minor corrections in results
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@ -747,14 +747,14 @@ For a given single excitation $m$, the explicit expressions of $\Delta \expval{\
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\section{Computational details}
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\section{Computational details}
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\label{sec:compdet}
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\label{sec:compdet}
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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.
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All the systems under investigation here have a closed-shell singlet ground state and we consider the lowest triplet state as reference for the spin-flip calculations adopting the unrestricted formalism throughout this work.
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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}].
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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 Hartree-Fock (UHF) starting point, and the {\GOWO} quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
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Note that the entire set of orbitals and energies is corrected.
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Note that the entire set of orbitals and energies is corrected.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h,Berger_2021}.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h,Berger_2021}.
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Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} excitation energies.
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Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} excitation energies.
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This is left for future work.
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This is left for future work.
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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}
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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}
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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$. \cite{Loos_2020e,Loos_2020h,Berger_2021}
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Besides, {\GOWO}@UHF and ev$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}
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In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
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In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
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Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
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Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
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@ -789,9 +789,9 @@ However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state
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For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved.
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For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved.
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The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) alongside the SF-CIS excitation energies (purple lines).
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The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) alongside the SF-CIS excitation energies (purple lines).
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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.
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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.
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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.
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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.
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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.
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At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effects included in the SF-dBSE scheme.
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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.
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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.
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The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
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The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
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@ -856,26 +856,26 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
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Our second example deals with the dissociation of the \ce{H2} molecule, which is a prototypical system for testing new electronic structure methods and, specifically, their accuracy in the presence of strong correlation (see, for example, Refs.~\onlinecite{Caruso_2013,Barca_2014,Vuckovic_2017,Li_2021}, and references therein).
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Our second example deals with the dissociation of the \ce{H2} molecule, which is a prototypical system for testing new electronic structure methods and, specifically, their accuracy in the presence of strong correlation (see, for example, Refs.~\onlinecite{Caruso_2013,Barca_2014,Vuckovic_2017,Li_2021}, and references therein).
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The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $(1\sigma_g)^2$ configuration.
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The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $(1\sigma_g)^2$ configuration.
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The variation of the excitation energies associated with the three lowest singlet excited states with respect to the elongation of the \ce{H-H} bond are of particular interest here.
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The variation of the excitation energies associated with the three lowest singlet excited states with respect to the elongation of the \ce{H-H} bond are of particular interest here.
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The lowest singly excited state $\text{B}\,{}^1 \Sigma_u^+$ has a $(1\sigma_g )(1\sigma_u)$ configuration, while the singly excited state $\text{E}\,{}^1 \Sigma_g^+$ and the doubly excited state $\text{F}\,{}^1 \Sigma_g^+$ have $(1\sigma_g ) (2\sigma_g)$ and $(1\sigma_u )(1\sigma_u)$ configurations, respectively.
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The lowest singly excited state $\text{B}\,{}^1 \Sigma_u^+$ has a $(1\sigma_g )(1\sigma_u)$ configuration, while the singly excited state $\text{E}\,{}^1 \Sigma_g^+$ and the doubly excited state $\text{F}\,{}^1 \Sigma_g^+$ have $(1\sigma_g ) (2\sigma_g)$ and $(1\sigma_u )^2$ configurations, respectively.
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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.
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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.
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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.
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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.
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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}).
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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}).
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All these calculations are performed with the cc-pVQZ basis.
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All these calculations are performed with the cc-pVQZ basis.
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The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$.
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The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as functions of $R(\ce{H-H})$.
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The EOM-CCSD reference energies are represented by solid lines.
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The EOM-CCSD reference energies are represented by solid lines.
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We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state in the dissociation limit with an error greater than $1$ eV, while CIS, unlike SF-CIS, is much more accurate around the equilibrium geometry.
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We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state in the dissociation limit with an error greater than $1$ eV, while CIS, unlike SF-CIS, is much more accurate around the equilibrium geometry.
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Similar observations can be made for the $\text{E}\,{}^1\Sigma_g^+$ state with a good description at the CIS level for all bond lengths.
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Similar observations can be made for the $\text{E}\,{}^1\Sigma_g^+$ state with a good description at the CIS level for all bond lengths.
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SF-CIS does not model accurately the $\text{E}\,{}^1\Sigma_g^+$ state before the avoided crossing, but the agreement between SF-CIS and EOM-CCSD is much satisfactory for bond length greater than $1.6$ \AA.
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SF-CIS does not model accurately the $\text{E}\,{}^1\Sigma_g^+$ state before the avoided crossing, but the agreement between SF-CIS and EOM-CCSD is much satisfactory for bond length greater than $1.6$ \AA.
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Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after, while this state is completely absent at the CIS level.
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Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after, while this state is completely absent at the CIS level.
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Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
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Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
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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.
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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 [$R(\ce{H-H}) \approx 1.5$ \AA] than at the EOM-CCSD level.
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In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
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In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
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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.
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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.
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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}
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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}
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Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions.
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Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions.
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Notably, one can see that the avoided crossing is not modeled at the SF-TD-BLYP level due to the lack of Hartree-Fock exchange.
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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.
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In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented.
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In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented.
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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.
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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.
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@ -886,7 +886,7 @@ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be foun
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The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom).
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The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom).
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Overall, one can see that $\expval{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
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Overall, one can see that $\expval{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
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For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval{\hS^2}$ values associated with the $\text{F}$ state
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For all spin-flip methods, the $\text{E}$ state is strongly spin contaminated as expected, while the $\expval{\hS^2}$ values associated with the $\text{F}$ state
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only deviate significantly from zero for short bond length and around the avoided crossing where it strongly couples with the spin contaminated $\text{E}$ state.
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only deviate significantly from zero for short bond length and around the avoided crossing where it strongly couples with the spin contaminated $\text{E}$ state.
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%%% FIG 2 %%%
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%%% FIG 2 %%%
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@ -921,7 +921,7 @@ only deviate significantly from zero for short bond length and around the avoide
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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}
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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}
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%with potential large spin contamination.
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%with potential large spin contamination.
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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.
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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.
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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).
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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).
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In this case, single-reference methods notoriously fail.
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In this case, single-reference methods notoriously fail.
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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.
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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.
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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.
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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.
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@ -1034,8 +1034,8 @@ In this article, we have presented the extension of the BSE approach of many-bod
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The present spin-flip calculations rely on a spin-unrestricted version of the $GW$ approximation and the BSE formalism with, on top of this, a dynamical correction to the static BSE optical excitations via an unrestricted generalization of our recently developed renormalized perturbative treatment.
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The present spin-flip calculations rely on a spin-unrestricted version of the $GW$ approximation and the BSE formalism with, on top of this, a dynamical correction to the static BSE optical excitations via an unrestricted generalization of our recently developed renormalized perturbative treatment.
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Taking the beryllium atom, the dissociation of the hydrogen molecule, and cyclobutadiene in two different geometries as examples, we have shown that the spin-flip BSE formalism can accurately model double excitations and seems to surpass systematically its spin-flip TD-DFT parent.
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Taking the beryllium atom, the dissociation of the hydrogen molecule, and cyclobutadiene in two different geometries as examples, we have shown that the spin-flip BSE formalism can accurately model double excitations and seems to surpass systematically its spin-flip TD-DFT parent.
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Further improvements could be obtained thanks to a better choice of the starting orbitals and their energies and we hope to investigate this in a forthcoming paper.
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Further improvements could be obtained thanks to a better choice of the starting orbitals and their energies and we hope to investigate this in a forthcoming paper.
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Techniques to alleviate the spin contamination in spin-flip calculations will also be explored in the near future.
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Techniques to alleviate the spin contamination in spin-flip BSE will also be explored in the near future.
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We hope to these new encouraging results will stimulate new developments around the BSE formalism to further establish it as a valuable \text{ab inito} alternative to TD-DFT for the study of molecular excited states.
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We hope to these new encouraging results will stimulate new developments around the BSE formalism to further establish it as a valuable \textit{ab inito} alternative to TD-DFT for the study of molecular excited states.
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\acknowledgements{
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\acknowledgements{
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