Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author has developed a new population analysis. The author introduced a simple and efficient extension of Hirshfeld population analysis, called scaled Hirshfeld (SH), in which neutral proatom densities are scaled to construct a promolecular density better adapted to the molecular electron density.

The study lacks of a good analysis of the new method.

I approve to publication if the author solves all these points:

 

1) The author has proved the existence of a unique solution in the SH approach. This solution is independent of the initial guess. However, the author must show numerically this affirmation!

This could be tested easily by running the SH calculations with a different guess than the Hirshfeld one (c_A=1). It could be done, for example with scaled atomic charges (twice or half).

 

2) In some iterative methods like in the iterative Hirshfeld (H-I) (J. Chem. Phys. 126, 144111 2007 DOI: 10.1063/1.2715563) occurs the bifurcation or ‘runaway charges’ problem (RSC Adv. 6, 47771 2016 DOI: 10.1039/c6ra04656h). For example in the H_2 triplet with 50 pm constrained bond length, H-I leads to more than one self-consistent solution, and these solutions can be controlled by the initial guess. The author must corroborate that the SH approach solves this problem at least with this H_2 triplet system!

 

3) The atomic charges are always an approximation to the electronic density and the nuclei of a system. It is desirable but not necessary that atomic charges reproduce the observable dipole moment of a system. The SH method reproduces better the dipole moment than the H method, this is important!

However, You should cite that other population analyses reproduce the dipole moment as these two:

 

Journal of Theoretical and Computational Chemistry 11, 163 (2012) DOI: 10.1142/S0219633612500113

J. Chem. Phys. 161, 144103 (2024) DOI: 10.1063/5.0224028

 

Then, the reader will see, there are other population analyses that reproduce the observable dipole moment.

 

4) The Hirshfeld method is robust with respect to the size of the basis sets. I suppose the SH approach is also robust. However, it will be very important for the article the knowledge of this fact!

I suggest to do a small Table with the correlation coefficients R^2 among different basis sets for the studied data-set with the SH method.

 

5) The mathematical expression that appears in the Label of Table 1 is incorrect!

It is the equation that calculates the error between the electrostatic potential (it from the nuclei and the electron density) and the estimated electrostatic potential from charges, in that expression missing a summation over the atoms A.

 

6) (a) Figure 1 clearly shows the relation of the H and SH charges with respect to the CM5 charges. However, it will be clearer if the author calculates the correlation coefficients R^2 among all population analyses studied in Table I.

 

(b) In relation with the relative magnitude of the atomic charges among the studied population analyses, one can look at the slope of the linear regression between them. This study will show the magnitude of the SH charges with respect to other population analyses.

 

This will serve to demonstrate the affirmation of the author:

 

“Hirshfeld method is known to systematically underestimate atomic charges and to perform poorly for modeling molecular interactions”

 

7) A flowchart of the new method will help to the reader. It will be useful for a direct implementation of the method in several programs!

Author Response

Please see the response attached alongside the updated manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper introduces “Scaled Hirshfeld (SH)” partitioning: within Hirshfeld’s 1977 stockholder framework, each neutral reference atom density is simply scaled by an iteratively optimized coefficient cA. They prove that SH is mathematically identical to the minimal-basis additive variational Hirshfeld (AVH-M), thereby endowing SH with an information-theoretic foundation, size-consistency and a unique solution. A fixed-point iteration solver is used; computational cost is comparable to original Hirshfeld. On a 168-molecule organic set, SH charges are systematically larger than Hirshfeld and reduce mean absolute errors for dipole moments and electrostatic potentials by ~39 % and ~34 %, respectively, approaching the empirical CM5 and the larger-basis AVH-B results.The study shows that a minimal, variational rescaling of neutral pro-atoms suffices to correct most Hirshfeld deficiencies, offering an immediate “drop-in” upgrade for force-fields and electrostatic models. But Specific revision requests:

1)Missing methodological detail: provide pseudo-code for the SH iteration (convergence threshold, maximum steps, initial guess, safeguard against negative coefficients) and supply a GitHub or permanent DOI repository to ensure reproducibility.

2)Insufficient data presentation: in addition to Table 1, include histograms or box-plots of errors and list the best- and worst-case molecules with their structures so readers can judge robustness.

3)Computational cost not discussed: report average CPU time (s/core) and peak memory for SH vs. Hirshfeld and AVH-B, and state the scaling with molecular size (linear? quadratic?).

4)Theoretical limitations unclear: because SH uses spherically averaged neutral atoms, discuss its performance for systems with large charge transfer or excited states and outline a roadmap for incorporating non-neutral or hybrid basis functions. 5)Figure & symbol consistency: font size in Fig. 1 axes is too small; unify “ext-KL” vs. “KL” notation; Eq. (11) overflows the margin—re-format. 

Comments on the Quality of English Language

About assessment of English quality: Generally clear and logically structured with accurate terminology. 

Author Response

Please see the attached document.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author presents a new population analysis called scaled Hirshfeld (SH), this tool seems to be robust. All tests that I have made on the first revision, now, are covered.

 

In the new method the neutral pro-atom densities (H) are scaled to produce the new scaled pro-atom densities, which are used in the weight function (w(r)). In the convergence it has only one solution for a ground or excited state density. This is fine. The resulting atomic charges show ability to approximate molecular electrostatics (dipole moment and electrostatic potential).

 

I approve to publication.

 

A minor error appears in the row 139: [?]

 

Congratulations!

Author Response

Thanks for your positive feedback and for noticing the corrupted reference on line 139, which appeared as [?] instead of [61]. This has been corrected.