Classification of Fullerene Isomers Using Local Topological Descriptors
Tamás Réti1, István László2, Enikő Bitay3, Tomislav Došlić4
1Széchenyi István University, H-9026 Györ, Egyetem tér 1., Hungary
2Budapest University of Technology and Economic, H-1521, Budapest, Hungary
3Sapientia University, 540485, Tirgu Mures/Corunca, Romania
4Faculty of Civil Engineering, Kačićeva 26, 10000 Zagreb, Croatia
[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
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hexagons occurring in edge coronas, (iii) QE can be estimated as a weighted linear function of edge-
parameters (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
992211E m...mmQ ε++ε+ε= (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:
75443322110201E mmmmmMQ µ+µ+µ+µ+µ+µ+µ= (2)
where 5461206060 98601 =ε−ε+ε=µ (3)
7.1902 =ε=µ (4)
1.0322 98611 −=ε+ε−ε−ε=µ (5)
9.34342 986522 =ε+ε−ε−ε+ε=µ (6)
8.25464 586533 =ε+ε−ε−ε+ε=µ (7)
3.12 6544 −=ε+ε−ε=µ (8)
5.12 9875 −=ε+ε−ε=µ (9)
448 Materials Science, Testing and Informatics V
Stability prediction using edge parameters
It was supposed that the fullerene stability can be predicted as a function of algebraically
independent edge parameters.
Isomer
Topological parameters Energy, QC
(eV) m1 m2 m3 m4 m7 Np Ω
C40:38 0 0 10 8 10 10 2,727 -342,031
C40:39 0 0 10 10 10 10 2,727 -341,631
C40:31 0 6 5 5 11 11 2,500 -341,438
C40:29 0 6 5 6 11 11 2,500 -341,345
C40:26 0 3 8 8 8 11 2,250 -341,094
C40:24 0 3 8 9 8 11 2,250 -341,022
C40:37 0 0 11 10 6 11 2,083 -340,636
C40:40 0 12 0 0 12 12 2,308 -340,580
C40:14 1 7 4 5 9 12 2,077 -340,476
C40:36 0 0 11 10 5 11 2,000 -340,431
C40:30 0 9 3 6 9 12 2,077 -340,304
C40:25 0 6 6 8 7 12 1,923 -340,277
C40:22 0 6 6 9 6 12 1,846 -340,230
C40:35 0 0 11 10 5 11 2,000 -340,196
C40:21 0 6 6 10 7 12 1,923 -340,151
C40:27 0 6 6 8 6 12 1,846 -340,126
C40:15 1 4 7 6 6 12 1,846 -339,943
C40:17 1 10 2 3 7 13 1,714 -339,884
C40:34 0 3 9 10 4 12 1,692 -339,827
C40:28 0 6 6 9 7 12 1,923 -339,777
C40:16 2 8 3 2 7 13 1,714 -339,645
C40:20 0 3 9 12 3 12 1,615 -339,627
C40:9 2 8 3 4 8 13 1,786 -339,614
C40:10 1 7 5 8 5 13 1,571 -339,558
C40:12 1 7 5 7 5 13 1,571 -339,370
C40:13 1 7 5 8 4 13 1,500 -339,347
C40:19 1 10 2 4 7 13 1,714 -339,292
C40:23 0 6 7 12 3 13 1,429 -338,690
C40:6 2 8 4 7 3 14 1,267 -338,624
C40:18 1 10 3 6 4 14 1,333 -338,341
C40:5 3 9 2 5 7 14 1,533 -338,332
C40:32 0 12 2 8 2 14 1,200 -338,270
C40:8 4 10 1 2 4 15 1,188 -338,113
C40:33 0 12 2 8 4 14 1,333 -337,922
C40:4 3 9 3 6 3 15 1,125 -337,348
C40:7 2 11 2 6 3 15 1,125 -337,330
C40:11 2 8 5 8 1 15 1,000 -336,642
C40:2 4 10 2 6 2 16 0,941 -336,489
C40:3 6 12 0 4 0 18 0,632 -335,193
C40:1 10 10 0 0 0 20 0,476 -333,806
Table 1 Topological parameters and relative energy of forty C40 isomers
For prediction purposes the following topological descriptor has been defined:
1Np1
m311
mmm1
m31 7
321
7 −+
+=−
+++
+=Ω (10)
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
Materials Science Forum Vol. 659 449
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 Topological parameters Energy,QC
(eV) m1 m2 m3 m4 m7 NP Ω
C66:4169 0 0 2 1 18 2 15.33 -583.0067
C66:4348 0 0 2 0 17 2 15.00 -582.8916
C66:4466 0 0 2 0 17 2 15.00 -582.7047
C66:4007 0 0 3 1 14 3 10.25 -582.3229
C66:3764 0 0 3 1 12 3 9.75 -582.3027
C66:4456 0 0 3 0 12 3 9.75 -582.1878
C66:4462 0 0 3 0 12 3 9.75 -582.1816
C66:4060 0 0 3 1 12 3 9.75 -582.1267
C66:4141 0 0 3 1 14 3 10.25 -582.1118
C66:4312 0 0 3 0 15 3 10.50 -582.0754
C66:4439 0 0 3 0 11 3 9.50 -582.0316
C66:3765 0 0 3 1 13 3 10.00 -582.0278
C66:3538 0 0 3 1 14 3 10.25 -582.0220
C66:4447 0 0 3 0 13 3 10.00 -581.9169
C66:4458 0 0 3 0 13 3 10.00 -5819087
C66:4331 0 0 3 0 13 3 10.00 -581.8906
C66:4454 0 0 3 0 10 3 9.25 -581.8632
C66:3824 0 0 3 1 15 3 10.50 -581.8594
C66:4434 0 0 3 0 13 3 10.00 -581.8251
C66:4369 0 0 3 0 12 3 9.75 -581.8133
C66:4388 0 0 3 0 15 3 10.50 -581.8098
C66:4410 0 0 3 0 9 3 9.00 -581.8034
C66:4444 0 0 3 0 12 3 9.75 -581.7878
C66:4398 0 0 3 0 9 3 9.00 -581.7731
C66:4409 0 0 3 0 11 3 9.50 -581.7640
C66:4455 0 0 3 0 13 3 10.00 -581.6897
C66:3473 0 0 3 1 16 3 10.75 -581.5661
C66:4449 0 0 3 0 14 3 10.25 -581.5501
C66:4433 0 0 4 0 9 4 7.00 -581.4675
C66:3961 0 0 4 1 10 4 7.20 -581.4670
C66:4441 0 0 3 0 14 3 10.25 -581.4669
C66:4316 0 0 4 0 13 4 7.80 -581.4382
C66:4297 0 0 4 1 10 4 7.20 -581.3990
C66:4346 0 0 4 0 11 4 7.40 -581.3902
C66:4244 0 0 4 1 8 4 6.80 -581.3872
C66:4313 0 0 4 0 11 4 7.40 -581.3737
C66:4430 0 0 4 0 8 4 6.80 -581.3698
C66:4381 0 0 4 0 8 4 6.80 -581.3404
C66:4008 0 0 4 1 11 4 7.40 -581.3177
C66:4349 0 0 4 0 9 4 7.00 -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
450 Materials Science, Testing and Informatics V
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 tight-
binding-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.
Materials Science Forum Vol. 659 451
Materials Science, Testing and Informatics V 10.4028/www.scientific.net/MSF.659 Classification of Fullerene Isomers Using Local Topological Descriptors 10.4028/www.scientific.net/MSF.659.447
DOI References
[5] D. Porezag, Th. Frauenheim, Th. Köhler, G. Seifert and R. Kaschner: Constitution of tight- inding-like
potentials on the basis of density-functional therory: Application to carbon, Phys. Rev. ol. B51, (1995) p.
12947-12957.
doi:10.1103/PhysRevB.51.12947 [7] QB. Yan, QR. Zheng and G. Su: Theoretical study on the structures, properties and pectroscopies of
fullerene derivatives C66X4 (X=H, F, Cl), Carbon, Vol. 45, (2007) p. 1821-1827.
doi:10.1016/j.carbon.2007.04.036 [5] D. Porezag, Th. Frauenheim, Th. Köhler, G. Seifert and R. Kaschner: Constitution of tight- binding-like
potentials on the basis of density-functional therory: Application to carbon, Phys. Rev. Vol. B51, (1995) p.
12947-12957.
doi:10.1103/PhysRevB.51.12947 [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.
doi:10.1039/a901600g [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.
doi:10.1016/j.carbon.2007.04.036