diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index bb2f3fb..4d42034 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -601,13 +601,6 @@ Throughout this work, all spin-flip calculations are performed with a UHF refere \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} 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}. @@ -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. 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 %%% -\begin{squeezetable} +%\begin{squeezetable} \begin{table*} \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. - 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}} \begin{ruledtabular} \begin{tabular}{lcccccccccc} - & \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVQZ} \\ - \cline{2-6} \cline{7-11} + & \mc{5}{c}{States} \\ +% & \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVQZ} \\ + \cline{2-6} %\cline{7-11} Method & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$ - & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ - & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$ & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\ +% & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$ +% & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\ \hline - 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-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-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-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-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-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-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424 \\ % 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)-x & & 2.866 & 6.581 & 7.664 & 8.612 - & & & & & \\ - 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) \\ + 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 \\ + 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{ruledtabular} - \fnt[1]{Value in the 6-31G basis taken from Ref.~\onlinecite{Casanova_2020}.} - \fnt[2]{Value in the 6-31G basis taken from Ref.~\onlinecite{Krylov_2001a}.} + \fnt[1]{Excitation energy taken from Ref.~\onlinecite{Casanova_2020}.} + \fnt[2]{Excitation energy taken from Ref.~\onlinecite{Krylov_2001a}.} \end{table*} -\end{squeezetable} +%\end{squeezetable} %%% %%% %%% %%% %%% FIG 1 %%%