Materials Science Forum Vol. 659 (2010) pp 447-451 © (2010) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.659.447
Classification of Fullerene Isomers Using Local Topological Descriptors Tamás Réti1, István László2, Enikő Bitay3, Tomislav Došlić4 1
Széchenyi István University, H-9026 Györ, Egyetem tér 1., Hungary
2
Budapest University of Technology and Economic, H-1521, Budapest, Hungary 3
Sapientia University, 540485, Tirgu Mures/Corunca, Romania
4
Faculty of Civil Engineering, Kačićeva 26, 10000 Zagreb, Croatia
1
2
3
4
[email protected],
[email protected],
[email protected],
[email protected]
Keywords: graph invariants, fullerene isomers, stability prediction Abstract. A method for the structural classification of fullerenes via graph invariants is presented. These graph invariants (called edge-parameters) represent the 9 different types of bonds existing in fullerenes between two neighbouring carbon atoms and they are also applicable to classify the fullerene isomers into equivalence classes. Discriminating performance of edge-parameters has been tested on the sets of C40 and C66 fullerene isomers. It is shown that the stability of C40 and C66 isomers can be efficiently predicted using a novel topological descriptor (Ω) defined as a function of four appropriately selected edge parameters. Introduction Starting with the extension of the concept detailed in Ref. [1], the aim of our investigations was to develop a general method which enables a more efficient classification of fullerenes. It is demonstrated that by analyzing the first neighbor environments of edges (edge coronas), algebraically independent edge-parameters (topological invariants) can be generated. These can be used to partition fullerene isomers into classes of equivalence and predict their stability. Discriminating performance of a novel topological descriptor has been tested on the set of C40 and C66 fullerene isomers. Edge parameters as topological invariants Alcami et al. developed a model devoted to estimate the enthalpy of formation (the energetic parameter QE) of traditional fullerenes Cn (n≤72) on the basis of 9 edge parameters generated from so-called edge coronas. These edge coronas represent the different first neighbor environments Ei (i=1,2,…9) of edges as shown in Fig.1 [1]. By definition, the i-th edge parameter mi=m(Ei) is identical to the number of edge-corona of type Ei. Consequently, Σ mi =M, where M stands for the total edge number of a traditional fullerene composed of pentagons and hexagons. It is easy to see that the pentagon adjacency index Np [2], (i.e. the number of edges between adjacent pentagons) can be simply calculated as a function of edge parameters: Np=m1 +m2 + m3. In the model outlined in Ref. [1], it was assumed that (i) every edge (i.e. every bond between two neighbor carbon atoms) represents a specific edge-energy value, (ii) edge energies are determined only by the edge-types, more exactly, by the local configurations of pentagons and All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 193.225.224.153-02/09/10,10:57:17)
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hexagons occurring in edge coronas, (iii) QE can be estimated as a weighted linear function of edgeparameters (m1, m2, …m9), where the positive weights are identical to the specific edge-energy values εj (1 ≤j ≤ 9) belonging to the 9 distinct edge-coronas. (See Fig1.) The edge-coronas have been previously considered in the context of perfect matching enumeration [3].
Fig. 1 Nine types of edge-coronas for traditional fullerenes According to the model, QE can be calculated as Q E = ε 1 m 1 + ε 2 m 2 + ... + ε 9 m 9
(1)
Specific edge–energy values εj given in Ref. [1] are as follows: ε1 = 19.8, ε2 = 17.6, ε3 = 10.3, ε4 = 15.7, ε5 = 12.4, ε6 = 7.8, ε7 = 6.2, ε8 = 4.7 and ε9 = 1.7. It has been verified that a linear interdependency can be found between these nine edge parameters, and if the number of total edge number (M) is fixed, there are only five parameters (m1, m2, m3, m4, m7) which are algebraically independent [4]. Consequently, QE can be calculated by the following simplified equation: Q E = µ 01 + µ 02 M + µ 1 m 1 + µ 2 m 2 + µ 3 m 3 + µ 4 m 4 + µ 5 m 7
(2)
µ 01 = 60 ε 6 + 60 ε 8 − 120 ε 9 = 546
(3)
µ 02 = ε 9 = 1 . 7
(4)
µ 1 = ε 1 − 2 ε 6 − 2 ε 8 + 3ε 9 = − 0 .1
(5)
µ 2 = ε 2 + 2 ε 5 − 4 ε 6 − 3ε 8 + 4 ε 9 = 3 .9
(6)
µ 3 = ε 3 + 4 ε 5 − 6 ε 6 − 4 ε 8 + 5ε 5 = 2 . 8
(7)
µ 4 = ε 4 − 2 ε 5 + ε 6 = − 1 .3
(8)
µ 5 = ε 7 − 2 ε 8 + ε 9 = − 1 .5
(9)
where
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Stability prediction using edge parameters It was supposed that the fullerene stability can be predicted as a function of algebraically independent edge parameters. Topological parameters Energy, QC
Isomer C40:38 C40:39 C40:31 C40:29 C40:26 C40:24 C40:37 C40:40 C40:14 C40:36 C40:30 C40:25 C40:22 C40:35 C40:21 C40:27 C40:15 C40:17 C40:34 C40:28 C40:16 C40:20 C40:9 C40:10 C40:12 C40:13 C40:19 C40:23 C40:6 C40:18 C40:5 C40:32 C40:8 C40:33 C40:4 C40:7 C40:11 C40:2 C40:3 C40:1
m1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 2 0 2 1 1 1 1 0 2 1 3 0 4 0 3 2 2 4 6 10
m2 0 0 6 6 3 3 0 12 7 0 9 6 6 0 6 6 4 10 3 6 8 3 8 7 7 7 10 6 8 10 9 12 10 12 9 11 8 10 12 10
m3 10 10 5 5 8 8 11 0 4 11 3 6 6 11 6 6 7 2 9 6 3 9 3 5 5 5 2 7 4 3 2 2 1 2 3 2 5 2 0 0
m4 8 10 5 6 8 9 10 0 5 10 6 8 9 10 10 8 6 3 10 9 2 12 4 8 7 8 4 12 7 6 5 8 2 8 6 6 8 6 4 0
m7 10 10 11 11 8 8 6 12 9 5 9 7 6 5 7 6 6 7 4 7 7 3 8 5 5 4 7 3 3 4 7 2 4 4 3 3 1 2 0 0
Np 10 10 11 11 11 11 11 12 12 11 12 12 12 11 12 12 12 13 12 12 13 12 13 13 13 13 13 13 14 14 14 14 15 14 15 15 15 16 18 20
Ω 2,727 2,727 2,500 2,500 2,250 2,250 2,083 2,308 2,077 2,000 2,077 1,923 1,846 2,000 1,923 1,846 1,846 1,714 1,692 1,923 1,714 1,615 1,786 1,571 1,571 1,500 1,714 1,429 1,267 1,333 1,533 1,200 1,188 1,333 1,125 1,125 1,000 0,941 0,632 0,476
(eV) -342,031 -341,631 -341,438 -341,345 -341,094 -341,022 -340,636 -340,580 -340,476 -340,431 -340,304 -340,277 -340,230 -340,196 -340,151 -340,126 -339,943 -339,884 -339,827 -339,777 -339,645 -339,627 -339,614 -339,558 -339,370 -339,347 -339,292 -338,690 -338,624 -338,341 -338,332 -338,270 -338,113 -337,922 -337,348 -337,330 -336,642 -336,489 -335,193 -333,806
Table 1 Topological parameters and relative energy of forty C40 isomers For prediction purposes the following topological descriptor has been defined: 31 + m 7 31 + m 7 −1 = −1 (10) 1 + m1 + m 2 + m 3 1 + Np It is should be noted that for the topological parameter Ω the inequality 0≤Ω≤60 holds. Since 0≤ Np ≤ 30 and 0≤ m7 ≤30, this implies that Ω=0 for fullerene C20 (dodecahedron) and Ω=60 for the buckminsterfullerene, only. In order to test the discriminating power of Ω, comparative tests were performed on the set of C40 and C66 isomers. The C40:n and C66:n isomer serial numbers were produced by the spiral computer program and all edge parameters were computed from the Schlegel diagram generated by the spiral codes [2]. Simultaneously, using Density Functional Tight-Binding (DFTB) method [5] we calculated the total energy values QC characterizing the relative stability of Ω=
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isomers. The number of topologically different C40 isomers is 40. All of them were generated and sorted in terms of the calculated total energy values. These energies and the corresponding topological parameters are summarized in Table 1. Isomer C66:4169 C66:4348 C66:4466 C66:4007 C66:3764 C66:4456 C66:4462 C66:4060 C66:4141 C66:4312 C66:4439 C66:3765 C66:3538 C66:4447 C66:4458 C66:4331 C66:4454 C66:3824 C66:4434 C66:4369 C66:4388 C66:4410 C66:4444 C66:4398 C66:4409 C66:4455 C66:3473 C66:4449 C66:4433 C66:3961 C66:4441 C66:4316 C66:4297 C66:4346 C66:4244 C66:4313 C66:4430 C66:4381 C66:4008 C66:4349
m1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Topological parameters m3 m4 m7 2 1 18 2 0 17 2 0 17 3 1 14 3 1 12 3 0 12 3 0 12 3 1 12 3 1 14 3 0 15 3 0 11 3 1 13 3 1 14 3 0 13 3 0 13 3 0 13 3 0 10 3 1 15 3 0 13 3 0 12 3 0 15 3 0 9 3 0 12 3 0 9 3 0 11 3 0 13 3 1 16 3 0 14 4 0 9 4 1 10 3 0 14 4 0 13 4 1 10 4 0 11 4 1 8 4 0 11 4 0 8 4 0 8 4 1 11 4 0 9
NP 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 3 4 4 4 4 4 4 4 4 4
Ω 15.33 15.00 15.00 10.25 9.75 9.75 9.75 9.75 10.25 10.50 9.50 10.00 10.25 10.00 10.00 10.00 9.25 10.50 10.00 9.75 10.50 9.00 9.75 9.00 9.50 10.00 10.75 10.25 7.00 7.20 10.25 7.80 7.20 7.40 6.80 7.40 6.80 6.80 7.40 7.00
Energy,QC (eV) -583.0067 -582.8916 -582.7047 -582.3229 -582.3027 -582.1878 -582.1816 -582.1267 -582.1118 -582.0754 -582.0316 -582.0278 -582.0220 -581.9169 -5819087 -581.8906 -581.8632 -581.8594 -581.8251 -581.8133 -581.8098 -581.8034 -581.7878 -581.7731 -581.7640 -581.6897 -581.5661 -581.5501 -581.4675 -581.4670 -581.4669 -581.4382 -581.3990 -581.3902 -581.3872 -581.3737 -581.3698 -581.3404 -581.3177 -581.2652
Table 2 Topological parameters and relative energy of the forty lowest energy C66 isomers As seen from Table 1, the most stable structures are isomers C40:38 and C40:39. Table 2 summarizes the calculated topological parameters and the corresponding energy values QC for the forty lowest energy C66 isomers. The number of topologically different C66 isomers is 4478. Among C66 fullerenes there are 3 isomers with lowest pentagon adjacency index (Np=2) and 26 isomers with Np=3. It appears that C66:4169 is the most stable. It is interesting to note that Np=m3 holds for the forty lowest energy C66 isomers. Comparing data included in Tables 1 and 2, it can be seen that topological descriptor Ω correlates highly with the computed total energy QC characterizing the relative stability of C40 and C66 isomers [6,7]. Summary and conclusions To characterize quantitatively the local combinatorial structure of lower fullerenes Cn with n≤70 a simple method has been suggested. The concept is based on the computation of 9 edge parameters generated from the so-called edge-coronas. For stability prediction purposes, a novel topological
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descriptor (Ω) has been defined. This includes not only the Np index, but an independent edge parameter (m7) as well. To test and evaluate the discriminating power of Ω the sets of C40 and C66 fullerene isomers have been chosen. In ranking the isomers according to their stability, the discriminating ability of Ω seems to be more efficient than that of pentagon adjacency index Np. Acknowledgements This work was supported by OTKA Foundation (no. K73776) and the Hungarian National Office of Research and Technology (NKTH) as a part of a Bilateral Cooperation Program (under contract no. HR-38/2008). References [1] M. Alcami, G. Sanchez, S. Diaz-Tendero, Y. Wang and F. Martin: Structural Patterns in Fullerenes Showing Adjacent Pentagons: C20 to C72, J. Nanosci. Nanotechnol. Vol.7, (2007) p. 1329-1338. [2] P. W. Fowler and D. E. Manolopoulos: An Atlas of Fullerenes, Calendron Press, Oxford, 1995. [3] T. Došlić: Importance and redundancy in fullerene graphs, Croat. Chem. Acta, Vol. 75 (2002) p. 869-879. [4] T. Réti, I. László and A. Graovac: Local Combinatorial Characterization of Fullerenes, in preparation. [5] D. Porezag, Th. Frauenheim, Th. Köhler, G. Seifert and R. Kaschner: Constitution of tightbinding-like potentials on the basis of density-functional therory: Application to carbon, Phys. Rev. Vol. B51, (1995) p. 12947-12957. [6] E. Albertazzi, C. Domene, P. W. Fowler, T. Heine, G. Seifert, C. Van Alsenoy and F. Zerbetto: Pentagon adjacency as a determinant of fullerene stability, Phys. Chem. Chem. Phys., Vol. 1, (1999) p. 2913-2918. [7] QB. Yan, QR. Zheng and G. Su: Theoretical study on the structures, properties and spectroscopies of fullerene derivatives C66X4 (X=H, F, Cl), Carbon, Vol. 45, (2007) p. 1821-1827.
