From c1d637bbee045fca08d0337e5d316bae11c71041 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 26 Aug 2020 16:57:38 +0200 Subject: [PATCH] response letter first draft and manuscript corrections --- BSEdyn.tex | 17 +++++----------- Response_Letter/Response_Letter.tex | 31 ++++++++++++++++++++--------- 2 files changed, 27 insertions(+), 21 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 8bc7927..743b00e 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -258,7 +258,7 @@ Double excitations play also a significant role in the correct location of the e In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet with dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019} Going beyond the static approximation is difficult and very few groups have been addressing the problem. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019} -Nonetheless, it is worth mentioning the seminal work of Strinati on core excitons in semiconductors, \cite{Strinati_1982,Strinati_1984,Strinati_1988} in which the dynamical screening effects were taken into account through the dielectric matrix, and where he observed an increase of the binding energy over its value for static screening and a narrowing of the Auger width below its value for a core hole. +Nonetheless, it is worth mentioning the seminal work of Strinati \titou{(who originally derived the dynamical correction to the BSE)} on core excitons in semiconductors, \cite{Strinati_1982,Strinati_1984,Strinati_1988} in which the dynamical screening effects were taken into account through the dielectric matrix, and where he observed an increase of the binding energy over its value for static screening and a narrowing of the Auger width below its value for a core hole. Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b} Indeed, studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pis$ transitions, but for the lowest $n \ra \pis$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b} @@ -715,15 +715,8 @@ This finally yields \end{equation} with $\Om{S}{\text{stat}} \equiv \Om{S}{(0)}$ and $\Delta\Om{S}{\text{dyn}} = Z_{S} \Om{S}{(1)}$. This is our final expression. - -%%% FIG 1 %%% -%\begin{figure} -% \includegraphics[width=\linewidth]{} -%\caption{ -%\label{fig:} -%} -%\end{figure} -%%% %%% %%% +\titou{As mentioned in Sec.~\ref{sec:intro}, the present perturbative scheme does not allow to access double excitations as one loses the dynamical nature of the screening by applying perturbation theory. +We hope to report a genuine dynamical treatment of the BSE in a forthcoming work.} In terms of computational cost, if one decides to compute the dynamical correction of the $M$ lowest excitation energies, one must perform, first, a conventional (static) BSE calculation and extract the $M$ lowest eigenvalues and their corresponding eigenvectors [see Eq.~\eqref{eq:LR-BSE-stat}]. These are then used to compute the first-order correction from Eq.~\eqref{eq:Om1-TDA}, which also require to construct and evaluate the dynamical part of the BSE Hamiltonian for each excitation one wants to dynamically correct. @@ -1146,12 +1139,12 @@ The BSE formalism is quickly gaining momentum in the electronic structure commun It now stands as a genuine cost-effective excited-state method and is regarded as a valuable alternative to the popular TD-DFT method. However, the vast majority of the BSE calculations are performed within the static approximation in which, in complete analogy with the ubiquitous adiabatic approximation in TD-DFT, the dynamical BSE kernel is replaced by its static limit. One key consequence of this static approximation is the absence of higher excitations from the BSE optical spectrum. -Following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored the BSE formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the screened Coulomb potential \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach coupled with the plasmon-pole approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} +Following Strinati's footsteps \titou{who originally derived the dynamical correction to the BSE}, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored the BSE formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the screened Coulomb potential \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach coupled with the plasmon-pole approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} In the present study, we have computed exactly the dynamical screening of the Coulomb interaction within the random-phase approximation, going effectively beyond both the usual static approximation and the plasmon-pole approximation. \titou{Dynamical corrections have been calculated using a renormalized first-order perturbative correction to the static BSE excitation energies following the work of Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} Note that, although the present study goes beyond the static approximation of BSE, we do not recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit. -However, we hope to report results on a genuinely dynamical approach in the near future.} +However, we hope to report results on a genuinely dynamical approach in the near future in order to access double excitations within the BSE formalism.} In order to assess the accuracy of the present scheme, we have reported a significant number of calculations for various molecular systems. Our calculations have been benchmarked against high-level CC calculations, allowing to clearly evidence the systematic improvements brought by the dynamical correction for both singlet and triplet excited states. We have found that, although $n \ra \pis$ and $\pi \ra \pis$ transitions are systematically red-shifted by $0.3$--$0.6$ eV thanks to dynamical effects, their magnitude is much smaller for CT and Rydberg states. diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex index 51909f5..a47be3c 100644 --- a/Response_Letter/Response_Letter.tex +++ b/Response_Letter/Response_Letter.tex @@ -82,7 +82,9 @@ be helpful to have all this in one paragraph. } ``It is important to note that, although all the studies mentioned above are clearly going beyond the static approximation of BSE, they are not able to recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit. However, it does permit to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations.'' To answer the reviewer's comment we have added a new paragraph to the concluding section, which reads: ``Dynamical corrections have been calculated using a renormalized first-order perturbative correction to the static BSE excitation energies following the work of Rohlfing and coworkers. Note that, although the present study goes beyond the static approximation of BSE, we do not recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit. -However, we hope to report results on a genuinely dynamical approach in the near future.''} +However, we hope to report results on a genuinely dynamical approach in the near future in order to access double excitations within the BSE formalism.'' +We have also added in Sec.~II.E. the following statement: ``As mentioned in Sec.~I, the present perturbative scheme does not allow to access double excitations as one loses the dynamical nature of the screening by applying perturbation theory. +We hope to report a genuine dynamical treatment of the BSE in a forthcoming work.''} \item {The authors emphasize that their $W$ is evaluated within RPA for @@ -131,14 +133,16 @@ same reason. The work is rigorous, carefully done and well presented. The dynamical corrections can be quiet sizeable and improve the BSE results considerably. With the full screened Coulomb interaction, the corrections become larger than previously reported for the plasmon pole model. Still, the dynamical corrections do not add significant computational effort, which makes the BSE scheme highly competitive for optical excitations of weakly to moderately correlated molecules. The methodology and the results will be highly interesting for the core readership of The Journal of Chemical Physics and I recommend publication after the small comments below have been addressed.} \\ - \alert{} + \alert{We would like to thank the reviewer for his/her kinds comments and recommending publication of the present manuscript.} \item {Throughout the manuscript I kept wondering, if the dynamical corrections had already been derived before. This is not particularly clear and I encourage the authors to clarify this. For example, in Section E are equations 32 to 42 new or were they already derived by Rohlfing and others? } \\ - \alert{Originally, the dynamical correction has been derived by Strinati [see Ref.~(2) for a detailed derivation].} + \alert{Equations 32 to 42 aren't new. + Originally, the dynamical correction has been derived by Strinati [see Ref.~(2) for a detailed derivation]. + We have clarified this point in two places in the revised manuscript (Introduction and Conclusion).} \item {In the introduction, the authors advocate the dynamical BSE corrections for double excitations. @@ -149,16 +153,24 @@ The work is rigorous, carefully done and well presented. The dynamical correctio So I wonder, does the perturbative approach taken for the dynamical corrections provide access to double excitations? In eq. 42, which is the final expression, all quantities carry an S index. This implies that dynamical corrections are only calculated for states that are already part of the static solution and no new states can be found with this approach. What is the potential then for applying dynamical corrections to double excitations? This should be clarified. } \\ - \alert{} + \alert{This comment is similar to one of the comment of Reviewer \#1. + The present perturbative treatment cannot access additional excitations, and we are definitely interested in pursuing in this direction in the near future. + This is clearly stated in the original version of our manuscript near the end of the Introduction: + ``It is important to note that, although all the studies mentioned above are clearly going beyond the static approximation of BSE, they are not able to recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit. However, it does permit to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations.'' + Nonetheless, to make it extra clear, we have added a new paragraph to the concluding section, which reads: ``Dynamical corrections have been calculated using a renormalized first-order perturbative correction to the static BSE excitation energies following the work of Rohlfing and coworkers. +Note that, although the present study goes beyond the static approximation of BSE, we do not recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit. +However, we hope to report results on a genuinely dynamical approach in the near future in order to access double excitations within the BSE formalism.'' +We have also added in Sec.~II.E. the following statement: ``As mentioned in Sec.~I, the present perturbative scheme does not allow to access double excitations as one loses the dynamical nature of the screening by applying perturbation theory. +We hope to report a genuine dynamical treatment of the BSE in a forthcoming work.''} \item {Equation 41 contains a derivative of $A^{(1)}(\Omega_S)$ with respect to $\Omega_S$. Is this derivative easy to compute? Is $A^{(1)}(\Omega_S)$ available in analytic form or is the derivative performed numerically? } \\ - \alert{Yes, this derivative is straightforward to compute, and it is computed at no extra cost basically. - The situation is very similar to the computation of the derivative of the $GW$ self-energy $\Sigma$ which is involved in the calculation of the spectral weight $Z$. - In the revised version of the manuscript, we have mentioned that this derivative can be computed easily at no extra cost.} + \alert{Yes, this derivative is straightforward to compute, and it is computed at no extra cost basically (as mentioned in the original manuscript, see Sec.~II.E.). + The situation is very similar to the computation of the derivative of the $GW$ self-energy $\Sigma$ which is involved in the calculation of the spectral weight $Z$.} +% In the revised version of the manuscript, we have mentioned that this derivative can be computed easily at no extra cost.} \item {Following up on my previous point, why does equation 40 have to be renormalised? @@ -181,7 +193,7 @@ The work is rigorous, carefully done and well presented. The dynamical correctio These should also be visible in the optical transition energies. Such a conical intersection would be a good test for a theory that goes beyond standard BSE and can tackle more correlated systems, as recently demonstrated, for example, for dynamical configuration interaction (DCI) theory, which also includes GW and BSE elements (see e.g. M. Dvorak, D. Golze, and P. Rinke, Physical Review Materials 3, 070801(R) (2019)). } \\ - \alert{} + \alert{This is an interesting comment. Although outside the scope of the present study, we hope to be able to check the existence of this conical intersection in the near future while working with a fully dynamical scheme (i.e., nor perturbatively).} \item {At the end of the Results and Discussion section the authors present an equation for a two level model (that does not have an equation number) to estimate when dynamical corrections are large and when not. @@ -191,7 +203,8 @@ The work is rigorous, carefully done and well presented. The dynamical correctio Or are they more pronounced, if $W$ has more structure in its frequency dependence? } \\ - \alert{} + \alert{This equation now has a number. + Xavier this is for you.} \end{itemize}