\documentclass[10pt]{letter}
\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}


\begin{document}

\begin{letter}%
{To the Editors of the Journal of Chemical Physics}

\opening{Dear Editors,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
\textit{``Dynamical Kernels for Optical Excitations''}.
\end{quote}
We thank the reviewers for their constructive comments.
Our detailed responses to their comments can be found below.
For convenience, changes are highlighted in red in the revised version of the manuscript. 

We look forward to hearing from you.

\closing{Sincerely, the authors.}

%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}

\begin{itemize}

	\item 
	{This paper presents a study of three different frequency-dependent kernels proposed in the literature that go beyond the static/adiabatic approximations of TDDFT and GW/BSE approaches to excitations. The presentation is very instructive and coherent, beginning by showing a universal aspect of these kernels (the context of Lowdin partitioning), before examining the performance of each on three simple molecular systems in a minimal basis. Although some of the main points discussed may not be new to experts in the field, the coherent placement of these issues together and clear explanations will be useful to the community. I am likely to recommend publication in JCP, once the following points are addressed. }
	\\
	\alert{}

	\item 
	{(1) The systems chosen are said to represent examples of valence, charge-transfer, and Rydberg excited states. Some explanation is required for why they represent charge-transfer and Rydberg excited states. For HeH+ example, the internuclear separation is taken to be near equilibrium, if I understand it correctly. What then is the charge-transfer character of the excited states, i.e. how much of a significant change in the charge distribution do the excited states have compared to the ground state? For the He example, if I recall correctly, the lowest double-excitation of He lies in the continuum, i.e. is not a bound state but rather a resonance. So I am not sure whether it is really accurate to say it is a Rydberg excited state. Did the authors check if the excitation energies they are obtaining for this state lie below the ionization threshold? Perhaps the finite basis set makes this resonance appear bound. }
	\\
	\alert{}
	
	\item 
	{(2) In the introduction it is stated that only the correlation part of the kernel is frequency-dependent. However I believe this is only true for two-electron systems. For the N electron case the exact-exchange kernel defined within TDDFT is frequency-dependent (see e.g. Hellgren and Gross, PRA 88, 052507 (2013) and references therein, Hesselmann and Goerling JCP 134, 034120 (2011)) }
	\\
	\alert{}
	
	\item 
	{(3) The authors comment that the D-TDDFT kernel (Eq 14) "is known to work best in the weak correlation regime where the true excitations have a clear single and double excitation character" (third last para end of sec IIIB). I found this a bit confusing. To me, weak correlation usually means that the ground-state is well-described by a single Slater determinant, so there are no very low-lying excitations. In this situation excitations of the system can have quite a mixed character, i.e. they need not be largely purely single excitations or largely purely double excitations, but could be, for example 50:50 mixtures of a single and double excitation. Perhaps the language just needs to be clarified in the sentence to reflect that what they mean is that one can define/quantify the single/double excitation character of the state? }
	\\
	\alert{}
	
	\item 
	{(4) In that same paragraph, when discussing the accuracy of the method for the excitations, it might be worth noting that in the cited reference 12 (Huix-Rotllant et al), it was observed that the best results seem to arise when using a hybrid kernel for the "static" part. }
	\\
	\alert{}
	
	\item 
	{Minor typos: 
Sec IIIB: "...same idea was taking further..."- taking should be taken 
Sec IV: "factitious" should be fictitious }
	\\
	\alert{}
 

\end{itemize}

%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}

\begin{itemize}
	
	\item 
	{The authors compare the performance of different ab initio methods to simulate the linear response
spectrum of systems of two interacting electrons, in particular their ability to describe double excitations
and accuracy to predict correct frequency of the single excitations. The three approaches under study use
a dynamical (frequency-dependent) kernel to compute the linear response, since as it is already known an
adiabatic exchange-correlation kernel can only reproduce single excitations.
The poles of the response function are computed using a matrix formulation within the 4-dimensional
vector space that describes a two-level system consisting of one valence and one conduction orbital
(HOMO and LUMO). Transition energies for H2, HeH+ and He using the dressed TDDFT as well as two
different dynamical kernels derived from BSE (including a perturbative treatment) are compared against
the exact solution.
The fact that dynamical(frequency-dependent) xc kernels are able to generate new poles in the linear
response was already observed in previous works which are cited in the manuscript. The authors revisit
this idea and test its effectiveness for different approximations to the dynamical kernel. The study is of
interest to the quantum chemistry community and the TDDFT community because it provides a careful
and useful comparison between different available dynamical kernels for systems for which the exact
spectrum can be computed numerically and used as reference. The manuscript provides a compact
introduction to each of the methods with references to relevant literature on the topic.}
	\\
	\alert{}

	\item 
	{1. The particularities of the chosen model systems should be better discussed. Since the compared
methods are approximations their performance is likely to depend on the system under study as
well as on the basis set.
For example the dressed TDDFT method is suppose to work best in the case of doubles strongly
coupled to single excitations which from the tables seems not to be the case in these systems
(w1updown and w2updown are not very close in energy) so it's rather remarkable how well
dressed TDDFT predicts the energy of w2updown.}
	\\
	\alert{}
	
	\item 
	{2. No explanation or reference is provided accompanying the statement that the model systems that
are chosen are prototypical of valence, charge-transfer and Rydberg excitations and weather these
different excitations can be well represented in the reduced (two-level) space used.}
	\\
	\alert{}
	
	\item 
	{3. The molecules H2 and HeH+ are studied at the equilibrium distance, right? It would be
interesting to assess the performance of the different dynamical kernel for stretched molecules.
If you claim the problem is an example of charge-transfer excitations then it would be good to
study the stretched molecules...
Would a larger basis set be needed in such case?}
	\\
	\alert{}
	
	\item 
	{4. The statement about the vanishing of the matrix element $<S|H|D>=0$ in H2 could be discussed
better or a reference added.}
	\\
	\alert{}
	
	\item 
	{5. The results for He are not discussed at all. Why is the matrix element $<S|H|D>=0$ vanishing for
H2 but not for He? Both systems share the same spatial symmetry (parity) ...
Can you really study Rydberg excitations using a minimal basis?}
	\\
	\alert{}
	
	\item 
	{6. It is also not discussed how the exact solution is computed. What is the exact Hamiltonian? Is the
eigensystem computed within the same minimal basis set? How are the excitation energies that
appear as 'exact' in the tables computed?}
	\\
	\alert{}
	
	\item 
	{7. It would be illustrative to represent the ground state and the single and double excitations within a
single-particle picture for each system (H2, HeH+,He), maybe with a little figure. For example for
HeH+ is the HOMO consisting of 1 e up and 1 e down (one electron closer to H+ the other
closer to He?) and then a single exc would be promoting one of these electrons to the LUMO and
the double promoting both electrons to the LUMO? That means that within the minimal basis of 2
orbitals both electrons have to do always the same, i.e. you can not describe the excitation of one
electron to the LUMO whereas the other one stays in the HOMO ? (of course this picture only
makes sense in the limit of weak interaction between particles)}
	\\
	\alert{}

	\item 
	{8. In order to address a larger community it would desirable to discuss briefly which orbitals are taking into account for each system when using the minimal basis. 
	Maybe that helps understand the results. 
	What is the difference between using STO-3G and Pople's 6-31G and why is one or the
other chosen?}
	\\
	\alert{}
	
	\item 
	{9. How does the dimension of the basis set affect the performance of the different dynamical kernel
under study?}
	\\
	\alert{}
	
	\item 
	{10. page 2, beginning of section A:
?two level quantum system made of 2 electrons in its singlet gs? is a bit misleading. The vector
space is 4-dimensional and if you place 2 electrons with opposite spin then the gs, which is a
singlet, consists of both electrons occupying the lowest orbital.}
	\\
	\alert{}
	
	\item 
	{11. In page 2, end of second paragraph: when you say 'same numerical examples' do you mean the
same systems (H2, HeH+, He)?}
	\\
	\alert{}
	
	\item 
	{Beginning of section B, 'one single and one double excitation'}
	\\
	\alert{}
	
	\item 
	{section B, by 'static problem' do you mean 'static kernel'?}
	\\
	\alert{}
 	
	\item 
	{Table I has different unities than the rest of the tables, which is a bit confusing. Why didn't you
use the same unities everywhere? Is the column called 'exact' in tables 2,3,4 computed from the
values given in table I? It is not just $n*(ev-ec)$ ($n=1$ for single, $n=2$ for double) since the single and
doubles are not a multiple of each other and also that would be $E_{HF}$',right? so how are the exact
transition energies computed? Do they correspond to: omega1 up down= <S|H|S> - <0|H|0> and
omega2 up down= <D|H|D> - <0|H|0> ?}
	\\
	\alert{}
	
	\item 
	{15. page 5, second para:
Do you mean by 'static excitations' w1updown and w1upup?}
	\\
	\alert{}
	

\end{itemize}
 
\end{letter}
\end{document}






