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

1reti@sze.hu, 2laszlo@eik.bme.hu,

3bitay.eniko@eme.ro, 4doslic@grad.hr

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

Materials Science Forum Vol. 659 (2010) pp 447-451Online available since 2010/Sep/13 at www.scientific.net© (2010) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/MSF.659.447

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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).

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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

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