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6
Manuscript/sfBSE-SI.tex
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@ -493,12 +493,12 @@ $4.$ & $0 .3189$ & $0 .8867$ & $0 .3328$ & $0 .2172$ & $0 .9453$ & $0 \
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=0.45\linewidth]{H2_CAM_B3LYP} \includegraphics[width=0.45\linewidth]{H2_BLYP}
\hspace{0.05\linewidth} \hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_B3LYP} \includegraphics[width=0.45\linewidth]{H2_B3LYP}
\vspace{0.025\linewidth} \vspace{0.025\linewidth}
\\ \\
\includegraphics[width=0.45\linewidth]{H2_BLYP} \includegraphics[width=0.45\linewidth]{H2_CAM_B3LYP}
\hspace{0.05\linewidth} \hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_WB97X_D} \includegraphics[width=0.45\linewidth]{H2_WB97X_D}
\vspace{0.025\linewidth} \vspace{0.025\linewidth}
@ -507,7 +507,7 @@ $4.$ & $0 .3189$ & $0 .8867$ & $0 .3328$ & $0 .2172$ & $0 .9453$ & $0 \
\hspace{0.05\linewidth} \hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_dBSE} \includegraphics[width=0.45\linewidth]{H2_dBSE}
\caption{ \caption{
Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)TD-B3LYP (top), (SF-) TD-BLYP (middle), and (SF-)dBSE (bottom) levels of theory. Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at various levels of theory.
The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip are represented as dashed and dotted lines, respectively. The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip are represented as dashed and dotted lines, respectively.
All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference. All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference.
\label{fig:H2}} \label{fig:H2}}

4
Manuscript/sfBSE.bib
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@ -28,7 +28,7 @@
author = {Chai, J. D. and Head-Gordon, M.}, author = {Chai, J. D. and Head-Gordon, M.},
date-added = {2021-02-25 09:23:14 +0100}, date-added = {2021-02-25 09:23:14 +0100},
date-modified = {2021-02-25 09:23:35 +0100}, date-modified = {2021-02-25 09:23:35 +0100},
journal = JCP, journal = {J. Chem. Phys.},
pages = {084106}, pages = {084106},
title = {Systematic Optimization of Long-Range Corrected Hybrid Density Functionals}, title = {Systematic Optimization of Long-Range Corrected Hybrid Density Functionals},
volume = 128, volume = 128,
@ -38,7 +38,7 @@
author = {Chai, J. D. and Head-Gordon, M.}, author = {Chai, J. D. and Head-Gordon, M.},
date-added = {2021-02-25 09:21:29 +0100}, date-added = {2021-02-25 09:21:29 +0100},
date-modified = {2021-02-25 09:23:17 +0100}, date-modified = {2021-02-25 09:23:17 +0100},
journal = PCCP, journal = {Phys. Chem. Chem. Phys.},
pages = {6615--6620}, pages = {6615--6620},
title = {Long-range Corrected Hybrid Density Functionals with Damped Atom--Atom Dispersion Corrections}, title = {Long-range Corrected Hybrid Density Functionals with Damped Atom--Atom Dispersion Corrections},
volume = 10, volume = 10,

28
Manuscript/sfBSE.tex
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@ -582,7 +582,7 @@ from which we obtain the following expressions for the spin-conserved and spin-f
At this stage, it is of particular interest to discuss the form of the spin-flip matrix elements defined in Eqs.~\eqref{eq:LR_BSE-Asf} and \eqref{eq:LR_BSE-Bsf}. At this stage, it is of particular interest to discuss the form of the spin-flip matrix elements defined in Eqs.~\eqref{eq:LR_BSE-Asf} and \eqref{eq:LR_BSE-Bsf}.
As readily seen from Eq.~\eqref{eq:LR_RPA-Asf}, at the RPA level, the spin-flip excitations are given by the difference of one-electron energies, hence missing out on key exchange and correlation effects. As readily seen from Eq.~\eqref{eq:LR_RPA-Asf}, at the RPA level, the spin-flip excitations are given by the difference of one-electron energies, hence missing out on key exchange and correlation effects.
This is also the case at the TD-DFT level when one relies on (semi-)local functionals. This is also the case at the TD-DFT level when one relies on (semi-)local functionals.
This explains why most of spin-flip TD-DFT calculations are performed with hybrid functionals containing a substantial amount of Hartree-Fock exchange as only the exact exchange integral of the form $\ERI{i_\sig j_\sig}{b_\bsig a_\bsig}$ survive spin-symmetry requirements. This explains why most of spin-flip TD-DFT calculations are performed with global hybrid functionals containing a substantial amount of Hartree-Fock exchange as only the exact exchange integral of the form $\ERI{i_\sig j_\sig}{b_\bsig a_\bsig}$ survive spin-symmetry requirements.
At the BSE level, these matrix elements are, of course, also present thanks to the contribution of $W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}$ as evidenced in Eq.~\eqref{eq:W_spectral} but it also includes correlation effects. At the BSE level, these matrix elements are, of course, also present thanks to the contribution of $W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}$ as evidenced in Eq.~\eqref{eq:W_spectral} but it also includes correlation effects.
%================================ %================================
@ -753,7 +753,7 @@ Further details about our implementation of {\GOWO} can be found in Refs.~\onlin
Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} excitation energies. Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} excitation energies.
This is left for future work. This is left for future work.
However, it is worth mentioning that, for the present (small) molecular systems, Hartee-Fock is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012} However, it is worth mentioning that, for the present (small) molecular systems, Hartree-Fock is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid (RSH) functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
Besides, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies, while {\GOWO} allows us to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$. \cite{Loos_2020e,Loos_2020h,Berger_2021} Besides, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies, while {\GOWO} allows us to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$. \cite{Loos_2020e,Loos_2020h,Berger_2021}
In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results. In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
@ -761,7 +761,7 @@ Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}. All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4} The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4}
Spin-flip TD-DFT calculations \cite{Shao_2003} (also performed with Q-CHEM 5.2.1) considering the BLYP, \cite{Becke_1988,Lee_1988} B3LYP, \cite{Becke_1988,Lee_1988,Becke_1993a} and BH\&HLYP \cite{Lee_1988,Becke_1993b} functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively. Spin-flip TD-DFT calculations \cite{Shao_2003} (also performed with Q-CHEM 5.2.1) considering the BLYP, \cite{Becke_1988,Lee_1988} B3LYP, \cite{Becke_1988,Lee_1988,Becke_1993a} and BH\&HLYP \cite{Lee_1988,Becke_1993b} functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively.
\alert{Additionally, we have performed spin-flip TD-DFT calculations considering the following the range-separated hybrid (RSH) functionals: CAM-B3LYP, \cite{Yanai_2004} LC-$\omega$PBE08, \cite{Weintraub_2009} and $\omega$B97X-D. \cite{Chai_2008a,Chai_2008b} \alert{Additionally, we have performed spin-flip TD-DFT calculations considering the following the RSH functionals: CAM-B3LYP, \cite{Yanai_2004} LC-$\omega$PBE08, \cite{Weintraub_2009} and $\omega$B97X-D. \cite{Chai_2008a,Chai_2008b}
In the present context, the main difference between these RSHs is their amount of exact exchange at long range: 75\% for CAM-B3LYP and 100\% for both LC-$\omega$PBE08 and $\omega$B97X-D.} In the present context, the main difference between these RSHs is their amount of exact exchange at long range: 75\% for CAM-B3LYP and 100\% for both LC-$\omega$PBE08 and $\omega$B97X-D.}
EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09} EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09}
As a consistency check, we systematically perform SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies. As a consistency check, we systematically perform SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
@ -788,7 +788,7 @@ Indeed, due to the lack of coupling terms in the spin-flip block of the SD-TD-DF
Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy and improves the description of both states. 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.
\alert{Comments on RSHs for Be.} \alert{Spin-flip TD-DFT calculations performed with CAM-B3LYP and $\omega$B97X-D are only slightly more accurate than their global hybrid counterparts, while SF-TD-LC-$\omega$PBE08 yields more significant improvements although it does not reach the accuracy of SF-(d)BSE.}
The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) 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 additional inclusion of correlation in the case of SF-BSE and SF-dBSE thanks to the introduction of static and dynamical screening, respectively. 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 static and dynamical screening, respectively.
@ -846,10 +846,10 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{fig1} \includegraphics[width=0.8\linewidth]{fig1}
\caption{ \caption{
Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained with the 6-31G basis at various levels of theory: Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained with the 6-31G basis at various levels of theory:
SF-TD-DFT \cite{Casanova_2020} (red), SF-CIS \cite{Krylov_2001a} (purple), SF-BSE (blue), SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). SF-TD-DFT (red), SF-CIS (purple), SF-BSE (blue), SF-ADC (orange), and FCI (black).
All the spin-flip calculations have been performed with an unrestricted reference. All the spin-flip calculations have been performed with an unrestricted reference.
\label{fig:Be}} \label{fig:Be}}
\end{figure*} \end{figure*}
@ -883,7 +883,8 @@ SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD
Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Vuckovic_2017,Cohen_2008a,Cohen_2008c,Cohen_2012} Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Vuckovic_2017,Cohen_2008a,Cohen_2008c,Cohen_2012}
Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions. Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions.
Notably, one can see that the $\text{E}\,{}^1\Sigma_g^+$ and $\text{F}\,{}^1 \Sigma_g^+$ states crossed without interacting at the SF-TD-BLYP level due to the lack of Hartree-Fock exchange. Notably, one can see that the $\text{E}\,{}^1\Sigma_g^+$ and $\text{F}\,{}^1 \Sigma_g^+$ states crossed without interacting at the SF-TD-BLYP level due to the lack of Hartree-Fock exchange.
\alert{Comments on RSHs for H2.} \alert{In the {\SI}, we also report the potential energy curves of \ce{H2} obtained with three RSHs (CAM-B3LYP, $\omega$B97X-D, and LC-$\omega$PBE08), which only brought a
modest improvement.}
In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented. In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented.
SF-BSE provides surprisingly accurate excitation energies for the $\text{B}\,{}^1\Sigma_u^+$ state with errors between $0.05$ and $0.3$ eV, outperforming in the process the standard BSE formalism. SF-BSE provides surprisingly accurate excitation energies for the $\text{B}\,{}^1\Sigma_u^+$ state with errors between $0.05$ and $0.3$ eV, outperforming in the process the standard BSE formalism.
@ -959,7 +960,7 @@ Nonetheless, it is pleasing to see that adding the dynamical correction in SF-dB
Then, CBD stands as an excellent example for which dynamical corrections are necessary to get the right chemistry at the SF-BSE level. Then, CBD stands as an excellent example for which dynamical corrections are necessary to get the right chemistry at the SF-BSE level.
Another interesting feature is the wrong ordering of the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states at the SF-B3LYP, SF-BH\&HLYP, and SF-CIS levels which give the former higher in energy than the latter. Another interesting feature is the wrong ordering of the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states at the SF-B3LYP, SF-BH\&HLYP, and SF-CIS levels which give the former higher in energy than the latter.
This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels. This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
\alert{Comments on RSHs for CBD.} \alert{Here again, one does not observe a clear improvement by considering RSHs instead of global hybrids (BH\&HLYP seems to perform particularly well in the case of CBD), although it is worth mentioning that RSH-based SF-TD-DFT calculations yield accurate excitation for the double excitation $1\,{}^1A_{g} \to 2\,{}^1A_{g}$ in the $D_{2h}$ geometry.}
%%% FIG 3 %%% %%% FIG 3 %%%
\begin{figure*} \begin{figure*}
@ -967,7 +968,8 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
\hspace{0.05\linewidth} \hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{fig3b} \includegraphics[width=0.45\linewidth]{fig3b}
\caption{ \caption{
Vertical excitation energies of CBD. Vertical excitation energies of CBD at various levels of theory:
SF-TD-DFT (red), SF-CIS (purple), SF-BSE (blue), SF-ADC (orange), and EOM-SF-CCSD (black).
Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data). Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data).
Right: $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3 A_{2g}$ state (see Table \ref{tab:CBD_D4h} for the raw data). Right: $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3 A_{2g}$ state (see Table \ref{tab:CBD_D4h} for the raw data).
All the spin-flip calculations have been performed with an unrestricted reference and the cc-pVTZ basis set. All the spin-flip calculations have been performed with an unrestricted reference and the cc-pVTZ basis set.
@ -1065,13 +1067,7 @@ This project has received funding from the European Research Council (ERC) under
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information available} \section*{Supporting information available}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Additional graphs comparing (SF-)TD-BLYP, (SF-)TD-B3LYP, and (SF-)dBSE with EOM-CCSD for the \ce{H2} molecule and raw data associated with Fig.~\ref{fig:H2}. Additional graphs comparing (SF-)TD-DFT and (SF-)dBSE with EOM-CCSD for the \ce{H2} molecule and raw data associated with Fig.~\ref{fig:H2}.
%output files associated with all the calculations performed in the present article.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{sfBSE} \bibliography{sfBSE}

6
Response_Letter/Response_Letter.tex
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@ -40,10 +40,10 @@ I recommend this manuscript for publication after the minor points addressed:}
{Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?} {Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?}
\\ \\
\alert{Following the excellent advice of Reviewer \#1, we have added data for the following range-separated hybrid functionals: CAM-B3LYP, LC-$\omega$HPBE, and $\omega$B97X-D. \alert{Following the excellent advice of Reviewer \#1, we have added data for the following range-separated hybrid functionals: CAM-B3LYP, LC-$\omega$HPBE, and $\omega$B97X-D.
CAM-B3LYP only has 75\% exact exchange at long range while LC-$\omega$PBE08 and $\omega$B97X-D have 100\% of HF exact exchange at longe range.
These results have been added to the corresponding Tables and Figures. These results have been added to the corresponding Tables and Figures.
In the case of \ce{H2}, we have chosen to add some of the graphs to the supporting information instead for the sake of clarity. For the sake of clarity, in the case of \ce{H2}, we have chosen to add some of the graphs to the supporting information instead.
In a nutshell, CAM-B3LYP does not really improved things and is less reliable than BH\&HLYP. In a nutshell, range-separated hybrids do not provide a clear improvement upon global hybrids and BH\&HLYP seems to be the best all-round performer.
Note that CAM-B3LYP only has 75\% exact exchange at long range while LC-$\omega$PBE08 and $\omega$B97X-D have 100\% of HF exact exchange at longe range.
All these results are discussed in the revised version of the manuscript.} All these results are discussed in the revised version of the manuscript.}
\item \item