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done for now with Be results

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Pierre-Francois Loos 2021-01-18 11:45:10 +01:00
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@ -601,13 +601,6 @@ Throughout this work, all spin-flip calculations are performed with a UHF refere
\label{sec:Be} \label{sec:Be}
%=============================== %===============================
%%%%%%%%%%%%%%%
%T2: I think it might be worth doing some calculations with a larger basis set (ie, aug-cc-pvqz).
% I've done a quick check and it seems to work much better and we could get some CIPSI excitation energies as reference.
% Also, I think we have much more spin contamination in this larger basis and it would be worth reporting it (for the reference and the excited state).
% No need to do evGW and qsGW, G0W0 (BSE and dBSE) is enough I guess + SF-ADC and SF-TD-DFT.
%%%%%%%%%%%%%%%
As a first example, we consider the simple case of the beryllium atom in a small basis (6-31G) which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020} As a first example, we consider the simple case of the beryllium atom in a small basis (6-31G) which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020}
Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration. Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration.
The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $D$ and $P$ spatial symmetries (respectively) are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}. The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $D$ and $P$ spatial symmetries (respectively) are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
@ -629,55 +622,43 @@ The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC me
Interestingly, SF-BSE and SF-ADC(2)-s have rather similar accuracies, except again for the $^1D$ state where SF-ADC(2)-s has clearly the edge over SF-BSE. Interestingly, SF-BSE and SF-ADC(2)-s have rather similar accuracies, except again for the $^1D$ state where SF-ADC(2)-s has clearly the edge over SF-BSE.
Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to FCI for this simple system with significant improvements for the lowest $^3P$ state and the $^1D$ doubly-excited state. Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to FCI for this simple system with significant improvements for the lowest $^3P$ state and the $^1D$ doubly-excited state.
\titou{Here comes the discussion for the larger basis.}
%%% TABLE I %%% %%% TABLE I %%%
\begin{squeezetable} %\begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \caption{
Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained for various methods with the 6-31G and aug-cc-pVQZ basis sets. Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained for various methods with the 6-31G basis set.
All the spin-flip calculations have been performed with a UHF reference. All the spin-flip calculations have been performed with a UHF reference.
The $\expval{S^2}$ value associated with each state is reported in parenthesis (when available). The $\expval*{S^2}$ value associated with each state is reported in parenthesis (when available).
\label{tab:Be}} \label{tab:Be}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccccccccc} \begin{tabular}{lcccccccccc}
& \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVQZ} \\ & \mc{5}{c}{States} \\
\cline{2-6} \cline{7-11} % & \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVQZ} \\
\cline{2-6} %\cline{7-11}
Method & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$ Method & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
& $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$
& $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
& $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\ & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\
% & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
% & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\
\hline \hline
SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013) SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013) \\
& () & () & () & () & () \\ SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006) \\
SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006) SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002) \\
& () & () & () & () & () \\ SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006) \\
SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002) SF-BSE@{\GOWO} & (0.004) & 2.399(1.999) & 6.191(0.023) & 7.792(1.000) & 9.373(0.013) \\
& () & () & () & () & () \\ SF-BSE@{\evGW} & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006) SF-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424 \\
& () & () & () & () & () \\
SF-BSE@{\GOWO} & (0.004) & 2.399(1.999) & 6.191(0.023) & 7.792(1.000) & 9.373(0.013)
& (0.021) & 2.286(1.994) & 5.181(0.187) & 6.481(1.000) & 7.195(0.719) \\
% SF-BSE@{\evGW} & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
% \alert{SF-BSE@{\qsGW}} & (0.102) & 2.532(2.000) & 6.241(1.873) & 7.668(1.000) & 9.417(0.217) \\
SF-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424
& & & & & \\
% SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\ % SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\
% SF-dBSE@{\qsGW} & & 2.335 & 6.317 & 7.689 & 9.470 \\ SF-ADC(2)-s & & 2.433 & 6.255 & 7.745 & 9.047 \\
SF-ADC(2)-s & & 2.433 & 6.255 & 7.745 & 9.047 SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 \\
& & & & & \\ SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618 \\
SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 FCI\fnm[2] & (0.000) & 2.862(2.000) & 6.577(0.000) & 7.669(2.000) & 8.624(0.000) \\
& & & & & \\ % & (0.000) & 2.718(2.000) & 5.277(0.000) & 6.450(2.000) & 7.114(0.000) \\
SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618
& & & & & \\
FCI\fnm[2] & (0.000) & 2.862(2.000) & 6.577(0.000) & 7.669(2.000) & 8.624(0.000)
& (0.000) & 2.718(2.000) & 5.277(0.000) & 6.450(2.000) & 7.114(0.000) \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Value in the 6-31G basis taken from Ref.~\onlinecite{Casanova_2020}.} \fnt[1]{Excitation energy taken from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{Value in the 6-31G basis taken from Ref.~\onlinecite{Krylov_2001a}.} \fnt[2]{Excitation energy taken from Ref.~\onlinecite{Krylov_2001a}.}
\end{table*} \end{table*}
\end{squeezetable} %\end{squeezetable}
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%%% FIG 1 %%% %%% FIG 1 %%%