Energy Decomposition Scheme for Rectangular Graphene Flakes
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe manuscript shows a simple expression of the total ground-state electronic energy E(m,n) of m x n rectangular graphene flakes at their DFTB3 optimized geometries, for 10 < m,n < 21.
Overall, the manuscript is interesting and can be useful for machine learning models. I have the following points to be addressed:
- I think some graphene flakes are not really planar, but they have a 3D geometry, being slightly bend, with some curvature. This issue should be explained in the manuscript.
- even if the E(m,n) is an important quantity, it can not predict other properties. For example, if the geometry is slightly changed, E(m,n) remains insensitive. Thus, maybe other parameters should be considered,
such as the averaged bond length and the averaged bond angles.
Author Response
We would like to thank the referee for the comments. Each of the comments (together with the essence of the original remark of the referee (in bold) and a short description of changes incorporated in the manuscript in order to address it (in italic)) is briefly discussed below. We hope that our answers satisfy the referee.
- Graphene flakes may have non-planar equilibrium geometry: We agree with the reviewer that some of the graphene flakes may be subject to Jahn-Teller distortions resulting in a non-planar geometry of the optimized flake. We suspect that this effect would be strongest for the no-hydrogen-teminated flakes, where the deficiencies in chemical saturation of carbon atoms give rise to quite complicated electronic structure of such flakes and, consequently, to its open-shell character and considerable degeneracies in the one-particle spectrum. The non-planarity deformations, allowing to lift this degeneracy and to increase the percentage of doubly-occupied levels, might be an important factor for lowering the total energy during the energy optimization. We suspect that for the class of the hydrogen-teminated graphene flakes studied in our paper this effect might be somewhat less-pronounced owing to the well-defined, chemically-saturated electronic nature of each of the flakes. An obvious way to verify whether a given flake is planar or non-planar in its equilibrium geometry is an inspection of the harmonic vibrational modes after the geometry optimization. Performing such a task for all the studied here flakes is formidable due to its computational complexity. However, to answer the referee's concern, we have performed DFTB+ harmonic frequency calculations for two intermediate-size flakes, Z(10,10) and Z(15,15), to see whether the non-planarity is an issue here. All the harmonic frequencies of Z(10,10) and Z(15,15) turned out to be positive, showing that Z(10,10) and Z(15,15) are in fact planar, despite of the fact that their electronic structure is metallic with 4 electrons distributed among 3 quasi-degenerate MOs with the occupation pattern 1.6:1.3:1.1 (for Z(10,10)) and 2 electrons distributed among 4 quasi-degenerate MOs with the occupation pattern 1.1:0.4:0.3:0.2 (for Z(15,15)). This two sets of calculations do not provide a formal proof that all the flakes studied by us are planar, but we feel that it is quite strong confirmation of the chemical saturation arguments given above. We believe that this particular discussion could be beneficial for a potential reader of our paper, so we decided to include it (almost verbatim) in the main body of the paper at the end of paragraph 1 of Section 3. The text added to the the manuscript reads: "The resulting equilibrium geometries of the optimized flakes were planar.
It is possible that some graphene flakes may be subject to Jahn-Teller distortions resulting in their non-planar geometry. This effect is probably strongest for flakes without hydrogen termination, where the deficiencies in chemical saturation of carbon atoms give rise to quite complicated electronic structure and, consequently, to its open-shell character and considerable degeneracies in the one-particle energy spectrum. The non-planarity deformations, allowing to lift this degeneracy and to increase the percentage of doubly-occupied levels, might be an important factor in lowering the total energy during the energy optimization. We suspect that for the class of the hydrogen-teminated graphene flakes studied in our paper, this effect might be somewhat less-pronounced owing to the well-defined, chemically-saturated electronic nature of each of the flakes. An obvious way to verify whether a given flake is planar or non-planar is an inspection of the harmonic vibrational modes. Performing such a task for all the studied here flakes is formidable due to its computational complexity. However, to answer one of the referee's comments, we have performed DFTB+ harmonic frequency calculations for two medium-size flakes, ${Z\left(10,10\right)}$ and ${Z\left(15,15\right)}$. All the harmonic frequencies of ${Z\left(10,10\right)}$ and ${Z\left(15,15\right)}$ turned out to be positive, showing that both these flakes are in fact planar, despite of the fact that their electronic structure is metallic with 4 electrons distributed among 3 quasi-degenarate MOs with the occupation pattern 1.6~:~1.3~:~1.1 for ${Z\left(10,10\right)}$ and with 2 electrons distributed among 4 quasi-degenarate MOs with the occupation pattern 1.1~:~0.4~:~0.3~:~0.2 for ${Z\left(15,15\right)}$. This two sets of calculations do not provide a proof that all the flakes studied by us are planar, but we feel that they provide quite strong argument supporting this idea."
- Energy is important, but can you consider other quantities? Every measurable physical property of a system can be defined as the derivative of the total energy E with respect to an appropriate external parameter. For example, the dipole moment is defined as the derivative of E with respect to the external electric field, polarizability - as the second derivative of E with respect to the external electric field, gradient - as the derivative of E with respect to the nuclei positions, frequencies - as the second derivative of E with respect to the nuclei positions, IR intensity - as the second derivative, once with respect to the nuclei positions and once with respect to the external electric field, Raman intensity - as the third derivative, once with respect to the nuclei positions and twice with respect to the external electric field, etc. In this respect, demonstration that the total energy can be expressed as a simple function of the parameters m and n is very important, because it indirectly and implicitly shows that also other quantities can be expressed as similar functions of the parameters m and n, even if the results of associated calculations are not presented in our manuscript. We are planning to extend the current research line to other quantities in the forthcoming papers. Actually, the effort associated with the current set of calculations was very extensive: it took more than 1 year of continuous calculations to be able to optimize all the geometries reported in the current paper and extract the energy dependence, allowing Hendra, a MS student in our laboratory to base his MS thesis on this set of calculations. The results are promising and interesting, as all the referees of the submitted manuscript agree, so we decided to extend this study also to other properties, which will constitute the topic of the MS thesis for a student starting her MS program this summer. We hope to present the results of this project in a forthcoming article.
Answering the referee's request to analyze also other quantities resulting from the MS thesis of Hendra, we decided to present the distributions of CC bond lengths and CCC angles in the analyzed flakes, with particular attention devoted to the trends observed in the transition from small to large flakes. The primary objective of this analysis is determination at which limiting value of the parameters m and n the finite-size flake starts displaying the properties of an infinite graphene sheet, as it was requested by another of the referees. The distributions of the CC bonds and CCC angles are shown in two new figures added to the manuscript, Figures 8 and 9. Discussion of these figures and limiting values of the CC bond lengths and CCC angles are presented in Section 5(ii). This discussion, including two new figures, is too long to be quoted here verbatim.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors presented a suprisingly simple fit function for the energy of rectangular zigzag graphene flakes depending only on their extensions n and m. If the fit is only done for larger flakes, then it yields reasonable good agreement and can be used to predict the energy of the flakes within the same parameter range the fitting has to be done.
I have four main points to criticize:
(1) The systems are chemically very similar, especially when the smaller flakes are excluded, there are no structural changes expected in the chemical bonds, just more bond/rings/hydrogens are added. Additionally the electronic structure for the larger flakes is already graphene-like (I suspect, as the authors stated once, that all larger ones are semimetallic), so it is not so surprising that the fit formula works. More interesting it would be, to really calculate the chemical energies mentioned under (iii) and see, whether this energies can be transferred between the systems.
(2) It is clearly seen in all figures, that the error or the fit grows linearly (or almost linearly with system size. Could one not add an correction and then predict really large flakes, which are not reachable for quantum chemical calculations.
(3) I do not know, how the study is connected to machine learning. It is a fit to selected functions. That is a publishable approach, so I do not understand why the authors wants to link it to machine learning.
(4) The authors predict the total energy of the graphene flake, but does it has any use to know the total energy of the system, that is not a measurable quantity.
A minor comment: In my opinion the introduction is quite extensive, and I do not really see the use of mentioning the models, which describe chemically different systems and different properties. Chemistry is full of approximated models, and of course, it is worth to develop such models. Perhaps the authors should concentrate more on chemical related systems like CNTs.
Overall the manuscript has some repetitions, perhaps it can be shortened in some parts, e.g. the discussion can be included partly in the results part and in the conclusion.
Overall the idea is worth to be published, but the relevance for measurable or transferable quantities should be discussed in more detail.
Author Response
We would like to thank the referee for the comments. Each of the comment (together with a short summary of the original comment and a brief description of modifications introduced to the text of the manuscript to accommodate the answer to the comment) is addressed below separately.
1a. It is not surprising that fit formula works for large flakes: Yes, we understand the referee's comment and on one side we fully agree with it. Yet, on the other side, we see how much work, time, and resources is required in order to compute the optimized geometry of large flake with DFTB, using the standard computational approach, and we cannot stop to be surprised that all this complexity can be replaced by simple, few-parameter model.
To accommodate this dichotomy in the manuscript, we have added the following sentence "On the other hand, in the limit $m,n\rightarrow \infty$, a rectangular graphene flake converges toward an infinite graphene sheet, physics of which is relatively simple, being sufficiently well described with a unit cell containing just two carbon atoms." to the Introduction section and modified the words "finding such a function can be in fact possible" to "finding an energy function $E=E\left(m,n\right)$ can be in fact possible" in the next sentence to merge the text with the change described above.
1b. Compute atom, bond, and ring energies and confirm their transferability: We thank the referee for this comment. This is indeed a very interesting idea and our primary concern to write the fragment (iii) of the Discussion section was motivated exactly by such a goal. Unfortunately, it turned out that in case of the class of rectangular flakes studied by us here, the 5 independent energetic parameters (e_C, e_H, e_CC, e_CH, e_ring) depend only on 3 functions of the structural parameters (1, m+n, and mn), and therefore the dependence cannot be disentangled uniquely. (There exist an infinite number of solutions to the underlying algebraic equations and choosing a single set of them cannot be performed uniquely.) Since an infinite graphene sheet could be build not only from rectangular flakes, but also from other types of shapes (e.g., hexagonal), it seems possible to disentangle the values of the energetic parameters e_C, e_H, e_CC, e_CH, and e_ring, if other families of graphene flakes with different shapes are included in the analysis. The promising candidates here are: prolate rectangular flakes, oblate rectangular flakes, and hexagonal flakes. Our plan is to perform similar analysis as presented in the current manuscript also for other families of flakes over the next two years. (The current project commenced 2 years ago and constituted the topic of MS thesis of Hendra; the calculations of optimized geometries of the flakes took over one year to finish.) We hope that the energy dependence on the structural parameters discovered for the other families will give us an opportunity to disentangle successfully the energetic parameters dependence and test their transferability to other types of polycyclic aromatic hydrocarbon systems.
This discussion was briefly summarized in the manuscript at the end of item (iii) in Section 5 in the following words: "In the future studies, we are planning to extend our analysis to other structured graphene flakes, including prolate rectangles $Pr(k, m, n)$, \cite{Cyvin_1988,Yen_1971} oblate rectangles $Ob(m, n)$, \cite{Cyvin_1988,Gutman_1985} and hexagons $O(k, m, n)$. \cite{Cyvin_1988,Witek_2021} We expect that distinct dependence of the total energy on the structural parameters for these structures will help to determine uniquely the five parameters $\epsilon_{\text{\tiny{H}}}$, $\epsilon_{\text{\tiny{C}}}$, $\epsilon_{\text{\tiny{O}}}$, $\epsilon_{\text{\tiny{CC}}}$, and $\epsilon_{\text{\tiny{CH}}}$ defined above."
2. Add a correction allowing to extrapolate the discovered formula to really large structures: We would be happy to follow the advice of the referee and design such a correction allowing to extrapolate our formula to really large systems that escape the possibility of treating them quantum-mechanically with the existing ab initio programs, but despite out earnest efforts we could not conceive such a correction. We are aware that our best formula works well maybe for structures larger by 50% than the largest exactly computed structure that has been included in the fitting process; for larger structures the linearly growing error mentioned by the referee will dominate and extort our predictions. The only way of avoiding this behavior is to include larger and larger structures in the analysis that would postpone the appearance of the error for even larger flakes or to design the correction term (with probably new, non-obvious dependence on the structural parameters n and m) that would allow to describe the departure (linearly growing error referred to by the referee) in a quantitative way. We believe that such a correction comprises the information about the finite edge of the flakes, with the total energy formula being a combination of the quadratic-like formula for the interior of the flake and the linear-like formula for the edge of the flake. Since we are not able to discover the analytic form of the correction, we announce the full set of energies and geometries associated with all the modeled systems to allow other researchers for an independent analysis possibly successfully resolving the analytic form of the sought correction.
This discussion has been summarized by adding the following two sentences at the end of item (i) in Section 5: "An interesting alternative here could be a theoretical analysis of the contributions to the total energy from the finite edge effects and possibly quantifying such an influence using non-obvious, new $(m,n)$-dependent basis functions. Such a development is expected to improve the description of small flakes and to permit extrapolation of the energy formula to really large values of $m$ and $n$ that presently escape the possibility of direct quantum chemical calculations."
3. What is the connection to machine learning? Machine learning approaches take as an input the topology of chemical connections in a given chemical system and return as an output the total energy of the system and a XYZ file corresponding to the optimal geometry of the system. The "knowing algorithm" of a machine learning scheme is dissipated over the neural connection (edge weights and vertices weights) of the underlying machine network of neural system. Each piece of information is dissolved in the network structure and every time when a prediction is made with a machine learning model, we cannot pinpoint the actual reason for such a prediction. On one hand, this approach is very familiar for humans, as their own brain works exactly in the same way. On the other hand, "machine knowing" goes against the principle of rationalization, so needed for progressing the development of science. With growing complexity of the system, one might expect that the machine learning knowledge of the system is growing too, possibly exponentially, soon overwhelming the "memory" and "reasoning" capabilities of a finite machine learning system. Our discovery shows that it might be not the case, as the rectangular graphene flakes tend to be described well by a simple function of the structural parameters. Such a discovery might be very important for further development of machine learning approaches, where a well-define deterministic algorithm could be used to describe the major contribution to the total energy and only the minuscule "correction" suggested by the referee in point 2 of the current answer will be effectively described by the intangible network of neural connections of the machine learning algorithm.
Since the other referees seem to acknowledge the connection between our paper and the machine learning approaches (referee 1 says "Overall, the manuscript is interesting and can be useful for machine learning models.") and since the answers to the referees' comments are available from the MDPI homepage as a part of this article, we have decided not no modify the manuscript to accommodate the current discussion, but instead make it available to an interested reader directly through the comments/answer to comments section of the manuscript.
4. Energy is not a physical observable; can you predict other measurable quantities? Indeed, the total energy E is not a measurable quantity, but every measurable physical property of a system can be defined as the derivative of E with respect to an appropriate external parameter. For example, the dipole moment is defined as the derivative of E with respect to the external electric field, polarizability - as the second derivative of E with respect to the external electric field, gradient - as the derivative of E with respect to the nuclei positions, frequencies - as the second derivative of E with respect to the nuclei positions, IR intensity - as the second derivative, once with respect to the nuclei positions and once with respect to the external electric field, Raman intensity - as the third derivative, once with respect to the nuclei positions and twice with respect to the external electric field, etc. In this context, the demonstration that the total energy can be expressed as a simple function of the parameters m and n is very important, because it indirectly and implicitly shows that also other quantities can be expressed as similar functions of the parameters m and n, even if the results of associated calculations are not presented in our manuscript. In order to answer the referee's request to analyze also other quantities resulting from the data produced by our research, we decided to present the distributions of CC bond lengths and CCC angles in the analyzed flakes, with particular attention devoted to the trends observed in the transition from small to large flakes. The primary objective of this analysis is determination at which limiting value of the parameters m and n the finite-size flake starts displaying the properties of an infinite graphene sheet, as it was requested by one of the referees. The results of the analysis are presented in two new figures added to the manuscript. Figure 8 shows the distributions of the CC bond lengths and CCC angles in square flakes Z(k,k) in the transition from small k to large k; Figure 9 shows analogous distributions for rectangular flakes Z(20,k) and Z(k,20). The discussion of the changes and the convergence of the distributions toward the uniform distributions expected for infinite graphene sheet are presented in section 5(ii) of the revised manuscript. We are planning to extend the current research line to other physical quantities in the forthcoming papers. The modifications, consisting of two new figures and a half-page excerpt of text, are too long to be quoted here. Instead, we suggest that the referee directly inspects the appropriate fragment of the revised manuscript.
Other remarks of the referee concerned the following aspects of the manuscript.
The introduction is too extensive and misses connection to other carbon based systems: We have inspected again the introduction section and indeed we found fragments that could be removed without compromising the writing. We have removed the following fragments: "A knowledge of a simple and compact expression for $E\left(m,n\right)$ would open a new vista for quantitative energy evaluation of graphene flakes practically without any numerical effort. Such a formula might provide a far-reaching insight into quantum mechanical description of complex molecules: the numerical parameters appearing in the expression for $E\left(m,n\right)$ might constitute important molecular invariants allowing us to understand better the analytical structure of many-body energies and wave functions." and "Following this logic, we see that if the geometry optimization for the simplest graphene flake (i.e., benzene) takes, say, 1 minute, then the calculation for a flake of the size $1000\times1000$ would take approximately 4 years. " in Section 1 of the original manuscript, and ", i.e., it was impossible to find parameters that could be used for all the 250 studied isomers. In this sense, the result obtained in the current study is more general, as the same energy formula can be used for a large collection of multiple zigzag chains with different number of carbon atoms" in Section 2. Two paragraphs of Section 2, starting from words "Before we describe the computational protocol..." and "Our main goal was to verify...", have been merged together, which allowed to reduce the content by some 20%; the resulting paragraph starts with words "The original reason to start working...". To bring to the attention of the reader that our approach is probably applicable to other carbon-based systems, a sentence "We believe that similar approach could be designed also for other carbon nanostructures, including graphene flakes of other shape, carbon nanotubes, and fullerenes." has been added at the end of Section 1.
Shorten the manuscript, avoid repetitions, and merge discussion with results: The modifications of the text related to removal of repetitions and shortening the exposition for Sections 1 and 2 have been described above. We believe that the remaining Sections 3, 4, and 5 do not contain any repetitions and cannot be shortened considerably. To merge the discussion with the results, we have performed the following changes
- The title of Section 4 was modified from "Results and Discussions" to "Construction of the fit".
- The title of Section 5 was modified from "Discussions" to "Results and Discussion".
- New Section 5 has been expanded considerably (by 100-200%) to accommodate all the new discussion needed to answer the referees' comments.
Reviewer 3 Report
Comments and Suggestions for Authors The authors present an interesting study concerning the possibility to predict the total energies of graphene nanoflakes using a small set of simple functions in the fitting procedure. The selected functions depend on the linear sizes (m,n) and have physical interpretations, like the flake perimeter (m+n), area (mn) or edge asymmetry (m-n). This approach has its limitations with respect to large system sizes. Additional functions (m^2,n^2,m/n,n/m) were added to reduce the prediction error.
The paper contains interesting observations concerning the simple dependence of the total energy on the parameters m,n. Various other examples of systems obeying simple rules were illustrated and this study can stimulate further investigations.
I would recommend the paper for publication, pending the following issues are addressed:
1. The complexity in the class of graphene nanoflakes should have a maximum for intermediate sizes. The larger systems are expected to behave more and more like ideal graphene. Is it possible to describe this limit as well ?
2. As the systems are relaxed, the outer bond lengths are reduced. I wonder in how far this effect affects the results and prediction accuracy (e.g. compare with unrelaxed geometry).
3. Also related to point nr. 2, the authors did not consider any passivation of the carbon nanoflakes. This would affect the bond lengths at the edges and also the charge distribution and total energy.
4. Do you expect that different XC functionals modify significantly the distribution seen in Fig. 2 ?
The quality of English is good.
Author Response
We would like to thank the referee for the comments.
The referee brings to our attention four issues to be addressed in the revised manuscript. We present briefly referee's concerns below together with our answer to each of them and a brief description of changes introduced to the manuscript in order to address them.
1. What is the limiting flake size for which it starts resembling graphene sheet? This is a very valuable and interesting question. We completely missed this aspect of research in our original contribution and we would like to thank the referee for bringing it to our attention. To answer this question we have performed the analysis of the distributions of the CC bond lengths and the CCC bond angles. In an infinite graphene sheet, all the bonds have exactly the same length (of approximately 1.42 A) and all the angles are equal to 120 degrees. In finite rectangular flakes, the bonds located at the edges of the rectangle and at its corners are subject to distortions associated with the finite size and different topology of connections than in the interior of the flake. We expect that for larger flakes, the interior indeed resembles an infinite graphene sheet, while the edges may show large departures from the limiting graphene values. The distributions of the CC bond lengths and the CCC bond angles are shown in two new figures introduced to the manuscript. Figure 8 shows how the distributions of the CC bond lengths and the CCC bond angles change for the square Z(k,k) structures in the transition k->infinity. (Here, k=2,...,20.) Figure 9 shows how analogous distributions change for the rectangular Z(20,k) and Z(k,20) structures. An extensive discussion of these results is presented in Section 5(ii). The figures and the discussion are too extensive to quote them here; we ask the referee to inspect directly to the relevant fragment of the revised manuscript. The presented distributions suggest that indeed the transition to the infinite graphene-like regime occurs for relatively small flakes. For the square structures, the interior of the Z(k,k) flakes is already graphenic for k=6; for rectangular flakes, the transition happens even faster. At the same time, the finite-edge effects are clearly visible in the distributions for all the studied flakes, even for k=20.
2. How large are the relaxation effects? This is again a very good question, but we are currently unable to answer it. Indeed, we started all our optimizations from idealized patches of a hexagonal lattice defined by the two indices m and n, but the lattice constant did not match the DFTB lattice constant of the graphene sheet so we do not have an access to the corresponding energy of unrelaxed graphene patch. Moreover, the graphene patch obtained in this way would not include the hydrogen termination. Positions of the hydrogen atoms, determining the distributions of the CH bond lenghts and the CCH bond angles, are quite dependent on the position in the edge, showing that the energy relaxation effects would have two principal components, one stemming from the departure of the carbon sublattice from the original ideal graphene sheet geometry at the corners and at the edges of the patch, and the second one stemming from the geometry relaxation of the initially arbitrarily-placed perpendicular and equidistant hydrogen atoms. Disentangling these two relaxation effects would require additional, three-step geometry optimization. In the first step, the hydrogens would be kept perpendicular to the edge with the CH distance corresponding (for example) to the CH distance in benzene, while the positions of the carbon atoms would be optimized allowing to quantify the departure from the ideal hexagonal lattice geometry formed by the carbon atoms of the graphene patch, as suggested by the referee. In the second step, the positions of the terminal hydrogens would be optimized (with carbon fixed), allowing to assess the energy effect associated with relaxation of the initially uniform distribution of CH bond lengths and CCH bond angles along the edges. Finally, in the third step the complete optimization of the flake would be performed to quantify the synergy in the relaxation of the carbon and hydrogen sublattices simultaneously. We expect that the relaxation of the carbon sublattice would have comparable magnitude to the relaxation of the hydrogen sublattice, while the second-order synergy effects would be smaller by an order of magnitude. To complete such an analysis, for each flake we would need four energies: E0, EC, EH, ECH, corresponding respectively to an unrelaxed patch, a carbon-relaxed patch, a carbon-and-then-hydrogen-relaxed patch, and a completely relaxed patch. At the moment, we have access only to the ECH energies, which we computed using large computer cluster during the last 14 months of calculations. Completing the analysis by computing the E0, EC, and EH energies would take another 2 years of research and probably would constitute a topic of another MS thesis in our laboratory. We consider the relaxation a very interesting and valuable direction of research, but, regretfully, we must postpone its numerical quantification until later, as the publication of the current paper would need to be delayed by another 2 years in order to incorporate the data requested by the referee. I believe that the referee would understand this situation and would allow us to postpone this part of the project until later, while publishing the currently available results in order to communicate them to the chemical community.
Since our results may stimulate other groups to perform similar research, we decided to add a paragraph to our manuscript explaining the geometry relaxation issue. It would be optimal if other groups performed their research including this aspect of analysis from the same beginning.
The following fragment has been added to the Discussion section as item (iv): "The energies used to construct the energy expression given by Eq.~(\ref{eq:EH}) in addition to the usual size and shape dependence encoded by the parameters $m$ and $n$ include also the contributions related to the geometry relaxation effects. In our study, all these components are treated collectively. It would be very interesting to consider the relaxation effects individually, for example by starting the geometry optimization from a rectangular patch of idealized infinite graphene sheet with uniform hydrogen termination. The relaxation effects can be divided into three types of contributions: 1)~those corresponding the the relaxation of the carbon sublattice, 2)~those corresponding to the relaxation of the hydrogen sublattice, and 3)~those corresponding to the synergic relaxation of both lattices simultaneously. Such an analysis is beyond the scope of the current study." to address this point.
3. Passivation of carbon nanoflakes: Indeed, this effect was not considered in the current study. In principle, the passivation of carbon nanoflakes could be performed in many different ways, e.g., by deposing carbon monolayers on the surface of metal or by using surfactants. In both cases, the computational complexity of the resulting models would permit for studying but the smallest considered here graphene flakes, possibly permitting an investigation of geometrical changes in the flakes upon the passivation, but not allowing for finding energy decomposition, which is the primary goal of the current study. We feel that passivation of carbon nanosurfaces and the structural relaxation associated with it (expected to by substantial in certain cases) could be better studied by using periodic boundary conditions with several graphene unit cells combine into larger supercell. Such a choice would describe also better the metal surfaces on which the deposition happens. Again, while this is an interesting and valid direction of research, an investigation in this direction has not been attempted in the current study.
4. Effect of the XC functional on the results: The DFTB method has been parameterized using the PBE Hamiltonians and there is no possibility of changing this choice of the XC functional within the current parameterization. However, we have performed additional DFT calculations for the the family of Z(m,1) flakes (i.e., polyacenes) using DFT with 4 different functionals. Interestingly, the qualitative features of these calculations did not depend strongly on the choice of the functional, but in several cases quite substantial departure for certain values of physical parameters (e.g., the magnitude of the HOMO-LUMO energy gap in polyacenes was about twice smaller) were observed for results obtained with pure functionals (PBE) vs. hybrid functionals (B3LYP, CAM-B3LYP and omega-B97XD). These results, constituting the other half of Hendra's MS thesis, will be announced soon (we currently prepare a manuscript to be submitted to PCCP), and the whole study will be devoted to to analyzing the stability of the wave functions and the geometric modulation of structural parameters in long polyacenes using DFTB and various flavors of DFT.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors significantly improved the manuscript and can be published as it is.
