Nonclassical Cn (n = 30–40, 50) fullerenes containing five-, six-, seven-member rings

Computational and Theoretical Chemistry 969 (2011) 35–43

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Nonclassical Cn (n = 30–40, 50) fullerenes containing five-, six-, seven-member rings Lingli Tang a,b, Linwei Sai a,b, Jijun Zhao b,c,⇑, Ruifeng Qiu a a

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China c Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China b

a r t i c l e

i n f o

Article history: Received 25 April 2011 Received in revised form 5 May 2011 Accepted 8 May 2011 Available online 13 May 2011 Keywords: Nonclassical fullerene Topological index Relative stability

a b s t r a c t Classical and heptagon-contained nonclassical fullerene isomers of C30–C40 and C50 are generated by Spiral algorithm and optimized using DFT methods. There are much larger number of nonclassical isomers than classical ones. Energies of isomers generally increase with the number of heptagons inside and those lowest-energy nonclassical isomers contain only one heptagon. For all fullerenes considered, there is no nonclassical isomer that energetically prevails over the lowest-energy classical counterpart. But the best nonclassical isomers are more stable than most classical ones. The validity of pentagon adjacency penalty rule (PAPR) and hexagon neighbor rule (HNR) is examined for the nonclassical fullerenes. PAPR is still roughly applicable while HNR almost breaks down. A new topological index U with better performance over PAPR and HNR is presented and its superiority is further confirmed on two subsets of C50 isomers. This U index can be considered as an extension of PAPR to nonclassical isomers with existence of heptagons. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Nonclassical fullerenes (NCFs) are cage-like carbon clusters containing other kinds of polygons in addition to the conventional fiveand six-member rings in the classical fullerenes (CFs) [1,2]. They are generally considered to be less stable than the CFs because of the extra local strain energy and further loss of p delocalization arising from the quadrangles, heptagons, and larger polygons [3]. However, some exceptional cases still exist. For example, two nonclassical isomers of C62, one with a heptagon and another with a square, were predicted to be more stable than all classical isomers [4–7]. Recently, fullerene derivatives of C58F17CF13 and C58F17 containing two seven-member rings (7MRs) were successfully isolated in experiment [8]. For smaller carbon clusters, a non-classical C22 fullerene cage with one 4MR and a C24 cage with two 4MRs were predicted as the ground state structures by theoretical calculations [9,10]. Compared to the comprehensive studies of CFs [5,6,11–16], less attention has been paid to NCFs. Using empirical and semi-empirical approaches like molecular mechanics, Hückel molecular orbital (HMO), and modified neglect of differential overlap (MNDO), Gao and Herdon showed some evidences that the nonclassical isomers with two squares can be more favorable than the classical ⇑ Corresponding author at: College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China. Tel.: +86 411 84709748; fax: +86 411 84706100. E-mail address: [email protected] (J. Zhao). 2210-271X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.comptc.2011.05.009

fullerenes when the fullerene size is less than 60 [17]. However, the reliability of these empirical calculations is questionable [17,18]. Using semi-empirical methods, C40 isomers consisting of squares, pentagons, hexagons, and those consisting of pentagons, hexagons, heptagons were investigated [18,19]. The corresponding generalized Stone–Wales map [20,21] involving some isomers with quadrangles, pentagons, hexagons and heptagons was afterward plotted [22]. Fowler et al. carried out a systematic study on all isomers of C40 and C48 including one octagon utilizing semiempirical methods [23]. All these studies reached almost the same conclusion, that is, no nonclassical isomer containing one or more squares, heptagons, octagons can have higher stability than their best classical counterparts, but many of them fall within the energy range spanned by the classical ones [18,19,22,23]. A generalized isolated pentagon rule (GIPR) was proposed for NCFs [19,23]: when a large ring is present, those low-energy structures maximize the number of their contacting pentagons and minimize the pentagon–pentagon adjacencies. Hernández et al. [24] conducted a systematic study on the CF isomers and one-heptagon NCF isomers from C30 to C70 using tight-binding model as well as DFT method. They found that those nonclassical one-heptagon isomers are competitive in energy with classical ones. In particular, a oneheptagon isomer was more stable than any of its classical counterparts for C62, in agreement with the results by Ayuela et al. [4]. Recently, Gan et al. performed DFT calculations on several isomers of C28 and C36 consisting of quadrangles, pentagons, hexagons, heptagons and proposed a model to predict the relative stability of polyhedron molecules of carbon [25]. Since only very limited amount of

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L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

isomers are considered, the applicability of the model for other fullerene sizes is uncertain. NCFs with heptagons may play a role in fullerene growth, fragmentation, and preparation of endohedral fullerene complexes. During the MD simulation, Li and Ning [26] observed frequent occurrence of NCFs (particularly those with heptagons) during the relaxation process of a C60 isomer which can geometrically rearrange to the Ih C60 isomer via 30 times of consecutive Stone– Wales rearrangements [20,21]. Hernández et al. presented three new mechanisms of incorporating carbon dimers into closed fullerene cages which resulted in formation of NCF isomers with heptagons [24]. Using semiempirical and ab initio methods, Murry et al. suggested a role for seven-member rings in the processes of fullerene annealing and fragmentation [27,28]. Xu and Scuseria simulated the fragmentation process from a perfect C60 (Ih) to a oneheptagon C58 isomer utilizing tight-binding molecular dynamics [29]. Employing Hartree–Fock and DFT methods, a one-heptagon C58 NCF isomer was considered as an ideal choice for preparation of endohedral fullerene complexes because it has larger diameter than conventional fullerenes and higher stability than most other candidates except a classical one. All candidates were generated by removing one to four adjacent atoms from the Ih C60 isomer [30–32]. Due to the existence of NCF isomers that are energetically competitive with or occasionally even superior to the CF isomers and their possible involvement in fullerene growth, fragmentation and preparation of endohedral fullerene complexes, a comprehensive investigation on these NCFs (especially those with heptagons) is crucial. So far, our knowledge on these NCFs is still rather limited. In particular, the relationship between the stability and the topology of these NCF/CF isomers is unknown. In this work, we systematically investigated the energies and relative stabilities of the CF isomers and NCF isomers with heptagons for C30–C40 and C50. We proposed a topological index that can effectively estimate the relative energies of both CF and NCF isomers. This new index prevails over the previously proposed pentagon adjacency penalty rule (PAPR) [33,34] and hexagon neighbor rule (HNR) [35]. Thus, it can be used to study the larger non-classical fullerenes for which a complete search of the most stable structure using first-principles are computational prohibited.

2. Methods Topological structures of CFs were generated using standard Spiral algorithm [36]. For those NCFs with heptagons, the standard Spiral algorithm can be used in a nearly straightforward way. In the standard algorithm, the sequences with only five and six are generated; while here we generated the sequences consisting of five, six and seven in order to introduce heptagons. The initial structures with desired topologies were firstly relaxed by empirical force field. All the fullerene structures were then optimized using

Table 1 Detailed numbers of CF and NCF isomers for C30–C40 and C50. Nm-7MR denotes the number of NCF isomers with m heptagons. NNCF (NCF) denotes total number of NCF (CF) isomers. Size

N1–

N2–7MR

N3–7MR

N4–7MR

N5–7MR

7MR

30 32 34 36 38 40 50

1 2 8 16 42 92 2784

0 3 5 23 57 182 12,767

0 0 1 5 21 82 27,108

0 0 0 2 1 30 27,579

0 0 0 0 0 0 12,895

N6–

N7–

7MR

7MR

0 0 0 0 0 0 2221

0 0 0 0 0 0 86

NNCF

NCF

1 5 14 46 121 386 85,440

3 6 6 15 17 40 271

density functional theory (DFT) methods and the energy sequence of these isomers was ranked. DFT calculations were performed using DMol3 program [37,38] within the generalized gradient approximation (GGA) with Perdew–Burke–Ernzerholf (PBE) functional [39] for exchange–correlation interaction. A double numerical basis set including d-polarization functions (DND) [37,38] was used. Self-consistent field (SCF) calculations were carried out with a convergence criterion of 105 Hartree on the total energy. Geometry optimization was performed using a convergence criterion of 2.0  105 Hartree on maximum energy gradient and 0.005 Å on maximum displacement for each atom. 3. Results and discussion 3.1. Number of isomers Table 1 summarizes the number of CF and NCF isomers for C30– C40 and C50. Generally speaking, the numbers of both types of isomers increase rapidly with fullerene size, whereas the amount of NCF ones increases much faster. Up to C50, the number of CF isomers is less than 1/300 of the number of NCF isomers. In other words, NCF is much more abundant than CF, which further increases the versatility of carbon fullerenes. For a given size of NCF, the number of isomers does not monotonically increase with the number of heptagons inside. It first increases, reaches saturation number at about half of the possible maximum number of heptagons, and then decreases. 3.2. Energies of isomers for C30–C40 Fig. 1 gives the relative energies of CF and NCF isomers with regard to the lowest-energy configuration for every fullerene size (C30–C40). All CF isomers and NCF isomers with heptagons are considered. In general, energies of NCF isomers increase with the number of heptagons. But this relationship is not rigorous. Many NCF isomers containing more heptagons are more stable than those with fewer heptagons. Within the size range considered, all of the best nonclassical isomers contain only one heptagon and they energetically prevail over most classical isomers. However, none of these NCF isomers is the lowest-energy configuration with regard to the best classical CF ones. It is generally accepted that introducing seven-member rings will raise the system’s energy due to extra local strain energy [3]. Additionally, a convex polyhedron with v vertex, e edges, f faces obeys the Euler theorem [36]:

v þf e¼2

ð1Þ

The trivalence of all carbon atoms on the fullerene cages forces the conditions:

e ¼ 3v =2

ð2Þ

f ¼ v =2 þ 2

ð3Þ

Let fn denote the number of n-sides polygons, we have:

X ð6  nÞfn ¼ 12

ð4Þ

n

Applying Eq. (4) to fullerenes consisting of pentagons, hexagons, and heptagons, following equation is obtained:

f5 ¼ f7 þ 12

ð5Þ

Therefore, introducing heptagons accompanies with increasing number of pentagons, which would further result in more number of adjacent pentagons and then raise the energy according to PAPR. On the other hand, as suggested by GIPR [19,23], formation of

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L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

6 One 7MR

1.5 1.0 0.5 0.0

Relative energy (eV)

Relative energy (eV)

2.0

Classical

-0.5

5 4 3

Two 7MR

2 One 7MR

1 0 -1

Classical

C 30

C 32

6

10

4 Two 7MR

3

Three 7MR

2 1 0

One 7MR

Relative energy (eV)

Relative energy (eV)

5

8 6 4

One 7MR

8

14

7

12

5 4

Four 7MR

3 Three 7MR

2 1 0 -1

Two 7MR One 7MR

Relative energy (eV)

Relative energy (eV)

C 34

6

C 38

Two 7MR

Classical --

C 36

10 8 6 4 Three 7MR Two 7MR

2 0

Classical

Three 7MR

2 0

Classical

Four 7MR

Classical

Four 7MR

One 7MR

C 40

Fig. 1. Energies of all CF and NCF isomers for C30–C40. All energies are relative to the most stable isomer for each fullerene size.

pentagon–heptagon pair on the fullerene cage may reduce the strain energy. This effect can explain that some NCF isomers with more heptagons are more stable than those with fewer heptagons. The structures of the best CF and NCF isomers for C30–C40 are displayed in Fig. 2. Coincidently, all of the best NCF isomers possess Cs symmetry. In the range of C30–C40, binding energies for both the best CF and NCF isomers increase with size (Fig. 3). The energy differences between the most favorable CF and NCF configurations do not vary monotonously with size and the smallest difference of only 0.021 eV/atom is found at C34. It is also noteworthy that all best CF isomers have the least number of adjacent pentagons among all isomers, consistent with PAPR. For the best NCF isomers, the number of adjacent pentagons is least among all NCF isomers and they all contain seven pentagon–heptagon pairs, which is the largest number they could possibly have. This also supports the GIPR [19,23]. 3.3. Topological indices It has been recognized that PAPR is effective to estimate relative stability of CF isomers. Another rule for CF isomers is HNR which was presented by Raghavachari [35] and was considered to per-

form better than PAPR in some cases [40]. The essential idea behind the HNR is that the hexagonal environments in a stable fullerene should be as close as possible to spread the steric strain P 2 P evenly. It can be quantified as H ¼ k k hk = k hk , where the socalled hexagon neighbor signature hk [35] is the number of hexagons with exact k hexagonal neighbors. For the NCF isomers with heptagons, it remains an open question whether these topological rules are still applicable. Fig. 4 shows the correlation between the number of adjacent pentagons (N55) and the relative energies of all CF and heptagoncontained NCF isomers for C30–C40. Generally speaking, the isomer energy rises with increasing number of adjacent pentagons for both CF and NCF isomers. For classical fullerenes, it was known that pentagon fusions will bring both steric and electronic disadvantages [3,14]. This effect is also valid for the NCF isomers. However, compared to the nearly linear dependence of relative energy with N55 for the CF isomers, the NCF data somewhat scatter. Thus, number of adjacent pentagons is an important but not the sole factor for determining the relative stability of NCF isomers. As a representative, Fig. 5 shows the relative energy versus index H for both CF and NCF isomers of C40. Despite the excellent

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L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

CF C30(C2v)

NCF C30(Cs)

CF C32(D3)

NCF C32(Cs)

CF C34(C2)

NCF C34(Cs)

CF C36(D6h)

NCF C36(Cs)

CF C38(C2)

NCF C38(Cs)

CF C40(D2)

NCF C40(Cs)

Binding energy (eV/atom)

Fig. 2. Structures of the best CF and NCF isomers for C30–C40.

ative energies of NCF isomers. On the one hand, heptagons introduce extra local strain energy that makes the system unstable [3]; on the other hand, they lead to more pentagon–heptagon adjacencies that are energetically favored as denoted by GIPR [19,23]. Thus, it would be desirable to induce a new topological index that is able to take into account the effect of heptagons. In this work, we define a so-called united index U as:

8.55 8.50

Classical 8.45 8.40

Non-classical

8.35

U ¼ N55  N57 þ 5N7

8.30 8.25 30

32

34

36

38

40

Fullerene size Fig. 3. Average binding energies of the best CF and NCF isomers for C30–C40.

performance for CF, the HNR breaks down in the case of NCF with heptagons. Interestingly, the CF and NCF data are clearly separated from each other, while the NCF ones distribute above those for CF isomers. In fact, for the classical fullerenes with only pentagons and hexagons, a high H index corresponds to large portion of graphitic patches on somewhere of the cage surface and also implies crowding of pentagons elsewhere, which would thus result in increase of energy. In the case of NCF isomers, because of the existence of heptagons, H not longer reflects the distribution of pentagons well particularly when the number of heptagons is more than one. In addition, for smaller sized NCF, those heptagon-contained isomers possess fewer hexagons compared with pentagons. Therefore, the index H which only involves hexagons is not sufficient to evaluate the relative stability of NCF isomers. Though neither PAPR nor index H concerns with heptagons, it is no doubt that heptagons play a crucial role in determining the rel-

ð6Þ

where N7 denotes the number of heptagons, N57 means the number of pentagon–heptagon pairs, and N55 is the number of adjacent pentagons. Using this united index, the lower U value, the higher relative stability of the fullerene isomer. Obviously, this empirical rule based on the united index in Eq. (6) is an extension of the wellknown PAPR originally proposed for the CF isomers [33,34]. In Fig. 6, we show the relative energies of fullerene isomers versus U index for C30–C40. Compared to the scattered data for PAPR in Fig. 4, the present empirical rule exhibits better performance for evaluating the relative stability of both CF and NCF isomers. This can be further quantified by the standard deviations of leastsquare linear fit for Figs. 4 and 6, which are summarized in Table 2. The standard derivations for U index are about 0.4 eV less than those for N55 by PAPR except that the difference is about 0.2 eV for C36. Such comparison clearly demonstrates that the present united index U is a more efficient way to evaluate the relative stability of fullerene isomers when there exist heptagons. It is noteworthy that an alternative U formula can also be obtained from numerical fitting, which might reproduce the data for small fullerenes (C30–C40) better. However, its transferability to the other systems would be questionable. Therefore, here we choose to use an empirical U formula following chemistry intuition, which is supposed to be more universal.

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L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

Relative energy (eV)

Relative energy (eV)

classical nonclassical

2.0

1.5

1.0

0.5

C 30

0.0

6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5

17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5

classical nonclassical

C32

14

16

18

20

N55 6

4 3 2

C34

1

nonclassical

6 5 4 3 2

C36

0 14

16

18

20

22

12

24

14

16

18

20

N55

nonclassical

6 5 4 3 2

C38

0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

N55

Relative energy (eV)

Relative energy (eV)

classical

1

22

24

26

28

30

N55

9

7

26

7

1

0

8

24

classical

8

nonclassical

Relative energy (eV)

Relative energy (eV)

9

classical

5

22

N55

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1

classical nonclassical

C40

10

12

14

16

18

20

22

24

26

28

30

N55

Fig. 4. Relative energies of isomers versus number of adjacent pentagons (N55) for C30–C40. All energies are relative to the most stable isomer for each fullerene size. CF and NCF isomers are discriminated by different shape dots: solid squares for CF and the hollow circles for NCF.

3.4. C50 isomers To further validate the current topological rule, we considered a large fullerene C50, which has totally 85,440 NCF isomers with possible number of heptagons ranging from one to seven. Due to the large number of isomers, here we chose two subsets of C50 isomers, i.e., the NCF isomers containing only one heptagon (Subset One); all the CF isomers, and those NCF isomers with U index less than 14 (Subset Two). The numbers of isomers are then reduced to 2784 and 3838 for the first and second subsets respectively, which are

still large enough to examine the relationship between the isomer energies and the topological indexes. Figs. 7 and 8 display the relative energies of C50 isomers versus various indexes (N55, U, H) for the first and second subsets, respectively. Similar to C40, index H performs excellently for the CF isomers of C50, but it does not work well for the heptagoncontained NCF isomers, especially for the Subset Two. As clearly shown by the scattering of data as well as the standard deviation of least-square linear fit, the united index U performs better than either the N55 by PAPR or the H by HNR for evaluating the relative

L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

Relative energy (eV)

40

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1

Table 2 Standard deviation (SD) of least-square linear fit to Figs. 4 and 6. Unit of SD is in eV.

classical nonclassical

SD (N55) SD (U)

0

2

4

6

8

10

12

H

14

16

SD: 2.13

Fig. 5. Relative energies of isomers versus index H for C40. All energies are relative to the most stable C40 isomer. CF and NCF isomers are discriminated by different shape dots: solid squares for CF and the hollow circles for NCF. SD (in unit of eV) is the standard deviation of least-square linear fit to the plot.

Relative energy (eV)

Relative energy (eV)

1.5 1.0 0.5

C30

32

34

36

38

40

0.57 0.242

0.794 0.36

0.846 0.409

0.943 0.787

0.934 0.56

1.078 0.74

stability of C50 isomers. Although the current two subsets only constitute a small portion of the entire isomer set, we expect that they can sample all of the isomers reasonably well. Thus, the superiority of U to N55 or H should still retain and might be even more significant on the entire set. For the isomers in the Subset One containing only one heptagon, the third term in Eq. (6) has no effect on differentiating the stability of different isomers. Thus the superiority of U to N55 shown in Fig. 7 can be attributed to the contribution of the second term – the number of pentagon–heptagon pairs. This supports the viewpoint of GIPR [19,23] that adjacency of pentagon and heptagon is

6

classical nonclassical

2.0

30

0.0

classical nonclassical

5 4 3 2 1

C32

0 15

17.0 17.5 18.0 18.5 19.0 19.5 20.0

16

17

18

U 9

classical nonclassical

5

8

4 3 2 1

C34

0 16

18

20

22

23

7 6 5 4 3 2

C36

1

12 14 16 18 20 22 24 26 28 30

24

U

classical nonclassical

6 5 4 3 2

C38

1 0

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

U

Relative energy (eV)

Relative energy (eV)

7

22

classical nonclassical

U

8

21

0 14

9

20

U

Relative energy (eV)

Relative energy (eV)

6

19

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1

classical nonclassical

C40 10 12 14 16 18 20 22 24 26 28 30 32 34

U

Fig. 6. Relative energies of isomers versus index U for C30–C40. All energies are relative to the lowest energy isomer for each fullerene size. CF and NCF isomers are discriminated by different shape dots: solid squares for CF and the hollow circles for NCF.

41

12

12

10

10

Relative energy (eV)

Relative energy (eV)

L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

8 6 4 2 0 8

10

12

14

16

N55

18

8 6 4 2 0

20

6

8

10

12

14

U

SD: 0.851

16

18

20

22

24

SD: 0.644

Relative energy (eV)

12 10 8 6 4 2 0 4

8

6

10

12

14

H

16

18

SD: 1.3

Fig. 7. Relative energies of one-heptagon NCF isomers of C50 versus N55, index U and H. All energies are relative to the most stable one-heptagon C50 isomer. SD (in unit of eV) is the standard deviation of least-square linear fit to each plot.

16 12 10 8 6 4 2

12 10 8 6 4 2

0

0

-2

-2

4

6

8

classical nonclassical

14

Relative energy (eV)

Relative energy (eV)

16

classical nonclassical

14

10 12 14 16 18 20 22

4

SD: 0.91

N55

16

8

10 12 14 16 18 20 22

U

SD: 0.732

classical nonclassical

14

Relative energy (eV)

6

12 10 8 6 4 2 0 -2 0

2

4

6

8

10 12 14 16 18 20 22 24

H

SD: 1.02

Fig. 8. Relative energies of all C50 classical isomers, and those nonclassical ones with U < 14 versus N55, index U and H. All energies are relative to the most stable isomer of C50. SD (in unit of eV) is the standard deviation of least-square linear fit to each plot. CF and NCF isomers are discriminated by different shape dots: solid squares for CF and the hollow circles for NCF.

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L. Tang et al. / Computational and Theoretical Chemistry 969 (2011) 35–43

C50 (D3,0.0eV)

C50 (Cs,1.68eV)

C50 (D5h,0.18eV)

C50 (C1,1.749eV)

C50 (Cs,1.811eV)

Fig. 9. Structures of two best CF and three best NCF isomers for C50. Relative energies to the lowest energy configuration and cage symmetries are given in the parentheses below each plot.

energetically favored. The increasing trend of relative energy with H in the Subset One is more pronounced than that in the Subset Two. However, the data are still rather scattered and the corresponding standard derivative is largest among the three topological indexes studied. In other word, even for the NCF isomers with only one heptagon, index H does not perform well in differentiating the stability of fullerene isomers. The equilibrium structures of two best CF and three NCF isomers from our DFT calculations on the Subset Two are depicted in Fig. 9, along with the cage symmetries and relative energies to the lowest-energy configuration. Similar to the smaller fullerenes, the best CF isomer of C50 is more stable than any of the NCF ones and the best NCF isomer contains only one heptagon.

U with inclusion of the number of heptagons, the number of pentagon–heptagon pairs, and the number of adjacent pentagons has been put forward to evaluate the relative stability of both classical and nonclassical fullerenes. Our theoretical calculations demonstrated that this topological rule using index U prevails over the PAPR for all fullerenes considered (C30–C40, C50). For the wellknown case of C62 [4], for which a one-heptagon nonclassical cage is most stable among all isomers, the U value for the most stable NCF isomer is three, which is the minimum value that all C62 CF and heptagon-contained NCF isomers could possibly have. Therefore, we argue that our index U is an effective index to differentiating the relative stability of both CF isomers and NCF isomers with heptagons.

4. Conclusion

Acknowledgement

For the medium-sized fullerenes of C30–C40 and C50, classical and nonclassical isomers containing heptagons were generated by Spiral algorithm and further optimized via DFT calculations. The correlation between the relative energies of all isomers of C30–C40 and the topologies of these fullerene cages was discussed. In general, introducing more heptagons in the fullerene cage would raise the strain energy and increase the number of adjacent pentagons, making the NCF isomer energetically unfavorable. However, there are exceptional cases that the nonclassical isomers with more heptagons are more stable than those with fewer heptagons, which might be partially attributed to the formation of the energetically favorable pentagon–heptagon pairs. For each sized fullerene, the lowest-energy NCF isomer contains only one heptagon. Although none of the NCF isomers is the lowest-energy configuration, the low-energy NCF isomers could be more stable than most of their classical counterparts. The applicability of the PAPR and HNR (previously proposed for classical fullerenes) on the nonclassical fullerenes was examined on the entire sets of CF and NCF isomers of C30–C40 as well as a portion of isomers of C50. For those heptagon-contained NCF, PAPR is still roughly applicable but its average derivation becomes larger; whereas HNR totally fails regardless of its excellent performance on the CF isomers. As an extension of PAPR, a new united index

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