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Pierre-Francois Loos 2021-02-26 21:22:58 +01:00
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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 %%%
\begin{figure*}
\includegraphics[width=0.45\linewidth]{H2_CAM_B3LYP}
\includegraphics[width=0.45\linewidth]{H2_BLYP}
\hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_B3LYP}
\vspace{0.025\linewidth}
\\
\includegraphics[width=0.45\linewidth]{H2_BLYP}
\includegraphics[width=0.45\linewidth]{H2_CAM_B3LYP}
\hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_WB97X_D}
\vspace{0.025\linewidth}
@ -507,7 +507,7 @@ $4.$ & $0 .3189$ & $0 .8867$ & $0 .3328$ & $0 .2172$ & $0 .9453$ & $0 \
\hspace{0.05\linewidth}
\includegraphics[width=0.45\linewidth]{H2_dBSE}
\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.
All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference.
\label{fig:H2}}

4
Manuscript/sfBSE.bib
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@ -28,7 +28,7 @@
author = {Chai, J. D. and Head-Gordon, M.},
date-added = {2021-02-25 09:23:14 +0100},
date-modified = {2021-02-25 09:23:35 +0100},
journal = JCP,
journal = {J. Chem. Phys.},
pages = {084106},
title = {Systematic Optimization of Long-Range Corrected Hybrid Density Functionals},
volume = 128,
@ -38,7 +38,7 @@
author = {Chai, J. D. and Head-Gordon, M.},
date-added = {2021-02-25 09:21:29 +0100},
date-modified = {2021-02-25 09:23:17 +0100},
journal = PCCP,
journal = {Phys. Chem. Chem. Phys.},
pages = {6615--6620},
title = {Long-range Corrected Hybrid Density Functionals with Damped Atom--Atom Dispersion Corrections},
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}.
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 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.
%================================
@ -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.
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}
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.
@ -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}.
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.
\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.}
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.
@ -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.
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.
\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).
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 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig1}
\includegraphics[width=0.8\linewidth]{fig1}
\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:
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.
\label{fig:Be}}
\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}
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.
\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.
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.
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.
\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 %%%
\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}
\includegraphics[width=0.45\linewidth]{fig3b}
\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).
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.
@ -1065,13 +1067,7 @@ This project has received funding from the European Research Council (ERC) under
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}.
%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.
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}.
%%%%%%%%%%%%%%%%%%%%%%%%
\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?}
\\
\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.
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.
In a nutshell, CAM-B3LYP does not really improved things and is less reliable than BH\&HLYP.
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.
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, range-separated hybrids do not provide a clear improvement upon global hybrids and BH\&HLYP seems to be the best all-round performer.
All these results are discussed in the revised version of the manuscript.}
\item