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Pierre-Francois Loos 2021-02-24 13:45:49 +01:00
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12
Manuscript/sfBSE.tex
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@ -461,7 +461,7 @@ This is where $GW$ schemes differ.
In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} a single iteration is performed, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} a single iteration is performed, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation} \begin{equation}
\label{eq:QP-eq} \label{eq:QP-eq}
\omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc} \omega = \eKS{p_\sig} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc}
\end{equation} \end{equation}
where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the KS level, and where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the KS level, and
\begin{equation} \begin{equation}
@ -598,9 +598,9 @@ However, our scheme goes beyond the plasmon-pole approximation as the spectral r
\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} \widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} + \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
\\ \\
\times \Bigg[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \times \Bigg[ \frac{1}{\omega - (\eGW{s_\sigp}{} - \eGW{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta}
\\ \\
+ \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg] + \frac{1}{\omega - (\eGW{r_\sigp}{} - \eGW{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg]
\end{multline} \end{multline}
The dBSE non-linear response problem is The dBSE non-linear response problem is
\begin{multline} \begin{multline}
@ -817,13 +817,13 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
\hline \hline
SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013) \\ SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013) \\
SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006) \\ SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006) \\
SF-TD-CAM-B3LYP\fnm[1] & (0.001) & 3.186(1.960) & 4.554(0.043) & 7.020(1.000) & 7.933(0.008) \\ \alert{SF-TD-CAM-B3LYP}\fnm[1] & (0.001) & 3.186(1.960) & 4.554(0.043) & 7.020(1.000) & 7.933(0.008) \\
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@ev$GW$ & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\ \alert{SF-BSE@ev$GW$} & (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@ev$GW$ & & 2.369 & 6.273 & 7.820 & 9.441 \\ \alert{SF-dBSE@ev$GW$} & & 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 \\

14
Response_Letter/Response_Letter.tex
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@ -39,17 +39,25 @@ I recommend this manuscript for publication after the minor points addressed:}
\item \item
{Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?} {Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?}
\\ \\
\alert{bla bla bla} \alert{Following the excellent advice of Reviewer \#1, we have added data for the long-range corrected CAM-B3LYP functional.
We believe that it is a choice with the other functionals (BLYP, B3LYP, and BH\&HLYP) already included in the manuscript.
As one can see...}
\item \item
{Figure 1: The similarity between SF-dBSE and SF-ADC(2)-s is more than simply the results. I would say that the two formulations are equivalent, and should lead to the same results in Figure 1 if the authors would have used GW/SF-dBSE instead of G0W0/SF-dBSE. Could the authors add these results based on GW and comment?} {Figure 1: The similarity between SF-dBSE and SF-ADC(2)-s is more than simply the results. I would say that the two formulations are equivalent, and should lead to the same results in Figure 1 if the authors would have used GW/SF-dBSE instead of G0W0/SF-dBSE. Could the authors add these results based on GW and comment?}
\\ \\
\alert{bla bla bla} \alert{The reviewer is right to mention similarities between the SF-dBSE and SF-ADC(2)-s schemes.
However, they are not strictly identical as ADC(2) includes second-order exchange diagrams which are not present in SF-dBSE@$GW$, even in the case of more elaborate schemes like ev$GW$ and qs$GW$.
To illustrate this and accordingly to the reviewer's suggestion, we have added the partially self-consistent SF-dBSE@ev$GW$ results as well as the fully self-consistent SF-dBSE@qs$GW$ results. As one can see, in the case of Be, there is not much differences between these schemes and the original SF-dBSE@$G_0W_0$ which nicely illustrates that HF eigenstates are actually are an excellent starling point in this particular case.
A discussion around these points have been also included in the revised version of the manuscript.}
\item \item
{Figure 1: The only difference between SF-ADC(2)-s and SF-ADC(2)-x is that the energy difference in the dynamic part of the BSE equation is corrected to first order. The equivalent thing in SF-BSE would be to add in Eq. 30 the direct and exchange corrections in the orbital energy difference appearing in the denominator of the second term (i.e., similar to using the diagonal part of Eq. 29a and 29c in the orbital energy difference of Eq. 30). Could the authors verify that?} {Figure 1: The only difference between SF-ADC(2)-s and SF-ADC(2)-x is that the energy difference in the dynamic part of the BSE equation is corrected to first order. The equivalent thing in SF-BSE would be to add in Eq. 30 the direct and exchange corrections in the orbital energy difference appearing in the denominator of the second term (i.e., similar to using the diagonal part of Eq. 29a and 29c in the orbital energy difference of Eq. 30). Could the authors verify that?}
\\ \\
\alert{bla bla bla} \alert{We thank the reviewer for mentioning this interesting fact. We were not aware of this.
Actually, this is already the case in SF-dBSE; the eigenvalues differences in the denominator of the second of Eq. 30 are $GW$ quasiparticle energies.
The $GW$ superscripts were missing in the original manuscript and they have been added.
We have performed SF-dBSE@$G_0W_0$ calculations replacing the $GW$ quasiparticle energies by the HF energies in the denominator of Eq. (30) but it does not seem to alter much the results in the case of Be.}
\item \item
{Figure 2: Could the authors discuss the kink in G0W0/SF-BSE and G0W0/SF-dBSE (in supporting) appearing at around 1.2 Angstroms between $1\Sigma_g^+$ and $1\Sigma_u^+$. It is really puzzling. Is it due to the lack of self consistency in the G0W0 approximation? What does GW/SF-BSE gives in this case?} {Figure 2: Could the authors discuss the kink in G0W0/SF-BSE and G0W0/SF-dBSE (in supporting) appearing at around 1.2 Angstroms between $1\Sigma_g^+$ and $1\Sigma_u^+$. It is really puzzling. Is it due to the lack of self consistency in the G0W0 approximation? What does GW/SF-BSE gives in this case?}