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JCTC_revision/GW-srDFT.bib
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@ -1,13 +1,36 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-02 22:34:02 +0100
%% Created for Pierre-Francois Loos at 2019-12-05 10:37:28 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Roca-Sanjuan_2006,
Author = {Daniel {Roca-Sanjuan} and Mercedes Rubio and Manuela Merchan and Luis {Serrano-Andres}},
Date-Added = {2019-12-05 10:35:36 +0100},
Date-Modified = {2019-12-05 10:37:20 +0100},
Doi = {10.1063/1.2336217},
Journal = {J. Chem. Phys.},
Pages = {084302},
Title = {Ab Initio Determination of The Ionization Potentials of {{DNA}} And {{RNA}} Nucleobases},
Volume = {125},
Year = {2006}}
@article{Govoni_2018,
Author = {Marco Govoni and Giulia Galli},
Date-Added = {2019-12-05 10:18:07 +0100},
Date-Modified = {2019-12-05 10:19:09 +0100},
Doi = {10.1021/acs.jctc.7b00952},
Journal = {J. Chem. Theory Comput.},
Pages = {1895--1909},
Title = {GW100: Comparison of Methods and Accuracy of Results Obtained with the WEST Code},
Volume = {14},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00952}}
@article{Adler_2007,
Author = {T. B. Adler and G. Knizia and H.-J. Werner},
Date-Added = {2019-10-26 20:55:14 +0200},

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JCTC_revision/GW-srDFT.tex
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@ -16,6 +16,8 @@
}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e}}
\newcommand{\eg}{\textit{e.g}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
@ -227,7 +229,7 @@ Explicitly correlated F12 correction schemes have been derived for second-order
However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}.
Note that the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
Note that the present basis set correction can be straightforwardly applied to other properties (\eg, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
The paper is organized as follows.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
@ -273,7 +275,7 @@ We assume that there exists a functional $\Omega^\Bas[\G{}{\Bas}]$ of $\Ne$-repr
\label{eq:FBn}
\end{equation}
The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function.
For example, we can choose for $\Omega^\Bas[\G{}{}]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
For example, we can choose for $\Omega^\Bas[\G{}{}]$ a Klein-like energy functional (see, \eg, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
\begin{equation}
\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
\label{eq:OmegaB}
@ -310,7 +312,7 @@ It leads the following Dyson equation
(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
\label{eq:Dyson}
\end{equation}
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $\vne(\br{})$, \textit{i.e.},
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $\vne(\br{})$, \ie,
\begin{equation}
(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} - \vne(\br{}) + \lambda) \delta(\br{}-\br{}'),
\end{equation}
@ -328,7 +330,7 @@ and
\n{}{}(\br{}) = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} e^{i \omega 0^+} \G{}{}(\br{},\br{},\omega).
\end{equation}
The solution of the Dyson equation \eqref{eq:Dyson} gives the Green function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$.
Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \textit{i.e.},
Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \ie,
\begin{align}
\lim_{\Bas \to \CBS} \bSig{}{\Bas} & = 0,
&
@ -342,6 +344,9 @@ The Dyson equation \eqref{eq:Dyson} can also be written with an arbitrary refere
where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
\titou{Note that the present basis set correction is applicable to any approximation of the self-energy (irrespectively of the diagrams included) without altering the CBS limit of such methods.
Consequently, it can be applied, for example, to GF2 methods (also known as second Born approximation \cite{Stefanucci_2013} in the condensed matter community) or higher orders. \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018}
Note, however, that the basis set correction is optimal for the \textit{exact} self-energy.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The {\GW} Approximation}
@ -424,16 +429,46 @@ In particular, the ionization potential (IP) and electron affinity (EA) are extr
\end{align}
where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Short-range correlation functionals}
\label{sec:srDFT}
%%%%%%%%%%%%%%%%%%%%%%%%
The fundamental idea behind the present basis set correction is to recongnise that the two-electron interaction $\abs*{\br{} - \br{}'}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $r_{12} = 0$.
We can therefore define an effective two-electron interaction which ``mimicks'' the Coulomb operator in an incomplete basis, \ie,
\begin{equation}
\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
=
\iint \n{2}{\Bas}(\br{},\br{}') W(\br{},\br{}') d\br{} d\br{}'
\end{equation}
A convenient choice is, for example,
\begin{equation}
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2}
\begin{cases}
f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}), & \n{2}{\Bas}(\br{}) \neq 0, \\
\infty, & \text{otherwise} \\
\end{cases}
\end{equation}
where, in the case of a single-determinant method (such as HF and KS-DFT),
\begin{equation}
f(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^\text{occ} \MO{p}(\br{}) \MO{q}(\br{}) \braket{pq}{ij} \MO{i}(\br{}) \MO{j}(\br{})
\end{equation}
and
\begin{equation}
\n{2}{\Bas}(\br{}) = \frac{[\n{}{\Bas}(\br{})]^2}{4}
\end{equation}
is the opposite-spin on-top pair density.
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Short-range correlation functionals}
\label{sec:srDFT}
%%%%%%%%%%%%%%%%%%%%%%%%
The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior~\cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals require a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial inhomogeneity of the basis-set incompleteness error and is computed using the HF or KS opposite-spin pair-density matrix in the basis set $\Bas$.
We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
The basis set corrected {\GOWO} quasiparticle energies are thus given by
\begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
@ -546,7 +581,7 @@ Unless otherwise stated, all the {\GOWO} calculations have been performed with t
The HF, PBE, and PBE0 calculations as well as the srLDA and srPBE basis set corrections have been computed with Quantum Package, \cite{QP2} which by default uses the SG-2 quadrature grid for the numerical integrations.
Frozen-core (FC) calculations are systematically performed.
The FC density-based basis set correction~\cite{Loos_2019} is used consistently with the FC approximation in the {\GOWO} calculations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \ie, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
@ -560,7 +595,7 @@ Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significa
\caption{
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares), and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dunning_1989} (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
The thick black line represents the CBS value obtained by extrapolation (see text for more details).
The green area corresponds to chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
The green area corresponds to chemical accuracy (\ie, error below $1$ {\kcal} or $0.043$ eV).
\label{fig:IP_G0W0_H2O}
}
\end{figure*}
@ -588,7 +623,7 @@ Additional graphs reporting the convergence of the IPs of each molecule of the G
Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (as well as Fig.~\ref{fig:IP_G0W0_H2O}) clearly evidence that the present basis set correction significantly increases the rate of convergence of IPs.
At the {\GOWO}@{\HF} (see Table \ref{tab:GW20_HF}), the MAD of the conventional calculations (\textit{i.e}, without basis set correction) is roughly divided by two each time one increases the basis set size (MADs of $0.60$, $0.24$, $0.10$, and $0.05$ eV going from cc-pVDZ to cc-pV5Z) with maximum errors higher than $1$ eV for molecules such as \ce{HF}, \ce{H2O}, and \ce{LiF} with the smallest basis set.
Even with the largest quintuple-$\zeta$ basis, the MAD is still above chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
Even with the largest quintuple-$\zeta$ basis, the MAD is still above chemical accuracy (\ie, error below $1$ {\kcal} or $0.043$ eV).
For each basis set, the correction brought by the short-range correlation functionals reduces by roughly half or more the MAD, RMSD, and MAX compared to the correction-free calculations.
For example, we obtain MADs of $0.27$, $0.12$, $0.04$, and $0.01$ eV at the {\GOWO}@HF+srPBE level with increasingly larger basis sets.
@ -613,7 +648,7 @@ The potential reasons for this could be: i) potential-based DFT corrections are
IPs (in eV) of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The extrapolation error is reported in parenthesis.
The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purposes.
\titou{Extrapolated {\GOWO}@PBE results obtained with plane wave basis sets, as well as CCSD(T) and experimental results are reported for comparison purposes.}
\label{tab:DNA_IP}
}
\begin{ruledtabular}
@ -631,15 +666,23 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
{\GOWO}@PBE\fnm[2] & def2-QZVP & $7.98$[$-0.18$] & $8.29$[$-0.16$] & $7.69$[$-0.18$] & $8.71$[$-0.16$] & $9.22$[$-0.16$] \\
{\GOWO}@PBE\fnm[3] & def2-TQZVP & $8.16(1)$ & $8.44(1)$ & $7.87(1)$ & $8.87(1)$ & $9.38(1)$ \\
\hline
CCSD(T)\fnm[4] & def2-TZVPP & $8.33$ & $9.51$ & $8.03$ & $9.08$ & $10.13$ \\
Experiment\fnm[5] & & $8.48$ & $8.94$ & $8.24$ & $9.20$ & $9.68$ \\
{\GOWO}@PBE\fnm[4] & plane waves & $8.12$ & $8.40$ & $7.85$ & $8.83$ & $9.36$ \\
{\GOWO}@PBE\fnm[5] & plane waves & $8.09(2)$ & $8.40(2)$ & $7.82(2)$ & $8.82(2)$ & $9.19(2)$ \\
\hline
CCSD(T)\fnm[6] & aug-cc-pVDZ & $8.40$ & $8.76$ & $8.09$ & $9.04$ & $9.43$ \\
CCSD(T)\fnm[7] & def2-TZVPP & $8.33$ & $9.51$ & $8.03$ & $9.08$ & $10.13$ \\
\hline
Experiment\fnm[8] & & $8.48$ & $8.94$ & $8.24$ & $9.20$ & $9.68$ \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{This work.}
\fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com} obtained with TURBOMOLE v7.0.}
\fnt[3]{Extrapolated values obtained from the def2-TZVP and def2-QZVP values.}
\fnt[4]{Reference \onlinecite{Krause_2015}.}
\fnt[5]{Experimental values are taken from Ref.~\onlinecite{vanSetten_2015} and correspond to vertical ionization energies.}
\fnt[4]{\titou{Extrapolated plane wave results from Ref.~\onlinecite{Maggio_2017} obtained with WEST.}}
\fnt[5]{\titou{Extrapolated plane wave results from Ref.~\onlinecite{Govoni_2018} obtained with VASP.}}
\fnt[6]{\titou{CCSD(T)//CCSD/aug-cc-pVDZ results from Ref.~\onlinecite{Roca-Sanjuan_2006}.}}
\fnt[7]{Reference \onlinecite{Krause_2015}.}
\fnt[8]{Experimental values are taken from Ref.~\onlinecite{vanSetten_2015} and correspond to vertical ionization energies.}
\end{table*}
%%% FIG 2 %%%

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@ -1,5 +1,5 @@
\documentclass[10pt]{letter}
\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e}
\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
@ -34,27 +34,35 @@ We look forward to hearing from you.
{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{bla bla.}
\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 ($W^{\beta}$) would deserve to be somehow explained in the present paper.}
{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{bla bla.}
\alert{We have largely expanded 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 $W(\mathbf{r}_1,\mathbf{r}_2)$.}
\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{bla bla.}
\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.}
\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{bla bla.}
\alert{This is indeed the case: the present basis-set correction can be apply to any self-energy.
However, the more accurate the self-energy, the faster the convergence, hence the choice of the GW self-energy in the present work.
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{bla bla.}
\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}
@ -63,34 +71,47 @@ We look forward to hearing from you.
\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}
{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{bla bla.}
\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{bla bla.}
\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.}
\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. }
{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{bla bla.}
\alert{Titou: I don't agree with this.}
\item
{Figures 1, and the corresponding figures in the supplementry are plotted on a linear scale of X. Personally I think it is much more instructive to plot agains X-3, which wil much clearer visualize convergence. }
{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{bla bla.}
\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 planewave basis set extrapolated results for the gw100 set. It may be interesting to include this in the comparison.}
{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{bla bla.}
\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{bla bla.}
\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.
The json file format fits the one provided by the GW100 website (\url{https://github.com/setten/GW100}).}
\end{itemize}
\end{letter}