ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50
Round 1
Reviewer 1 Report
I appreciate having access to the extensive data contained in this paper.
Author Response
We would like to thank the referee for the favorable opinion.
The reviewer did not ask for any modifications to the submitted manuscript.
Reviewer 2 Report
Additional explanation how the ZZ polynomials related with physical/chemical properties will increase interests of the readers. In mathematics, a merit to use generating functions is to see the ring structures which reflects operations to the objects, but there is no such information in the manuscript. I would recommend the authors to provide some information of those two.
Author Response
The reviewer has two comments to our paper:
- Add chemical/physical connections for the computed ZZ polynomials.
- Comment on algebraic properties of generating functions.
We have followed very closely the first suggestion of the referee. To this end, we have optimized geometries of the fullerene cages for each studied isomers of C20 - C50 and we have correlated the thus obtained thermodynamic stability indicators with the number K of Kekule structures and the number C of Clar covers. The data obtained in this way is plotted in a form of three additional figures added to the original manuscript. The first figure gives a scattered plot of points corresponding to the (energy-per-atom,K) pairs. The second figure gives an analogous plot for the (energy-per-atom,K) pairs. Finally, the third plot corresponds to a distribution of the (energy-per-atom,K) pairs devided additionally for subsets corresponding to distinct Clar numbers. A discussion of these new data is added to the manuscript in a form of Section 4 titled "Discussion". We believe that this addition duly reflects the requested changes.
As for the second comment, concerning adding a paragraph describing algebraic properties of generating functions, we have some reservations. Namely, most of generating functions correspond to infinite sequences and until seminal work of Gian-Carlo Rota in 1970s it was not clear what is the formal status of power series possibly diveregent everywhere (from the analysis point of view). Rota's work on connecting generating functions (as formal power series) to so-called Scheffer sequences and introducing formal algebra of such sequences was an advent of umbral calculus, (see for eample the book "The Umbral Calculus" by Steven Roman) an extremely interesting and fruitful discipline on the intersection of algebra and combinatorics. Of course, it is possible to add a paragraph describing this development together with a brief discussion of umbral algebra to the manuscript, but we feel that it would be a superficial step, because all the ZZ polynomials correspond to FINITE sequences, for which the questions of existence of limits and convergence are trivially satisfied, not leaving any doubts about formally impeccable character of used by us objects. On one hand then, there is no strong need to discuss formal properties of inifnite Scheffer sequences because all our sequences are finite, and on the other hand we are afraid that such a discussion might actually scare off mathematically not trained scientists rather than invite them to embrace the concept of ZZ polynomials. If possible, we would like to politely ask the reviewer to alleviate his/her second comment as it might be possible that following the suggestion might cause more harm rather than advantage for the potential readership of our manuscript.
Reviewer 3 Report
The manuscript entitled "ZZ polynomials for isomers of (5,6)-fullerenes Cn with n=20–50" is an interesting study and covers all isomers of small (5,6)-fullerenes with n=20–50. As a complete study it will be a very helpful initial information for the further more advanced quantum-chemical calculations. In my opinion the manuscript should be accepted, however, some minor changes should be applied.
- Section - "conclusions" is missing.
- Authors claim that no errors have been done in the program:"Few of the computed ZZ polynomials have been verified by pencil-and-paper calculations to make sure that no programming errors are present in the used subroutines."and refer the reader to the Ref. 67 and 68. The link given does not lead to the source code: https://bitbucket.org/solccp/zzdecomposer_binary/downloads/. Therefore I suggest to include the source code of the program as Supplementary Information so everybody can check if there are truly no errors. Of course, one trusts the Authors but the possibility to cross check the data is one of the conditions for the further usage as a benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes.
Author Response
The reviewer requested from us 2 actions:
- Adding a conclusion section.
- Providing computer program used for the calculations.
We have added a short conclusion section at the end of the manuscript.
We would be happy to share the program with all interested scientists. To this end, we have added a sentence to the paper, saying that the code can be obtained upon request from the authors.
Round 2
Reviewer 2 Report
NA
