce{CO} & $^1\Pi(n \ra \pis)$ \\
& $^1\Sigma^-(\pi \ra \pis)$ \\
& $^1\Delta(\pi \ra \pis)$ \\
& $^3\Pi(n \ra \pis)$ \\
& $^3\Sigma^+(\pi \ra \pis)$\\
& $^3\Delta(\pi \ra \pis)$ \\
& $^3\Sigma_u^-(\pi \ra \pis)$ \\
\ce{HNO} & \\
\ce{CH2O} & \\
\ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ \\
& $^1B_{1u}(\pi \ra \pis)$ \\
& $^1B_{1g}(\pi \ra 3p)$ \\
& $^3B_{1u}(\pi \ra \pis)$ \\
& $^3B_{3u}(\pi \ra 3s)$ \\
& $^3B_{1g}(\pi \ra 3p)$ \\
\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 \\
\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 \\
& $^3B_1(n \ra 3s)$ & & 8.14 & 7.98 & -0.15 & 1.014 & & & & & & 7.25 & 6.86 & 7.20 & 6.91 & 7.28 \\
& $^3A_2(n \ra 3p)$ & & 9.97 & 9.89 & -0.07 & 1.008 & & & & & & 9.24 & 8.72 & 9.20 & 8.77 & 9.26 \\
& $^3A_1(n \ra 3s)$ & & 10.28 & 10.13 & -0.15 & 1.012 & & & & & & 9.54 & 9.15 & 9.49 & 9.20 & 9.56 \\
\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 \\
& $^1\Delta_u(\pi \ra \pis)$ & & 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 \\
& $^1\Sigma_u^+$(R) & & 13.98 & 13.91 & -0.07 & 1.008 & & & & & & 13.75 & 13.07 & 13.26 & 12.89 & 13.12 \\
& $^1\Pi_u$(R) & & 14.24 & 14.21 & -0.03 & 1.002 & & & & & & 14.52 & 14.00 & 13.67 & 13.96 & 13.49 \\
& $^3\Sigma_u^+(\pi \ra \pis)$ & & 9.50 & 8.46 & -1.04 & 1.060 & & & & & & 8.20 & 8.15 & 7.66 & 8.19 & 7.68 \\
& $^3\Pi_g(n \ra \pis)$ & & 9.85 & 9.27 & -0.58 & 1.050 & & & & & & 8.33 & 8.20 & 8.09 & 8.19 & 8.04 \\
& $^3\Delta_u(\pi \ra \pis)$ & & 10.19 & 9.24 & -0.95 & 1.060 & & & & & & 9.30 & 9.25 & 8.91 & 9.30 & 8.87 \\
& $^3\Sigma_u^-(\pi \ra \pis)$ & & 10.89 & 10.06 & -0.82 & 1.058 & & & & & & 10.29 & 10.23 & 9.83 & 10.29 & 9.68 \\
\ce{CO} & $^1\Pi(n \ra \pis)$ & 16.46 & 9.54 & 9.19 & -0.34 & 1.029 & & & & & & 8.78 & 8.69 & 8.59 & 8.64 & 8.49 \\
& $^1\Sigma^-(\pi \ra \pis)$ & & 10.25 & 9.90 & -0.35 & 1.023 & & & & & & 10.13 & 10.03 & 9.99 & 10.30 & 9.99 \\
& $^1\Delta(\pi \ra \pis)$ & & 10.71 & 10.39 & -0.32 & 1.023 & & & & & & 10.41 & 10.30 & 10.12 & 10.60 & 10.12 \\
& $^1\Sigma^+$(R) & & 11.88 & 11.85 & -0.03 & 1.005 & & & & & & 11.48 & 11.32 & 11.22 & 11.11 & 10.94 \\
& $^1\Sigma^+$(R) & & 12.39 & 12.37 & -0.02 & 1.003 & & & & & & 11.71 & 11.83 & 11.75 & 11.63 & 11.49 \\
& $^1\Pi$(R) & & 12.37 & 12.32 & -0.05 & 1.004 & & & & & & 12.06 & 12.03 & 11.96 & 11.83 & 11.69 \\
& $^3\Pi(n \ra \pis)$ & & 8.10 & 7.33 & -0.77 & 1.055 & & & & & & 6.51 & 6.45 & 6.36 & 6.42 & 6.30 \\
& $^3\Sigma^+(\pi \ra \pis)$ & & 9.61 & 9.04 & -0.57 & 1.037 & & & & & & 8.63 & 8.54 & 8.34 & 8.72 & 8.45 \\
& $^3\Delta(\pi \ra \pis)$ & & 10.20 & 9.69 & -0.50 & 1.036 & & & & & & 9.44 & 9.33 & 9.23 & 9.56 & 9.30 \\
& $^3\Sigma_u^-(\pi \ra \pis)$ & & 10.79 & 10.38 & -0.42 & 1.034 & & & & & & 10.10 & 10.01 & 9.81 & 10.27 & 9.82 \\
& $^3\Sigma_u^+$(R) & & 11.48 & 11.38 & -0.10 & 1.010 & & & & & & 10.98 & 10.83 & 10.71 & 10.60 & 10.45 \\
\ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ & & & & & & & & & & & 7.35 & 7.34 & 7.42 & 7.29 & 7.35 \\
& $^1B_{1u}(\pi \ra \pis)$ & & & & & & & & & & & 7.95 & 7.91 & 8.02 & 7.92 & 7.91 \\
& $^1B_{1g}(\pi \ra 3p)$ & & & & & & & & & & & 8.01 & 7.99 & 8.08 & 7.95 & 8.03 \\
& $^3B_{1u}(\pi \ra \pis)$ & & & & & & & & & & & 4.62 & 4.59 & 4.46 & 4.59 & 4.53 \\
& $^3B_{3u}(\pi \ra 3s)$ & & & & & & & & & & & 7.26 & 7.23 & 7.29 & 7.19 & 7.24 \\
& $^3B_{1g}(\pi \ra 3p)$ & & & & & & & & & & & 7.97 & 7.95 & 8.03 & 7.91 & 7.98 \\
\ce{CH2O} & $^1A_2(n \ra \pis)$ & 12.00 & 5.03 & 4.68 & -0.35 & 1.027 & & & & & & 4.04 & 3.92 & 4.01 & 4.07 & 3.97 \\
& $^1B_2(n \ra 3s)$ & & 7.87 & 7.85 & -0.02 & 1.001 & & & & & & 6.64 & 6.50 & 7.23 & 6.56 & 7.18 \\
& $^1B_2(n \ra 3p)$ & & 8.76 & 8.72 & -0.04 & 1.003 & & & & & & 7.56 & 7.53 & 8.12 & 7.57 & 8.07 \\
& $^1A_1(n \ra 3p)$ & & 8.85 & 8.84 & -0.01 & 1.000 & & & & & & 8.16 & 7.47 & 8.21 & 7.52 & 8.18 \\
& $^1A_2(n \ra 3p)$ & & 8.87 & 8.85 & -0.02 & 1.002 & & & & & & 8.04 & 7.99 & 8.65 & 8.04 & 8.64 \\
& $^1B_1(\si \ra \pis)$ & & 10.18 & 9.77 & -0.42 & 1.032 & & & & & & 9.38 & 9.17 & 9.28 & 9.32 & 9.19 \\
& $^1A_1(\pi \ra \pis)$ & & 10.05 & 9.81 & -0.24 & 1.026 & & & & & & 9.08 & 9.46 & 9.67 & 9.54 & 9.48 \\
& $^3A_2(n \ra \pis)$ & & 5.53 & 5.05 & -0.47 & 1.049 & & & & & & 3.58 & 3.46 & 3.56 & 3.59 & 3.57 \\
& $^3A_1(\pi \ra \pis)$ & & 8.15 & 7.32 & -0.83 & 1.067 & & & & & & 6.27 & 6.20 & 5.97 & 6.30 & 6.05 \\
& $^3B_2(n \ra 3s)$ & & 7.51 & 7.54 & 0.03 & 0.994 & & & & & & 6.66 & 6.39 & 7.08 & 6.44 & 7.03 \\
& $^3B_2(n \ra 3p)$ & & 8.62 & 8.61 & -0.00 & 0.998 & & & & & & 7.52 & 7.41 & 7.94 & 7.45 & 7.92 \\
& $^3A_1(n \ra 3p)$ & & 8.75 & 8.69 & -0.06 & 1.007 & & & & & & 7.68 & 7.40 & 8.09 & 7.44 & 8.08 \\
& $^3B_1(n \ra 3d)$ & & 8.82 & 8.82 & -0.01 & 1.000 & & & & & & 8.57 & 8.39 & 8.43 & 8.52 & 8.41 \\
\end{tabular}
\end{ruledtabular}
\end{table*}