# zbMATH — the first resource for mathematics

The geometry of $$C_{60}$$: a rigorous approach via molecular mechanics. (English) Zbl 1353.82072
The fullerenes have attracted an immerse amount of attention in the scientific literature. The identification of their three-dimensional structure; the study of their chemical properties, including aromaticity, solubility, and electrochemistry; and their application in medicine and pharmacology have developed into the new branch of Fullerene Chemistry. Usually a central question concerning fullerenes is their stability, either from the electrochemical, or the mechanical standpoint. Stability is believed to be a key factor in explaining why just a few fullerenes isomers out of a theoretically predicted wide variety have actually been revealed. Among these the fullerene $$C_{60}$$ experimentally discovered in 1985 is remarkably stable and considerably amounts of these molecules have been detected in the interstellar space. Here a rigorous analysis of the geometric structures and the stability properties of the $$C_{60}$$ molecule is given. The analysis set within the variational frame of molecular mechanics. This consists of modeling molecular configurations in terms of classical mechanics: atomic relations are described by classical interaction potentials between atomic positions. The bibliography of the article consists of 55 references.

##### MSC:
 82D25 Statistical mechanical studies of crystals 82D80 Statistical mechanical studies of nanostructures and nanoparticles
##### Keywords:
crystals; nanostructures; nanoparticles
CHARMM; GROMOS
Full Text:
##### References:
 [1] G. Albarrán, V. A. Basiuk, E. V. Basiuk, and J. M. Saniger, Stability of interstellar fullerenes under high-dose $$γ$$-irradiation, Adv. Space Res., 33 (2004), pp. 72–75. [2] N. L. Allinger, Molecular Structure, Understanding Steric and Electronic Effects from Molecular Mechanics, John Wiley & Sons, Hoboken, NJ, 2010. [3] X. Blanc and C. Le Bris, Periodicity of the infinite-volume ground state of a one-dimensional quantum model, Nonlinear Anal., 48 (2002), pp. 791–803. · Zbl 0992.82043 [4] X. Blanc and M. Lewin, The crystallization conjecture: A review, EMS Surv. Math. Sci., 2 (2015), pp. 219–306. · Zbl 1341.82010 [5] D. E. Botchvar and E. G. Galpern, About hypothetical cystems: Carbododecahedron, s-icosahedron, and carbo-s-icosahedron, Dokl. Akad. Nauk, 209 (1973), p. 610 (in Russian). [6] D. W. Brenner, Empirical potential for hydrocarbons for use in stimulating the chemical vapor deposition of diamond films, Phys. Rev. B, 42 (1990), pp. 9458–9471. [7] B. R. Brook, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4 (1983), pp. 187–217. [8] M. Clark, R. D. Cramer III, and N. Van Opdenbosch, Validation of the general purpose tripos $$5.2$$ force field, J. Comput. Chem., 10 (1989), pp. 982–1012. [9] E. Davoli, P. Piovano, and U. Stefanelli, Sharp $$N^{3/4}$$ law for the minimizers of the edge-isoperimetric problem on the triangular lattice, submitted, 2015. [10] E. Davoli, P. Piovano, and U. Stefanelli, Wulff Shape Emergence in Graphene, Math. Models Methods Appl. Sci., to appear. · Zbl 1355.82073 [11] W. E and D. Li, On the crystallization of 2D hexagonal lattices, Comm. Math. Phys., 286 (2009), pp. 1099–1140. · Zbl 1180.82191 [12] W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), pp. 241–297. · Zbl 1106.74019 [13] R. S. Elliott, N. Triantafyllidis, and J. A. Shaw, Stability of crystalline solids — I: Continuum and atomic lattice considerations, J. Mech. Phys. Solids, 54 (2006), pp. 161–192. · Zbl 1120.74392 [14] B. Farmer, S. Esedoḡlu, and P. Smereka, Crystallization for a Brenner-like potential, Comm. Math. Phys., to appear, 2016. [15] L. Flatley, M. Taylor, A. Tarasov, and F. Theil, Packing twelve spherical caps to maximize tangencies, J. Comput. Appl. Math., 254 (2013), pp. 220–225. · Zbl 1292.52019 [16] L. Flatley and F. Theil, Face-centered cubic crystallization of atomistic configurations, Arch. Ration. Mech. Anal., 218 (2015), pp. 363–416. · Zbl 1335.82006 [17] G. Friesecke and F. Theil, Validity and failure of the Cauchy–Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), pp. 445–478. · Zbl 1084.74501 [18] C. S. Gardner and C. Radin, The infinite-volume ground state of the Lennard-Jones potential, J. Stat. Phys., 20 (1979), pp. 719–724. [19] W. F. van Gunsteren and H. J. C. Berendsen, Groningen Molecular Simulation (GROMOS) Library Manual, BIOMOS b.v., Groningen, 1987. [20] R. C. Haddon, $$π$$-electrons in three dimensions, Acc. Chem. Res., 21 (1988), pp. 243–249. [21] R. C. Haddon and L. T. Scott, $$π$$-orbital conjugation and rehybridation in bridged annulenes and deformed molecules in general: $$π$$-orbital axis vector analysis, Pure $$\&$$ Appl. Chem., 58 (1986), pp. 137–142. [22] G. C. Hamrick and C. Radin, The symmetry of ground states under perturbation, J. Stat. Phys., 21 (1979), pp. 601–607. [23] R. Heitman and C. Radin, Ground states for sticky disks, J. Stat. Phys., 22 (1980), pp. 281–287. [24] T. L. Hill, On steric effects, J. Chem. Phys., 14 (1946), 465. [25] T. L. Hill, Steric effects. I. Van der Waals potential energy curves, J. Chem. Phys., 16 (1948), 399. [26] R. D. James, Objective structures, J. Mech. Phys. Solids, 54 (2006), pp. 2354–2390. · Zbl 1120.74312 [27] S. Jansen, W. König, B. Schmidt, and F. Theil, Low temperature Lennard-Jones chains, Oberwolfach Rep., 35 (2015), pp. 243–245. [28] H. W. Kroto, The stability of the fullerenes $$C_n$$, with n=24, 28, 32, 36, 50, 60, and 70, Nature, 329 (1987), pp. 529–531. [29] H. W. Kroto, $$C_{60}$$: Buckminsterfullerene, the celestial sphere that fell to earth, Angew. Chem. Int. Ed. Engl., 31 (1992), pp. 111–129. [30] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, $$C_{60}$$: Buckminsterfullerene, Nature, 318 (1985), pp. 162–163. [31] E. G. Lewars, Computational Chemistry, 2nd ed., Springer, New York, 2011. · Zbl 1367.92005 [32] E. Mainini, H. Murakawa, P. Piovano, and U. Stefanelli, Carbon-nanotube geometries: Analytical and numerical results, Discrete Cont. Dyn. Syst. Ser. S, 2016, to appear. · Zbl 1362.82059 [33] E. Mainini, H. Murakawa, P. Piovano, and U. Stefanelli, Carbon-nanotube geometries as optimal configurations, in preparation, 2016. · Zbl 1394.82028 [34] E. Mainini, P. Piovano, and U. Stefanelli, Finite crystallization in the square lattice, Nonlinearity, 27 (2014), pp. 717–737. · Zbl 1292.82043 [35] E. Mainini, P. Piovano, and U. Stefanelli, Crystalline and isoperimetric square configurations, Proc. Appl. Math. Mech., 14 (2014), pp. 1045–1048. · Zbl 1292.82043 [36] E. Mainini and U. Stefanelli, Crystallization in carbon nanostructures, Comm. Math. Phys., 328 (2014), pp. 545–571. · Zbl 1391.82058 [37] S. L. Mayo, B. D. Olafson, and W. A. Goddard, DREIDING: A generic force field for molecular simulations, J. Phys. Chem., 94 (1990), pp. 8897–8909. [38] E. Osawa, Superaromaticity, Kagaku (Kyoto), 25 (1970), pp. 854–863. [39] C. Radin, The ground state for soft disks, J. Stat. Phys., 26 (1981), pp. 365–373. [40] C. Radin, Classical ground states in one dimension, J. Stat. Phys., 35 (1983), pp. 109–117. [41] A. K. Rappé and C. J. Casewit, Molecular Mechanics across Chemistry, University Science Books, Sausalito, CA, 1997. [42] D. Sfyris, Phonon, Cauchy-Born and homogenized stability criteria for a free-standing monolayer graphene at the continuum level, Eur. J. Mech A–Solid, 55 (2016), pp. 134–148. · Zbl 1406.74160 [43] L. Schen and J. Li, Equilibrium structure and strain energy of single-walled carbon nanotubes, Phys. Rev. B, 71 (2005), 165427. [44] U. Stefanelli, Stable Carbon Configurations, submitted, 2016. [45] F. H. Stillinger and T. A. Weber, Computer simulation of local order in condensed phases of silicon, Phys. Rev. B, 8 (1985), pp. 5262–5271. [46] S. J. Stuart, A. B. Tutein, and J. A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions, J. Chem. Phys., 112 (2000), pp. 6472–6486. [47] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37 (1988), pp. 6991–7000. [48] F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006), pp. 209–236. · Zbl 1113.82016 [49] W. J. Ventevogel, On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle, Phys. A., 92 (1978), pp. 343–361. [50] W. J. Ventevogel and B. R. A. Nijboer, On the configuration of systems of interacting particle with minimum potential energy per particle, Phys. A., 98 (1979), pp. 274–288. [51] W. J. Ventevogel and B. R. A. Nijboer, On the configuration of systems of interacting particle with minimum potential energy per particle, Phys. A., 99 (1979), pp. 565–580. [52] H. J. Wagner, Crystallinity in two dimensions: A note on a paper of C. Radin, J. Stat. Phys., 33 (1983), pp. 523–526. [53] P. K. Weiner and P. A. Kollman, AMBER: Assisted model building with energy refinement. A general program for modeling molecules and their interactions, J. Comput. Chem., 2 (1981), pp. 287–303. [54] F. H. Westheimer and J. E. Mayer, The theory of the racemization of optically active derivatives of diphenyl, J. Chem. Phys., 14 (1946), 733. [55] C. S. Yannoni, P. P. Bernier, D. S. Bethune, G. Meijer, and J. R. Salem, NMR determination of the bond lengths in $$C_{60}$$, J. Amer. Chem. Soc., 113 (1991), pp. 3190–3192.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.