minor corrections in results

Manuscript/sfBSE.tex

@@ -747,14 +747,14 @@ For a given single excitation $m$, the explicit expressions of $\Delta \expval{\
 \section{Computational details}
 \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.
-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}].
+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.
+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}].
 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,Berger_2021}.
 
 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, 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, 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}
 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.
@@ -789,9 +789,9 @@ However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state
 For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved. 
 
 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 static and dynamical screening, respectively.
 Overall, the SF-CIS and SF-BSE excitation energies are closer to FCI than the SF-TD-DFT ones, except for the lowest triplet state where the SF-TD-BH\&HLYP excitation energy is more accurate probably due to error compensation.
-At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect included in the SF-dBSE scheme.
+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.
 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.
@@ -856,26 +856,26 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
 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).
 The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $(1\sigma_g)^2$ configuration. 
 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. 
-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. 
+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. 
 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.
 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 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 functions of $R(\ce{H-H})$.
 The EOM-CCSD reference energies are represented by solid lines.
 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.
 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.
 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. 
 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.
 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 [$R(\ce{H-H}) \approx 1.5$ \AA] than at the EOM-CCSD level.
 
 In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
 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{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 avoided crossing is not modeled 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.
 
 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.
@@ -886,7 +886,7 @@ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be foun
 
 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).
 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.
-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 
+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 
 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.
 
 %%% FIG 2 %%%
@@ -921,7 +921,7 @@ only deviate significantly from zero for short bond length and around the avoide
 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.
 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 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).
+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.
 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.
@@ -1034,8 +1034,8 @@ In this article, we have presented the extension of the BSE approach of many-bod
 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.
 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.
 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.
-Techniques to alleviate the spin contamination in spin-flip calculations will also be explored in the near future.
-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.
+Techniques to alleviate the spin contamination in spin-flip BSE will also be explored in the near future.
+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{