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Manuscript/sfBSE.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-17 22:10:38 +0100 %% Created for Pierre-Francois Loos at 2021-01-18 20:20:29 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -353,7 +353,8 @@
@article{Berger_2021, @article{Berger_2021,
author = {J. Arjan Berger and Pierre-Fran{\c c}ois Loos and Pina Romaniello}, author = {J. Arjan Berger and Pierre-Fran{\c c}ois Loos and Pina Romaniello},
date-added = {2020-12-09 09:59:26 +0100}, date-added = {2020-12-09 09:59:26 +0100},
date-modified = {2021-01-17 20:12:04 +0100}, date-modified = {2021-01-18 20:20:17 +0100},
doi = {10.1021/acs.jctc.0c00896},
journal = {J. Chem. Theory Comput.}, journal = {J. Chem. Theory Comput.},
pages = {191}, pages = {191},
title = {Potential energy surfaces without unphysical discontinuities: the Coulomb-hole plus screened exchange approach}, title = {Potential energy surfaces without unphysical discontinuities: the Coulomb-hole plus screened exchange approach},

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\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext} \newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SI}{\textcolor{blue}{supplementary material}} \newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}} \newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}} \newcommand{\T}[1]{#1^{\intercal}}

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Manuscript/sfBSE.tex
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@ -566,8 +566,7 @@ For a given single excitation $m$, the explicit expressions of $\Delta \expval{\
\section{Computational details} \section{Computational details}
\label{sec:compdet} \label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All the systems under investigation here are closed shell and we consider a triplet reference state. All the systems under investigation have a closed-shell electronic structure and we consider the lowest triplet state as reference for the spin-flip calculations adopting the unrestricted formalism throughout this work.
We then adopt the unrestricted formalism throughout this work.
%The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point. %The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
%For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}]. %For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
%Note that, in any case, the entire set of orbitals and energies is corrected. %Note that, in any case, the entire set of orbitals and energies is corrected.
@ -575,13 +574,13 @@ We then adopt the unrestricted formalism throughout this work.
%Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies. %Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
The {\GOWO} calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, and the {\GOWO} quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}]. The {\GOWO} calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, and the {\GOWO} quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
Note that the entire set of orbitals and energies is corrected. Note that the entire set of orbitals and energies is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}. Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h,Berger_2021}.
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, HF 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, HF 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}
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$GW$. 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.
%\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. 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}.
@ -589,7 +588,7 @@ The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(
Spin-flip TD-DFT calculations \cite{Shao_2003} 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, and are also performed with Q-CHEM 5.2.1. Spin-flip TD-DFT calculations \cite{Shao_2003} 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, and are also performed with Q-CHEM 5.2.1.
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 the 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 the SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
Throughout this work, all spin-flip calculations are performed with a UHF reference. Throughout this work, all spin-flip and spin-conserved calculations are performed with a UHF reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Results}
@ -601,13 +600,14 @@ Throughout this work, all spin-flip calculations are performed with a UHF refere
\label{sec:Be} \label{sec:Be}
%=============================== %===============================
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}
It 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 symmetries 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\%. On the left side of Fig.~\ref{fig:Be}, we report SF-TD-DFT excitation energies (red lines) 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. 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. Indeed, 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. 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.
@ -615,7 +615,7 @@ For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significan
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 the 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 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. 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. At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect included in the SF-dBSE scheme.
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. 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. The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
@ -645,9 +645,9 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
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-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-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@{\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-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@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424 \\
% SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\ SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\
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 \\ SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 \\
SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618 \\ SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618 \\
@ -666,8 +666,8 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
\includegraphics[width=\linewidth]{Be} \includegraphics[width=\linewidth]{Be}
\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 for 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 for various levels of theory:
SF-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). 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).
All the spin-flip calculations have been performed with a UHF reference. All these spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}} \label{fig:Be}}
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -684,7 +684,7 @@ The lowest singly excited state $\text{B}\,{}^1 \Sigma_u^+$ has a $(1\sigma_g )(
Because these latter two excited states interact strongly and form an avoided crossing around $R(\ce{H-H}) = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state. Because these latter two excited states interact strongly and form an avoided crossing around $R(\ce{H-H}) = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state.
Note that this avoided crossing is not visible with non-spin-flip methods restricted to single excitations (such as CIS, TD-DFT, and BSE) as these are ``blind'' to double excitations. Note that this avoided crossing is not visible with non-spin-flip methods restricted to single excitations (such as CIS, TD-DFT, and BSE) as these are ``blind'' to double excitations.
Three methods, in their standard and spin-flip versions, are studied here (CIS, TD-BH\&HLYP and BSE) and are compared to the reference EOM-CCSD excitation energies (that is equivalent to FCI in the case of \ce{H2}). Three methods, in their standard and spin-flip versions, are studied here (CIS, TD-BH\&HLYP and BSE) and are compared to the reference EOM-CCSD excitation energies (that is equivalent to FCI in the case of \ce{H2}).
All these calculations are performed in the cc-pVQZ basis, and both the spin-conserved and spin-flip calculations are performed with an unrestricted reference. All these calculations are performed with the cc-pVQZ basis.
The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$. The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$.
The EOM-CCSD reference energies are represented by solid lines. The EOM-CCSD reference energies are represented by solid lines.
@ -695,27 +695,27 @@ Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before
Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations. Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
At the SF-CIS level, the avoided crossing between the $\text{E}$ and $\text{F}$ states is qualitatively reproduced and placed at a slightly larger bond length than at the EOM-CCSD level. At the SF-CIS level, the avoided crossing between the $\text{E}$ and $\text{F}$ states is qualitatively reproduced and placed at a slightly larger bond length than at the EOM-CCSD level.
In the center panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results. In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}. Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}.
SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD-BH\&HLYP excitation energies of the $\text{B}$ and $\text{E}$ states are only trustworthy around equilibrium but inaccurate at dissociation. SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD-BH\&HLYP excitation energies of the $\text{B}$ and $\text{E}$ states are only trustworthy around equilibrium but inaccurate at dissociation.
Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{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}
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.
A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}.
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.
However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with error ranging from half an eV to $1.6$ eV \titou{(spin-contamination of the SF-BSE wave function?)}. However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with error ranging from half an eV to $1.6$ eV \titou{(spin-contamination of the SF-BSE wave function?)}.
Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit. Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit.
Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\Sigma_g^+$ doubly-excited state, resulting in an avoided crossing around $R(\ce{H-H}) = 1.6$ \AA. Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\Sigma_g^+$ doubly-excited state, resulting in an avoided crossing around $R(\ce{H-H}) = 1.6$ \AA.
A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions.
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure} \begin{figure}
\includegraphics[width=1\linewidth]{H2_CIS.pdf} \includegraphics[width=\linewidth]{H2_CIS.pdf}
\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf} \includegraphics[width=\linewidth]{H2_BHHLYP.pdf}
\includegraphics[width=1\linewidth]{H2_BSE.pdf} \includegraphics[width=\linewidth]{H2_BSE.pdf}
\caption{ \caption{
Excitation energies of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states (with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state) of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory. Excitation energies of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states (with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state) of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory.
The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip results 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 unrestriced reference. All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference.
The raw data are reported in the {\SI}. The raw data are reported in the {\SI}.
\label{fig:H2}} \label{fig:H2}}
\end{figure} \end{figure}
@ -728,29 +728,28 @@ Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\
Cyclobutadiene (CBD) is an interesting example as the electronic character of its ground state can be tuned via geometrical deformation. \cite{Balkova_1994,Levchenko_2004,Manohar_2008,Karadakov_2008,Li_2009,Shen_2012,Lefrancois_2015,Casanova_2020,Vitale_2020} Cyclobutadiene (CBD) is an interesting example as the electronic character of its ground state can be tuned via geometrical deformation. \cite{Balkova_1994,Levchenko_2004,Manohar_2008,Karadakov_2008,Li_2009,Shen_2012,Lefrancois_2015,Casanova_2020,Vitale_2020}
%with potential large spin contamination. %with potential large spin contamination.
In its $D_{2h}$ rectangular $A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals. In the $D_{2h}$ rectangular geometry of the $A_g$ singlet ground state, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals.
However, in its $D_{4h}$ square-planar $A_{2g}$ triplet state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals (\ie, singlet open-shell state). However, in the $D_{4h}$ square-planar geometry of the $A_{2g}$ triplet state, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals (\ie, singlet open-shell state).
In this case, single-reference methods notoriously fail. In this case, single-reference methods notoriously fail.
Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations. Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.
The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ states of CBD have been extracted from Ref.~\onlinecite{Manohar_2008} and have been obtained at the CCSD(T)/cc-pVTZ level. The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ states of CBD have been extracted from Ref.~\onlinecite{Manohar_2008} and have been obtained at the CCSD(T)/cc-pVTZ level.
EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. For comparison purposes, EOM-SF-CCSD and SF-ADC excitation energies have been extracted from Ref.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}, respectively.
All of them have been obtained with a UHF reference like the SF-BSE calculations performed here. All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
Tables~\ref{tab:CBD_D2h} and \ref{tab:CBD_D4h} report excitation energies (with respect to the singlet ground state) obtained at the $D_{2h}$ and $D_{4h}$ geometries, respectively, for several methods using the spin-flip \textit{ansatz}. Tables~\ref{tab:CBD_D2h} and \ref{tab:CBD_D4h} report excitation energies (with respect to the singlet ground state) obtained at the $D_{2h}$ and $D_{4h}$ geometries, respectively, for several methods using the spin-flip \textit{ansatz}.
For comparison purposes, we also report SF-ADC and EOM-SF-CCSD excitation energies from Ref.~\onlinecite{Lefrancois_2015} and Ref.~\onlinecite{Manohar_2008}, respectively.
All these results are represented in Fig.~\ref{fig:CBD}. All these results are represented in Fig.~\ref{fig:CBD}.
For each geometry, three excited states are under investigation: For each geometry, three excited states are under investigation:
i) the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states of the $D_{2h}$ geometry; i) the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of the $D_{2h}$ geometry;
ii) the $1\,{}^3 A_{2g}$, $2\,{}^1 A_{1g}$ and $1\,{}^1 B_{2g}$ states of the $D_{4h}$ geometry. ii) the $1\,{}^3 A_{2g}$, $2\,{}^1 A_{1g}$, and $1\,{}^1 B_{2g}$ states of the $D_{4h}$ geometry.
It is important to mention that the $2\,{}^1A_{1g}$ state of the rectangular geometry has a significant double excitation character, \cite{Loos_2019} and is then hardly described by second-order methods [such as CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} or EOM-CCSD \cite{Koch_1990,Stanton_1993,Koch_1994}] and remains a real challenge for third-order methods [as, for example, ADC(3), \cite{Trofimov_2002,Harbach_2014,Dreuw_2015} CC3, \cite{Christiansen_1995b} or EOM-CCSDT \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}]. It is important to mention that the $2\,{}^1A_{1g}$ state of the rectangular geometry has a significant double excitation character, \cite{Loos_2019} and is then hardly described by second-order methods [such as CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} or EOM-CCSD \cite{Koch_1990,Stanton_1993,Koch_1994}] and remains a real challenge for third-order methods [as, for example, ADC(3), \cite{Trofimov_2002,Harbach_2014,Dreuw_2015} CC3, \cite{Christiansen_1995b} or EOM-CCSDT \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}].
Comparing the present SF-BSE@{\GOWO} results for the rectangular geometry (see Table \ref{tab:CBD_D2h}) to the most accurate ADC level, \ie, SF-ADC(3), we have a difference in the excitation energy of $0.017$ eV for the $1\,^3B_{1g}$ state. Comparing the present SF-BSE@{\GOWO} results for the rectangular geometry (see Table \ref{tab:CBD_D2h}) to the most accurate ADC level, \ie, SF-ADC(3), we have a difference in excitation energy of $0.017$ eV for the $1\,^3B_{1g}$ state.
This difference grows to $0.572$ eV for the $1\,^1B_{1g}$ state and then shrinks to $0.212$ eV for the $2\,^1A_{g}$ state. This difference grows to $0.572$ eV for the $1\,^1B_{1g}$ state and then shrinks to $0.212$ eV for the $2\,^1A_{g}$ state.
Overall, adding dynamical corrections via the SF-dBSE@{\GOWO} scheme does not improve the accuracy of the excitation energies [as compared to SF-ADC(3)] with errors of $0.052$, $0.393$, and $0.293$ eV for the $1\,^3B_{1g}$, $1\,^1B_{1g}$, and $2\,^1 A_{g}$ states, respectively. Overall, adding dynamical corrections via the SF-dBSE@{\GOWO} scheme does not improve the accuracy of the excitation energies [as compared to SF-ADC(3)] with errors of $0.052$, $0.393$, and $0.293$ eV for the $1\,^3B_{1g}$, $1\,^1B_{1g}$, and $2\,^1 A_{g}$ states, respectively.
Now, looking at Table \ref{tab:CBD_D4h} which gathers the results for the square-planar geometry, we see that, at the SF-BSE@{\GOWO} level, the first two states are wrongly ordered with the triplet $1\,^3B_{1g}$ state lower than the singlet $1\,^1A_g$ state. Now, looking at Table \ref{tab:CBD_D4h} which gathers the results for the square-planar geometry, we see that, at the SF-BSE@{\GOWO} level, the first two states are wrongly ordered with the triplet $1\,^3B_{1g}$ state lower than the singlet $1\,^1A_g$ state.
(The same observation can be made at the SF-TD-B3LYP level.) (The same observation can be made at the SF-TD-B3LYP level.)
This is certainly due to the poor Hartree-Fock reference which lacks opposite-spin correlation and it could be potentially alleviated by using a better starting point for the $GW$ calculation, as discussed in Sec.~\ref{sec:compdet}. This is certainly due to the poor Hartree-Fock reference which lacks opposite-spin correlation and this issue could be potentially alleviated by using a better starting point for the $GW$ calculation, as discussed in Sec.~\ref{sec:compdet}.
Nonetheless, it is pleasing to see that adding the dynamical correction in SF-dBSE@{\GOWO} not only improves the agreement with SF-ADC(3) but also retrieves the right state ordering. Nonetheless, it is pleasing to see that adding the dynamical correction in SF-dBSE@{\GOWO} not only improves the agreement with SF-ADC(3) but also retrieves the right state ordering.
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.
@ -767,7 +766,7 @@ Then, CBD stands as an excellent example for which dynamical corrections are nec
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%%
%%% TABLE ?? %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{ \caption{
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state. Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
@ -798,7 +797,7 @@ Then, CBD stands as an excellent example for which dynamical corrections are nec
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE ?? %%% %%% TABLE III %%%
\begin{table} \begin{table}
\caption{ \caption{
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state. Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
@ -833,7 +832,7 @@ Then, CBD stands as an excellent example for which dynamical corrections are nec
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \label{sec:ccl}
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\alert{Here comes the conclusion.}
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\acknowledgements{ \acknowledgements{
@ -841,6 +840,11 @@ We would like to thank Xavier Blase and Denis Jacquemin for insightful discussio
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).} This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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\section*{Supporting information available}
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Additional graphs comparing (SF-)TD-BLYP, (SF-)TD-B3LYP, and (SF-)dBSE with EOM-CCSD for the \ce{H2} molecule, raw data associated with Fig.~\ref{fig:H2}, output files associated with all the calculations performed in the present article.
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%\section*{Data availability statement} %\section*{Data availability statement}
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