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Julien Toulouse
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@ -761,7 +761,7 @@ The estimated exact energies are based on a fit of experimental data and obtaine
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\caption{
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Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets.
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The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
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The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}.
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The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07} \alert{which one is correct?}.
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\label{fig:F2}}
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\end{figure*}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -772,7 +772,7 @@ The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} sys
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We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{N2}, \ce{O2}, and \ce{F2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table \ref{tab:d0}.
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Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have be anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
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Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
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It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the accuracy of the atomization energy slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI atomization energy and very close to chemical accuracy.
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Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected.
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@ -29,132 +29,60 @@
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\sffamily
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\section*{\sffamily Curing basis-set convergence of wave-function theory using density-functional theory: a systematically improvable approach}
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\section*{\sffamily A basis-set error correction based on density-functional theory for strongly correlated molecular systems}
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We would like to thank the reviewers for their carefull reading of our manuscript, and we reply in the present document to their remarks and criticisms.
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\subsection*{Comments of reviewer 1 and reply}
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\subsubsection*{Comments of reviewer 1}
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The authors propose a method to account for short‐range electron correlation using DFT techniques.
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The idea makes sense. I have long been frustrated (and intrigued) by the fact that DFT handles short‐
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range correlation effortlessly while wavefunction theory (WFT) requires a large basis set and a lot of
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computational work. The main problem of all combined WFT/DFT methods is double counting. The
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authors partially eliminate it by replacing the Coulomb operator with its long‐range component, tailored
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to the basis set (which can describe correlation only on the scale of the distance between the closest
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nodes and larger.)
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I have only a few comments.
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\begin{itemize}
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\item The method proposed is simpler than using explicit interelectronic (F12) coordinates. I wonder how
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much the latter cost computationally. The theory is certainly complicated and there are still a few loose
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ends in R12 or F12 theory but does it slow down the calculations significantly? The authors could add a
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comment on this. F12 has the advantage that it eliminates double counting rigorously but I am not sure
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how expensive it is. The method described in the paper is virtually free on the cost scale of a large
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correlated calculation but adds some of the disadvantages of DFT, mainly noise from numerical
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quadrature.
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\item Page 1, right column, bottom. The authors repeat the misleading claim that “DFT abandons the
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complex ... wavefunction for the simple one‐body density.” This is valid for orbital‐free DFT which is not
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yet working. It should be toned down.
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\item P. 3, above equ. (4) change “give” to “gives”.
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\item P. 4, right column, line 12 from below. The phrase “projected in a basis” is perhaps better if replaced
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by “projected to a basis”.
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\item I suspect that the CIPSI energy for N + at the AQZ level in Table II (‐54.020414) has a problem. The
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energy difference between QMC and CIPSI is generally in the 5 th digit, as pointed out in the paper. For
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this figure, the deviation is in the 3 rd digit. Please check it again.
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\item I suggest that the authors switch to the IUPAC (and IUPAP) abbreviation for the atomic unit of energy,
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E h (and mE h ).
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\item In its current form, the method is only applicable to atoms, as the local range separation parameter
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depends on the distance from the nucleus. The authors should indicate how it could be generalized to
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molecules.
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\end{itemize}
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Reviewer 1 Evaluations:\\
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Recommendation: Optional revision\\
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New Potential Energy Surface: No\\
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Overall Rating (required): Top 5-25\% - significant, novel, and impactful contribution of broad interest \\
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The manuscript presents an extension of the density-functional-based basis-set
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correction scheme, by some of the current authors, to strongly correlated systems. This
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basis-set correction scheme is a beautiful and powerful combination of density-functional
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and wave-function theoretical concepts and practices.
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All the basis-set corrections are applied to high-quality approximations of FCI
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results that are computationally costly. A natural question arises, what would be the
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effect of these basis-set corrections if the computational method were of lower quality.
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The reader would benefit, if there were at least a discussion of this issue in the
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manuscript.
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A few very minor remarks:\\
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- The reference numbering is not always consecutive\\
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- “have be” $\to$ “have been” (in the 4 th sentence of the 3 rd paragraph of IIIC)\\
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- In the caption of Fig. 4, the last two sentences are seemingly contradictory.
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\subsection*{Reply to reviewer 1}
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\begin{itemize}
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\item[] "The main problem of all combined WFT/DFT methods is double counting. The
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authors partially eliminate it by replacing the Coulomb operator with its long‐range component, tailored
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to the basis set (which can describe correlation only on the scale of the distance between the closest
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nodes and larger.)" \\
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In the case of standard range-separated DFT (RS-DFT), there is by construction no double counting of the electronic correlation effects as it relies on a clean splitting of the interaction, which is not the case for most of the combined WFT/DFT methods. The practical problems in RS-DFT are the approximated functionals or the ansatz used for the wave function part. Nevertheless, in the present work, as the effective interaction in the basis set is not perfectly represented by the fit that we proposed here (see equation (30) page 7), some double counting can still be present.
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\item "The method proposed is simpler than using explicit interelectronic (F12) coordinates. I wonder how
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much the latter cost computationally. The theory is certainly complicated and there are still a few loose
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ends in R12 or F12 theory but does it slow down the calculations significantly? The authors could add a
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comment on this. F12 has the advantage that it eliminates double counting rigorously but I am not sure
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how expensive it is. The method described in the paper is virtually free on the cost scale of a large
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correlated calculation but adds some of the disadvantages of DFT, mainly noise from numerical
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quadrature.
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" \\
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The authors are aware of several F12 theories but are not expert in that field, so we preferred not to comment too much on the computational cost of F12, specially on the latest developments. Regarding the avoidance of double counting, the reviewer could refer to the very recent study on the practical effect of the truncation of the one-particle basis set in approximated CCSD-F12 approaches:
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M. K. Kesharwani, N Sylvetsky, A. K\"ohn, D. P. Tew, and J. M. L. Martin, Journal of Chemical Physics 149, 154109 (2018).
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Nonetheless, even if our approach is conceptually simpler than F12 theory, as it is at its early stage of development, it is hard to compare the computational cost with F12 theories considering the amount of efforts involved in that field.
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\item "Page 1, right column, bottom. The authors repeat the misleading claim that “DFT abandons the
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complex ... wavefunction for the simple one‐body density.” This is valid for orbital‐free DFT which is not
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yet working. It should be toned down. "\\
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From a purely formal point of view, we agree that only orbital-free DFT abandons the wave function, as the Kohn-Sham Slater determinant is still needed for the computation of the kinetic part in KS-DFT.
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Nonetheless, from a WFT perspective (which is the overall point of view of the paper), KS-DFT is drastically simpler than WFT as it does not rely on the accurate description of the complex and non local two-body density matrix. This is why we stated that "DFT abandons the complex ... wave function for the simple one-body density", and also the sentence in page 1 does not refer to a specific variant of DFT as it refers to the formal implication of the Hohenberg-Kohn theorem.
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\item "P. 3, above equ. (4) change “give” to “gives”." \\
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The typo was corrected.
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\item "P. 4, right column, line 12 from below. The phrase “projected in a basis” is perhaps better if replaced
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by “projected to a basis”. \\
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We thank the author for the stylistic proposal. After inquiry, we found that the most adequate sentence is "projected on" (which is equivalent to "projected onto") which was changed in the text.
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\item "I suspect that the CIPSI energy for N + at the AQZ level in Table II (‐54.020414) has a problem. The
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energy difference between QMC and CIPSI is generally in the 5 th digit, as pointed out in the paper. For
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this figure, the deviation is in the 3 rd digit. Please check it again." \\
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We also found strange the discrepancy between our CIPSI energy and the \textit{i}-FCIQMC result in the case of N$^+$ in the aug-cc-pVQZ basis set. Nonetheless, we double checked our result and pushed our calculations quite far (since $|E_{PT2}| < 0.1 mH$) and obtained essentially the same energy and a variational energy lower than the one in \textit{i}-FCIQMC. As a consequence, considering that the initiator approximation of FCIQMC necessary introduces a bias and that we obtained a lower variational energy than the \textit{i}-FCIQMC, we believe that the problem is on the side of \textit{i}-FCIQMC.
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\item "I suggest that the authors switch to the IUPAC (and IUPAP) abbreviation for the atomic unit of energy,
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E h (and mE h )."\\
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We believe that the present abbreviations for the atomic unit of the energies reported here do not introduce any inconvenience for the clarity of the exposition, therefore we keep the present abbreviations.
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\item "In its current form, the method is only applicable to atoms, as the local range separation parameter
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depends on the distance from the nucleus. The authors should indicate how it could be generalized to
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molecules." \\
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I think that the reviewer 1 misunderstood some aspects of the procedure proposed here, as there are no restrictions in its mathematical formulation for the special cases of atomic systems.
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Indeed, the definition of the effective electron-electron interaction projected on a basis (see equations (21) to (27) page 5, or appendix A for a more detailed derivation) relies on an expectation value of the Coulomb interaction over a very general wave function which can represent an atom or a molecule. The derived effective interaction (see equation 27) is a general function from ${\rm I\!R}^{6}\rightarrow {\rm I\!R}$, which can be computed in any physical system.
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We chose to study atomic systems as these systems are basically free from static correlation effects, exhibit large basis set errors for the calculations of energy differences and are well understood from a physical point of view (which enable us for instance the detailed analysis of the last section). Incoming works will assess the validity of this approach on molecular systems.
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\end{itemize}
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\subsection*{Comments of reviewer 2 and reply}
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\subsubsection*{Comments of reviewer 2}
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Reviewer 2 Evaluations:\\
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Recommendation: Optional revision\\
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New Potential Energy Surface: No\\
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Overall Rating (required): Top 5 \% - highly significant, novel, and impactful contribution of broad interest\\
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Reviewer 2 (Comments to the Author):\\
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This is a very high-quality manuscript, in which a DFT-based correction for finite basis-set errors in wavefunction theory is extended to the case of strongly correlated molecular systems, by looking at molecular dissociation curves. The beauty of the theory behind this DFT basis-set correction is that it can be applied to any wave function method. The authors use their very smart idea (published in previous works) of a suitable mapping of the interaction projected in the finite basis to a long-range only interaction (erf(mu*r)/r), determining the value of the local range separated parameter mu by matching the interaction at r=0.
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Here the authors further extend the DFT part of their theory by using explictilyt the on-top (OT) pair density dependence of the short-range functionals of multideterminant range separated DFT to correct the basis set error. They also show that the OT dependence can fully eliminate the spin-polarization dependence.
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The idea is very neat and well explained, and the results are excellent. I recommend publication with the minor revisions below.
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In the present manuscript by Giner et al. the authors present a novel method to correct for basis set incompleteness errors in Full CI (FCI) calculations. Their method is based on the local mapping of an effective (basis set dependent) electron-electron interaction to a range separation (RS) parameter as used in RS density functionals. By combining the proposed local mapping and the derived formalism it is possible to introduce a perturbative basis set correction for FCI calculations. The obtained results for spherical systems demonstrate convincingly that their novel method achieves a rapid basis set convergence.
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The manuscript is very well written and the work is of great interest to the electronic structure theory community. I recommend publication of the manuscript.
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1) The authors should comment more the results of Table 1 for atomization energies. Although their theory is guaranteed to converge to the exact result for the complete basis set limit, the results of table 1 show that for atomization energies this does not always occur in a monotinic way. There are cases (H$_{10}$ and N$_2$ with the OT functionals, O$_2$ with all the functionals and F$_2$ with augmented basis with all the functionals) in which the results are worse for a quadruple-zeta basis than for triple zeta. Could this be due to the fact that when augmenting the basis the OT pair density from the CASSCF wavefunction does not necessarily improve but might oscillate around the Coulomb cusp creating artefacts in the extrapolated OT? After all the basis sets are optimized on the energy and do not really care about the cusp. Or is the effect due to a different accuracy of the correction for the molecule and the atoms? It would be nice if the authors could say something on that.
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I only have a few optional suggestions and questions:
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\begin{itemize}
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\item The present work does not draw any comparison to F12-like approaches in the results section. I suggest for this or future studies to compare results obtained using the present approach to results obtained using FCI + [2]R12 (as proposed by Kong and Valeev [J. Chem. Phys. 135, 214105 (2011)]), or FCI + CT (as proposed in Yanai and Shiozaki [J. Chem. Phys. 136, 084107 (2012)]). It seems that the present approach is computationally simpler and more efficient than F12-like approaches (please correct me if I'm wrong). Therefore it would be interesting to compare these approaches to each other in terms of accuracy.
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\item The authors have investigated atoms only. Can the present approach be directly transferred to non-spherical systems, or do the authors expect difficulties? Presenting results for binding curves would be desirable for this or future studies.
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\item Typo:\\
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p.5 : "which means that its does not..." -> "which means that it does not..."
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\end{itemize}
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2) Appendix A 1: eqs. A3, as discussed by the authors may break down for the case of degenearcy of the isolated fragments. The authors comment that the spin degeneracy can be avoided by considering the functionals without spin polarization and that the spatial degeneracy can be also cured by considering for the fragment the same element of the ensemble as for the suprasystem.
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However I still wonder if this is ok. For the spin I agree with the authors, but for the spatial degeneracy I still have some doubts. Could the authors comment at least qualitatively on what would happen for C$_2$ and B$_2$ for example? Is the DFT-based basis set correction going to give the same result for the spherically-averaged C or B atom than for the atoms with the occupied p orbital oritented alsong the bond axis? Or does the DFT-based correction show on this the same problems of standard DFT?
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\subsection*{Reply to reviewer 2}
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\begin{itemize}
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\item "The present work does not draw any comparison to F12-like approaches in the results section. I suggest for this or future studies to compare results obtained using the present approach to results obtained using FCI + [2]R12 (as proposed by Kong and Valeev [J. Chem. Phys. 135, 214105 (2011)]), or FCI + CT (as proposed in Yanai and Shiozaki [J. Chem. Phys. 136, 084107 (2012)]). It seems that the present approach is computationally simpler and more efficient than F12-like approaches (please correct me if I'm wrong). Therefore it would be interesting to compare these approaches to each other in terms of accuracy." \\
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\item[1]
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We completely agree with the reviewer 2 that our approach is directly comparable to those proposed by Kong and Valeev, as both approaches can be seen as a post-WFT calculation which aims to account for the incompleteness of the basis set used in quantum chemistry. More generally, we indeed plan to perform a comparative study with F12 approaches which will help us to better understand the strengths and limitations of our new approach. \\
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Regarding the method proposed by Yanai and Shiozaki, it is, to our understanding, not straightforwardly comparable to the present work as it deals with a two-body Hamiltonian which is modified by the presence of Slater-geminals. Therefore, it introduces a coupling between the correlation effects in the basis set (at whatever level of treatment selected) and those introduced by the explicit correlation factor. It is then more of a "perturb then diagonalize" approach, whether our approach if closer to a "diagonalize then perturb", as the one of Kong and Valeev. Current developments will try to make the procedure self consistent by carrying the minimization over all possible densities instead of doing the approximation of the FCI density as it is done here (see equation (14)).
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\item "The authors have investigated atoms only. Can the present approach be directly transferred to non-spherical systems, or do the authors expect difficulties? Presenting results for binding curves would be desirable for this or future studies." \\
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\item[2]
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As mentioned to reviewer 1, we investigated atomic systems only in order to focus our attention on the slow convergence of the dynamical correlation with the basis set (no static correlation involved). Also, these systems are quite well understood and the spherical symmetry has allowed us to perform a simple physical analysis of the behavior of our approach (the last section of the paper).
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Of course, the approach can be transferred to non-spherical systems since the effective interaction is derived from the expectation value of the Coulomb interaction on a general wave function which can represent molecular systems, and since the functionals used in RS-DFT are perfectly suited for the treatment of molecules. We have already performed calculations on homo- and hetero-nuclear molecules (atomization energies of the G2 sets of molecules) and the presentation and analysis of the results will be the subject of a forthcoming paper.
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\item Typo:\\
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p.5 : "which means that its does not..." -> "which means that it does not..." \\
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The typo was fixed.
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\end{itemize}
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\end{document}
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