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\begin{letter}%
{To the Editors of the Journal of Chemical Theory and Computation}

\opening{Dear Editors,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
\textit{``A Density-Based Basis-Set Incompleteness Correction for GW Methods''}.
\end{quote}
We would to 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 hope that you will agree that our manuscript is now suitable for publication in JCTC.

We look forward to hearing from you.

\closing{Sincerely, the authors.}

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

\begin{itemize}

	\item 
	{The authors present an exciting piece of work with what seems to be an efficient, and ``possibly  simpler'' than F12-RI, technique  to accelerate convergency with respect to basis set size in the case of GW quasiparticle energy calculations. The DFT-based correction towards the CBS limit was introduced in previous papers in the case of the total correlation energy. Considering the total energy as a functional of the one-body Green's function allows to bridge total energies and self-energy using functional derivative techniques. The results are rather impressive, demonstrating that corrected triple-zeta calculations are equivalent for small systems to quintuple-zeta ones. 
	The paper is thus clearly within the scope of the Journal of Chemical Theory and Computation, presents original work that may prove useful to a large community. The referee recommends publication provided that the authors seriously consider the following suggestions/questions.}
	\\
	\alert{We thank the reviewer for his/her support.
	We have taken all his/her comments into account and our response to these comments can be found below.}

	\item 
	{The main criticism as a reader is that all details of the construction of the total energy correction to the ``finite-size basis difference'' with respect to the CBS limit is absent from the paper (very short Section II-C).  The authors refer the reader to previous publications (mainly [57]) dealing with total energies in a CCSD(T) quantum chemistry wavefunction framework with which the Green's function community may not be very familiar with.  In particular the construction of a local range-separation parameter related to the diagonal of the ``effective'' 2-electron-operator-in-a-basis  ($W^{B}$) would deserve to be somehow explained in the present paper.}
	\\
	\alert{We have included a new subsection (Section II.C.) to include additional details about the present basis set correction.
	In particular, the construction of the range-separation function $\mu(\mathbf{r})$ is detailed as well as the corresponding effective two-electron operator $W(\mathbf{r}_1,\mathbf{r}_2)$.}
	We have also expanded Section II.D. to add more details about the short-range correlation functionals.
	Their corresponding potentials are reported in the Supporting Information.

	\item 
	{Following the previous question, and from a pragmatic point of view, what is needed as an input to construct this basis-set-incompleteness correction, namely this effective local potential of Eq. [31] ? Again the answer is present in equations 4-9 of Ref. [57] but could be summarised in the present paper and possibly simplified in the present case of a perturbation theory based on a input mono-determinental Kohn-Sham or HF description of the many-body wavefunction. This may also give an hint on the cost (scaling) and complexity of the approach. }
	\\
	\alert{As mentioned above, we now provide all the equations in the single determinant case to construct $\mu(\mathbf{r})$, the main ingredient (alongside the density) of the present short-range correlation functionals.
	The formal scaling of the present approach is now quickly discussed in Section III.
	}
	
	\item 
	{As a corollary to this comment, the referee is still surprised that one may build a ``universal'' correction, in a sens that the same correction would apply to any approximation to the self-energy (if the referee understands correctly ...) whatever the diagrams used. If this is a correct statement, this should be emphasised and probably better commented.}	
	\\
	\alert{This is indeed the case: the present basis-set correction can be apply to any self-energy.
	However, the basis-set correction only corrects for the basis-set errors and not for the self-energy errors within the basis set.
	We have clarified this point in the revised version of the manuscript (end of Section II.A).}	

	\item 
	{Minor: The referee is somehow surprised by the IP CCSD(T) values for cytosine and uracil in Table III which are noticeably much larger than the experiment, in contrast with the other nucleobases.  As a matter of fact the CCSD(T) values by Roca-Sanjuan et al (JCP 2006) agree reasonably with the values reported by the authors for adenine, guanine, thymine, but are completely off for cytosine and uracil. Could the authors check and potentially comment.}
	\\
	\alert{After double checking the  CCSD(T) IP values of cytosine and uracil, we can affirm that they have been correctly extracted from the work of Krause, Harding, and Klopper (see Table III).
	An important point is that these CCSD(T) IPs have been obtained with the def2-TZVPP basis set (with the geometries of the GW100 test set which have been optimized at the PBE/def2-QZVP level of theory), a basis set which is larger than the aug-cc-pVDZ basis considered by Roca-Sanjuan et al.
	Note also that they have been computed on CCSD/aug-cc-pVDZ structures.
	Consequently, we believe that the CCSD(T) computed by the Klopper group are superior as the basis set effect is usually dominant in the computation of such properties (the geometrical effects are usually rather small).
	Nonetheless, we now report both sets of values in Table III and comment on their differences in the main text.}
	
\end{itemize}

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

\begin{itemize}
	\item 
	{I enjoyed reading the manuscript and am of the opinion that it presents an important contribution to the field addressing one of the main bottlenecks that the GW approach is infamous for, the slow basis set convergence. 
	There are a few issues however that the authors should address}
	\\
	\alert{We thank the referee for these kinds comments.
	His/her comments are addressed below.}
	
	\item 
	{The authors discuss GW in depth in sections II.A and II.B. For me however the novelty in this paper is all about what is in section II.C. We are given references there but to me C should be extended to provide more information.}
	\\
	\alert{As already mentioned in the answer to Reviewer \#1, we have significantly extended this section in order to provide additional details about the present basis set correction. In particular, we provide the working equations to compute all the key quantities in the case of a single-determinant such as KS-DFT and HF.
	See the new Section II.C. and expanded Section II.D.}
	
	\item 
	{In Section III the authors mention that the infinitesimal eta is put to 0. 
	This is physically incorrect. 
	eta is a positive infinitesimal and cannot be just put to zero. 
	Numerically it has been shown that indeed the self energy becomes discontinuous by doing so. 
	This is the main reason for the low quality rating. }
	\\
	\alert{Physically, $\eta$ ensures the correct time ordering when passing from $(t-t')$ to $\omega$ by a Fourier transform.
	It is an infinitesimal which should be put to zero at the very end of the calculation.
	In practice, this is not possible as one would have to solve everything analytically. 
	Therefore, one has to make a choice: set it equal to zero from the start or set it equal to a small value.
	So physically, one could say that both are equally correct (or equally incorrect).
	As a numerical check, we have performed additional calculations with very small $\eta$ values and the corresponding quasiparticle energies are nearly identical.
	Moreover, we do not have any numerical issues in our calculations.}	
	\item 
	{Figures 1, and the corresponding figures in the supplementary are plotted on a linear scale of X. 
	Personally I think it is much more instructive to plot against X$^{-3}$, which will much clearer visualize convergence. }	
	\\
	\alert{As suggested by the referee, we have tried to plot the corresponding graphs against X$^{-3}$ but we believe that the graphs are nicer as it was before.
	Therefore, we would prefer to stick with the original graphs if it's OK.}	
	
	\item 
	{The comparison in table II is made against a CBS limit extrapolating local basis sets. 
	10.1021/acs.jctc.7b00952 also provides plane wave basis set extrapolated results for the gw100 set. 
	It may be interesting to include this in the comparison.}	
	\\
	\alert{The work of Govoni and Galli ({10.1021/acs.jctc.7b00952}) reports indeed CBS values with a plane wave basis.
	However, their data are computed with the PBE functional while Table I and II are computed with HF and PBE0 respectively.
	Therefore, these data are not directly comparable.
	However, we have included their values for the nucleobases (alongside other plane wave values obtained with VASP) in Table III and cited the corresponding references.}
	
	\item 
	{Finally, as a very general point I thinks paper reporting large amounts of data, where the amount of data in this paper for me clearly qualifies as large, should also provide the data in a machine readable format. a json, hdf5 of netcdf4 format would be a good standard, a csv would be minimal.}	
	\\
	\alert{Accordingly to the reviewer's comment, we now provide all the data in txt and json format in addition to the pdf file gathering all the supporting information.}\end{itemize}

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