diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index f683a56..4d42034 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -601,76 +601,64 @@ 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}. 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}). -Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy. +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. +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. 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). -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. -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. -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. -Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies. +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 additional inclusion of correlation in the case of SF-BSE and SF-dBSE thanks to the introduction of the static and dynamical screening, respectively. +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. +At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect. +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 %%% -\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-pVTQ} \\ - \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) - & (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) - & (0.009) & 3.205(1.000) & () & () & () \\ - SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002) - & (0.005) & 2.799(1.976) & () & () & () \\ - SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006) - & (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-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 - & & 2.415 & 5.360 & 6.431 & 7.206 \\ - SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 - & & 2.692 & 5.256 & 6.411 & 7.098\\ - 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) \\ + 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 %%% @@ -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. \label{tab:CBD_D2h}} \begin{ruledtabular} - \begin{tabular}{lccc} + \begin{tabular}{lrrr} & \mc{3}{c}{Excitation energies (eV)} \\ \cline{2-4} Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\ \hline - SF-TD-B3LYP\fnm[3] &1.750 &2.260 &4.094 \\ - SF-TD-BH\&HLYP\fnm[3] &1.583 &2.813 & 4.528\\ - SF-CIS\fnm[1] & & & \\ - 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(dT)\fnm[1] &1.475 &3.215 &4.176 \\ - 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(3)\fnm[2] & 1.456&3.285 &4.334 \\ - SF-BSE@{\GOWO}\fnm[3] & 1.438 & 2.704 &4.540 \\ - SF-dBSE@{\GOWO}\fnm[3] & 1.403 &2.883 &4.621 \\ + SF-TD-B3LYP\fnm[3] & $1.750$ & $2.260$ & $4.094$ \\ + SF-TD-BH\&HLYP\fnm[3] & $1.583$ & $2.813$ & $4.528$ \\ + SF-CIS\fnm[1] & $1.521$ & $3.836$ & $5.499$ \\ + 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(dT)\fnm[1] & $1.475$ & $3.215$ & $4.176$ \\ + 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(3)\fnm[2] & $1.456$ & $3.285$ & $4.334$ \\ + SF-BSE@{\GOWO}\fnm[3] & $1.438$ & $2.704$ & $4.540$ \\ + SF-dBSE@{\GOWO}\fnm[3] & $1.403$ & $2.883$ & $4.621$ \\ \end{tabular} \end{ruledtabular} \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. \label{tab:CBD_D4h}} \begin{ruledtabular} - \begin{tabular}{lccc} + \begin{tabular}{lrrr} & \mc{3}{c}{Excitation energies (eV)} \\ \cline{2-4} Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\ \hline - SF-TD-B3LYP \fnm[3] &-0.020 & 0.486& 0.547\\ - SF-TD-BH\&HLYP\fnm[3] &0.048 & 1.282&1.465 \\ - SF-CIS\fnm[1] &0.317 & 3.125&2.650 \\ - 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(dT)\fnm[1] &0.098 &1.456 &1.853 \\ - 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(3)\fnm[2] &0.083 &1.621 &1.930 \\ - SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\ - SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\ + SF-TD-B3LYP\fnm[3] & $-0.020$ & $0.486$ & $0.547$ \\ + SF-TD-BH\&HLYP\fnm[3] & $0.048$ & $1.282$ & $1.465$ \\ + SF-CIS\fnm[1] & $0.317$ & $3.125$ & $2.650$ \\ + 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(dT)\fnm[1] & $0.098$ & $1.456$ & $1.853$ \\ + 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(3)\fnm[2] & $0.083$ & $1.621$ & $1.930$ \\ + SF-BSE@{\GOWO}\fnm[3] & $-0.049$ & $1.189$ & $1.480$ \\ + SF-dBSE@{\GOWO}\fnm[3] & $0.012$ & $1.507$ & $1.841$ \\ \end{tabular} \end{ruledtabular} \fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}