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@ -601,76 +601,64 @@ 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}.
The left side of Fig.~\ref{fig:Be} (red lines) reports SF-TD-DFT excitation energies obtained with the BLYP, B3LYP, and BH\&HLYP functionals, which correspond to an increase of exact exchange from 0\% to 50\%. The left side of Fig.~\ref{fig:Be} (red lines) reports SF-TD-DFT excitation energies obtained with the BLYP, B3LYP, and BH\&HLYP functionals, which correspond to an increase of exact exchange from 0\% to 50\%.
As mentioned in Ref.~\onlinecite{Casanova_2020}, the $^3P(1s^2 2s^1 2p^1)$ and the $^1P(1s^2 2s^1 2p^1)$ states are degenerate at the SF-TD-BLYP level and their excitation energies are given by the $2s$--$2p$ orbital energy difference due to the lack of coupling terms in the spin-flip block of the SD-TD-DFT equations (\textit{vide supra}). As mentioned in Ref.~\onlinecite{Casanova_2020}, the $^3P(1s^2 2s^1 2p^1)$ and the $^1P(1s^2 2s^1 2p^1)$ states are degenerate at the SF-TD-BLYP level.
Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy. Due to the lack of coupling terms in the spin-flip block of the SD-TD-DFT equations (see Subsec.~\ref{sec:BSE}), their excitation energies are given by the energy difference between the $2s$ and $2p$ orbitals and both states are strongly spin contaminated.
Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy and improves the description of both states.
However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state is still off by $1.6$ eV as compared to the FCI reference. However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state is still off by $1.6$ eV as compared to the FCI reference.
For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved. For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved.
The center part of Fig.~\ref{fig:Be} (blue lines) shows the SF-BSE results alongside the SF-CIS excitation energies (purple lines). The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) alongside the SF-CIS excitation energies (purple lines).
All of these are computed with 100\% of exact exchange with the inclusion of correlation in the case of SF-BSE thanks to the introduction of the screening. All of these are computed with 100\% of exact exchange with the additional inclusion of correlation in the case of SF-BSE and SF-dBSE thanks to the introduction of the static and dynamical screening, respectively.
They are close the FCI results, because of the fact that there is one hundred percent exact exchange in the BSE method, with an error of $0.02$-$0.6$ eV depending on the scheme of SF-BSE. Overall, the SF-CIS and SF-BSE excitation energies are closer to FCI than the SF-TD-DFT ones, except for the lowest triplet state where the SF-TD-BH\&HLYP excitation energy is more accurate probably due to error compensation.
For the last excited state $^1D(1s^2 2p^2)$ the largest error is $0.85$ eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination. At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect.
Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies. Note that the exact exchange seems to spin purified the $^3P(1s^2 2s^1 2p^1)$ state while the singlet states at the SF-BSE level are slightly more spin contaminated than their SF-CIS counterparts.
The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
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.
%%% 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-pVTQ} \\ & \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) \\
& (0.012) &2.998 (1.000) & () & () & () \\ 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) \\
& (0.009) & 3.205(1.000) & () & () & () \\ 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) \\
& (0.005) & 2.799(1.976) & () & () & () \\ 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 \\
& (0.005) & 2.059(2.000) &5.580 (0.178) & () & () \\
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
& & 2.177 & 5.380 & 6.806 & \\
% 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 \\
& & 2.415 & 5.360 & 6.431 & 7.206 \\ 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) \\
& & 2.692 & 5.256 & 6.411 & 7.098\\ % & (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
& & & 5.221 & & \\
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}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% FIG 1 %%% %%% FIG 1 %%%
@ -790,22 +778,22 @@ So here we have an example where the dynamical corrections are necessary to get
All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set. All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
\label{tab:CBD_D2h}} \label{tab:CBD_D2h}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lrrr}
& \mc{3}{c}{Excitation energies (eV)} \\ & \mc{3}{c}{Excitation energies (eV)} \\
\cline{2-4} \cline{2-4}
Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\ Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\
\hline \hline
SF-TD-B3LYP\fnm[3] &1.750 &2.260 &4.094 \\ SF-TD-B3LYP\fnm[3] & $1.750$ & $2.260$ & $4.094$ \\
SF-TD-BH\&HLYP\fnm[3] &1.583 &2.813 & 4.528\\ SF-TD-BH\&HLYP\fnm[3] & $1.583$ & $2.813$ & $4.528$ \\
SF-CIS\fnm[1] & & & \\ SF-CIS\fnm[1] & $1.521$ & $3.836$ & $5.499$ \\
EOM-SF-CCSD\fnm[1] &1.654 & 3.416&4.360 \\ EOM-SF-CCSD\fnm[1] & $1.654$ & $3.416$ & $4.360$ \\
EOM-SF-CCSD(fT)\fnm[1] & 1.516& 3.260&4.205 \\ EOM-SF-CCSD(fT)\fnm[1] & $1.516$ & $3.260$ & $4.205$ \\
EOM-SF-CCSD(dT)\fnm[1] &1.475 &3.215 &4.176 \\ EOM-SF-CCSD(dT)\fnm[1] & $1.475$ & $3.215$ & $4.176$ \\
SF-ADC(2)-s\fnm[2] & 1.573& 3.208& 4.247\\ SF-ADC(2)-s\fnm[2] & $1.573$ & $3.208$ & $4.247$ \\
SF-ADC(2)-x\fnm[2] &1.576 &3.141 &3.796 \\ SF-ADC(2)-x\fnm[2] & $1.576$ & $3.141$ & $3.796$ \\
SF-ADC(3)\fnm[2] & 1.456&3.285 &4.334 \\ SF-ADC(3)\fnm[2] & $1.456$ & $3.285$ & $4.334$ \\
SF-BSE@{\GOWO}\fnm[3] & 1.438 & 2.704 &4.540 \\ SF-BSE@{\GOWO}\fnm[3] & $1.438$ & $2.704$ & $4.540$ \\
SF-dBSE@{\GOWO}\fnm[3] & 1.403 &2.883 &4.621 \\ SF-dBSE@{\GOWO}\fnm[3] & $1.403$ & $2.883$ & $4.621$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.} \fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
@ -821,22 +809,22 @@ So here we have an example where the dynamical corrections are necessary to get
All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set. All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
\label{tab:CBD_D4h}} \label{tab:CBD_D4h}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lrrr}
& \mc{3}{c}{Excitation energies (eV)} \\ & \mc{3}{c}{Excitation energies (eV)} \\
\cline{2-4} \cline{2-4}
Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\ Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
\hline \hline
SF-TD-B3LYP \fnm[3] &-0.020 & 0.486& 0.547\\ SF-TD-B3LYP\fnm[3] & $-0.020$ & $0.486$ & $0.547$ \\
SF-TD-BH\&HLYP\fnm[3] &0.048 & 1.282&1.465 \\ SF-TD-BH\&HLYP\fnm[3] & $0.048$ & $1.282$ & $1.465$ \\
SF-CIS\fnm[1] &0.317 & 3.125&2.650 \\ SF-CIS\fnm[1] & $0.317$ & $3.125$ & $2.650$ \\
EOM-SF-CCSD\fnm[1] &0.369 & 1.824& 2.143\\ EOM-SF-CCSD\fnm[1] & $0.369$ & $1.824$ & $2.143$ \\
EOM-SF-CCSD(fT)\fnm[1] & 0.163&1.530 &1.921 \\ EOM-SF-CCSD(fT)\fnm[1] & $0.163$ & $1.530$ & $1.921$ \\
EOM-SF-CCSD(dT)\fnm[1] &0.098 &1.456 &1.853 \\ EOM-SF-CCSD(dT)\fnm[1] & $0.098$ & $1.456$ & $1.853$ \\
SF-ADC(2)-s\fnm[2] & 0.266& 1.664& 1.910 \\ SF-ADC(2)-s\fnm[2] & $0.266$ & $1.664$ & $1.910$ \\
SF-ADC(2)-x\fnm[2] & 0.217&1.123 &1.799 \\ SF-ADC(2)-x\fnm[2] & $0.217$ & $1.123$ & $1.799$ \\
SF-ADC(3)\fnm[2] &0.083 &1.621 &1.930 \\ SF-ADC(3)\fnm[2] & $0.083$ & $1.621$ & $1.930$ \\
SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-BSE@{\GOWO}\fnm[3] & $-0.049$ & $1.189$ & $1.480$ \\
SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\ SF-dBSE@{\GOWO}\fnm[3] & $0.012$ & $1.507$ & $1.841$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.} \fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}