Stability of fullerenes with four-membered rings

12May 1995

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 237 (1995) 239-245

ELSEVIER

Stability of fullerenes with four-membered rings D. Babi6, N. Trinajsti6 The Rugjer Bogkovi~ Institute, P.O. Box 1016, 41001 Zagreb, Croatia

Received 12 January 1995; in final form 28 February 1995

Abstract

Fullerenes with four-membered rings with up to 60 atoms were systematically generated and their stability examined by the conjugated circuits (CC) model and the topological resonance energy (TRE) model. As none of the models accounts for strain, which is important in spherically shaped conjugated systems with four-membered rings, the results are only qualitative. Values of resonance energy obtained by the two models show remarkable disagreement. Isomers selected as the most stable are quite different in dependence on the criterion used. The structures selected by TRE possess a small number or no four-membered rings (for C6o, buckminsterfullerene was selected), while those selected by the CC model sometimes possess greater numbers (up to 6) of four-membered rings, paired by a common edge.

1. Introduction

It is commonly accepted that fullerenes cannot possess any other than five- and six-membered rings. Solid arguments for this assertion were given by Schmalz et al. [1]. Some authors theoretically considered the possible presence of seven-membered rings and experimental findings were also reported [2,3]. Gao and Herndon [4] investigated fullerenes with four-membered rings. They chose five fullerene isomers with sizes ranging from 32 to 60 atoms and constructed for each isomer a similar one with two four-membered rings in the structure. The results for ,n-electron energy obtained by the SCF-UHF method indicated a greater stability of isomers with fourmembered rings in three pairs of isomers. Strain was assessed by molecular mechanics and the results showed greater strain in isomers with four-membered rings, as expected. Nevertheless, A H ° values calculated by molecular mechanics and M N D O - U H F indicated that in a few cases there is slightly greater

stability of the structures containing four-membered rings over their analogues with only five- and sixmembered rings. Although the results are not fully consistent, they clearly show that fullerenes with four-membered rings cannot be dismissed in advance. In our preliminary study we considered the same isomers as in Ref. [4] using two well-known methods for conjugated systems: the conjugated circuits model (CC) [5] and the topological resonance energy (TRE) method [6,7]. Although considerably simple, both models were successfully applied in a number of cases [8,9]. The results we obtained were fully parallel to the SCF-UHF values from Ref. [4]. In the present study we report the results obtained by systematic examination of all 4,5,6-fullerenes with up to 60 carbon atoms. By 4,5,6-fullerenes we denote isomers with allowed (not necessarily present) four-membered ring(s) in the structure. Due to their simplicity, CC and TRE models are convenient means for the systematic examination of fullerenes on a

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240

D. Babi(, N. Trinajsti( / Chemical Physics Letters 237 (1995) 239-245

large scale. However, some caution is needed when they are applied to strained systems. These two models do not account for strain which is quite an important factor in spherically shaped conjugated molecules. In an earlier study [10] it was shown that a larger strain (implied by the greater number of abutting pentagons) is accompanied by a smaller resonance energy. Nevertheless, the stability of isomers with four-membered rings would be much more sensitive to strain effects. Therefore, the results presented here should be considered as preliminary and qualitative.

of clusters the algorithm is expected to be complete, as verified for 5,6-fullerenes with up to 70 atoms [11,12]. The generation was performed by starting from each possible ring size, that is, from either a 4-, 5- or 6-membered ring. Already for C36 there is an isomer which could be obtained only by starting from a hexagon. The numbers of 4,5,6-isomers generated for sizes from 12 to 60 atoms are given in Table 1. W e have stopped at 60 atoms because the time needed for the generation of the isomers became too big and the number of generated isomers was threatening to outgrow our computational capacities. One may note that the presence of four-membered rings greatly enriches the world of fullerenes.

2. Generation of isomers Isomers were generated by the method described in Refs. [11,12]. It was modified in order to include the possibility of four-membered rings, but otherwise the program was the same. The generating algorithm may be incomplete because it includes a hypothesis which is not yet proven, but for relatively small sizes

Table 1 The number of isomers classified according to number of atoms (n) and number of quadrangles (q) q

n

20 22 24 26 28 6 3 1 3 5 2 5 4 4 10 11 25 3 5 11 14 2 2 3 11 1 0 1 1 0 1 0 1

3 10 29 34 13 3 1

30

3 8 55 50 30 5 2

32

34

36

38

2 8 3 7 16 21 25 27 71 102 129 198 78 126 203 280 50 91 133 236 10 20 37 57 3 6 6 15

7 45 225 438 334 109 17

40 7 38 342 601 544 163 40

23 32 59 93 153 230 374 536 820 1175 1735 q t/

42 6 5 4 3 2 1 0

5 59 421 851 759 278 45

44

46

48

50

14 6 12 12 72 82 93 135 549 661 911 1014 1146 1616 2041 2796 1167 1574 2332 3107 406 656 951 1416 89 116 199 271

52 13 112 1342 3610 4245 1995 437

54 10 165 1619 4623 5703 2929 580

56 23 203 1962 5815 7876 3953 924

58

60

12 19 204 230 2277 2975 7629 9103 9950 13456 5647 7475 1205 1812

2418 3443 4711 6539 8751 11854 15629 20756 26924 35070

3. Computation of resonance energies Resonance energies according to the CC model, RE(CC), are easily computed by using the method described in Ref. [10]. However, TRE could only be calculated by using the recently developed algorithm [13] for the computation of the matching polynomial. Nevertheless, this calculation is still a time-consuming task and it could not be performed for all the generated isomers due to their great number. As we were only interested in the most stable isomers, we hoped that RE(CC) might serve as a pilot quantity for locating the most stable isomers according also to TRE. Thus we intended to compute TRE values only for the top ten isomers selected by RE(CC), and to verify the correlation between TRE and RE(CC) a posteriori. This expectation showed not to be true. The computed TRE and RE(CC) values were largely uncorrelated. W e have checked this on a sample of all 374 isomers of 4,5,6-fullerenes with 32 atoms. The resuits are shown in Fig. 1. Fortunately, w e found a surprisingly good correlation between TRE and E~. The values computed for the same isomers as above are plotted in Fig. 2. This relation between TRE and E,~ is quite unexpected and deserves a separate study. Preliminary investigation shows that Eref, which is subtracted from E~ in order to get TRE, varies so slightly in the set of fixed size isomers that it can be well approximated by a constant.

D. Babi~, N. Trinajsti6/ Chemical Physics Letters 237 (1995) 239-245

241

0,04

0.02

o

0.00

°°~o~ -0.02

°

o o

°oo°~-

o~ oo

o

0.00

°°

o

r~ -0.02

o

[..,

o

o

-0.04 -o.o4

¢i~o

~:~

°4° o

-0.06

-0.06 o

o

-0.08

~

0.00

I

i

0.02

~

I

i

I

0.04

~

I

0.06

0.06

-0.08

i

,

1.46

0.10

I 1.48

,

I

i

1.SO

I

1.52

,

I

1.54

,

1.56

E~/e

RE(CC)/e* Fig. 1. TRE of 374 isomers of 4,5,6-fullerenes with 32 atoms plotted against RE(CC). Both resonance energies are normalized to the number of Ir electrons. T R E / e is expressed in /3 units and R E ( C C ) / e * denotes the R E ( C C ) / e relative to that of graphite.

Thence we performed a screening for the most stable isomers by computing their E~ values, and then computed TRE values only for the top ten

Fig. 3. TRE/e plotted against E,,/e for the isomers having the greatest E,,/e with sizes ranging from 32 to 60 atoms. For each (even) size the ten topmostisomerswere selected.

isomers of each size. A good correlation between the and T R E / e for selected isomers (shown in Fig. 3) justifies using E~ for screening purposes.

E~/e

0.02

4. Results 0.(~

oof °

-0.02

o

0

-0.04!

-0.06

-0.08 1.46

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I

1.48

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I

1.50

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I

1.52

L54

Enle Fig. 2. T R E / e of the same 374 isomers of 4,5,6-fullerenes with 32 atoms plotted against their E , , / e . Both quantities are given in fl units.

The results obtained by using RE(CC) and TRE appear rather disparate. The overlap between the ten most stable isomers selected by using these two quantities separately is negligible and only occurs for the smallest isomers. Clearly, this overlap must occur for smaller cages where there are a fewer number of isomers. So far we have not found a reasonable explanation for this discrepancy. The two models are derived on quite different grounds: TRE relies on a somewhat arbitrary assumption on the suppression of delocalization by removing cyclic components from Sachs graphs [6,7], and RE(CC) was derived on empirical [5] and semi-empirical [14] levels. Recently, the foundations of the CC model were also established in the framework of VB theory [15] at least for (alternant) benzenoid systems, for which no contradiction was previously observed between these two models.

D. Babi(, N. Trinajsti£ / Chemical Physics Letters 237 (1995) 239-245

242

Table 2 4,5,6-fullerenes selected as the most stable by TRE criterion n

Spiral code

E~/e

RE(CC)/e *

TRE/e

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

455565656454 4555656565554 44566656566454 446566566565554 4556656565656455 45656566656554556 466665555555566664 4566656555665666464 45656566656655665645 456665666656656455646 4566666656566466565654 46666565656566465646566 456665666656656555656466 4666655665665665656565646 46666565656566565656566664 466665666566656555655565666 4666656566666656555655566656 46666565656666565656656666465 466665656565666666666565656564 4566656656666666665665656565646 56666656565656566565656565666665

1.4870 1.4907 1.5073 1.5150 1.5187 1.5184 1.5271 1.5258 1.5298 1.5320 1.5355 1.5398 1.5392 1.5405 1.5459 1.5428 1.5446 1.5482 1.5493 1.5486 1.5527

0.1189 0.0636 0.1475 0.2858 0.1628 0.2721 0.1528 0.2045 0.4231 0.3042 0.3057 0.4996 0.4013 0.4223 0.5819 0.4251 0.4961 0.6246 0.5894 0.5696 0.7151

-0.04002 -0.03469 -0.02111 -0.01201 -0.00719 -0.00741 0.00109 -0.00111 0.00367 0.00537 0.00872 0.01289 0.01322 0.01447 0.01969 0.01735 0.01905 0.02185 0.02304 0.02243 0.02738

We have found that TRE and RE(CC) are better correlated for 5,6-fullerenes (a comparison was done for all C50 fullerenes). Also we have found that E~

and RE(CC) are well correlated for 5,6-fullerene isomers with no abutting pentagons. Having in mind the relationship between E~ and TRE, one may

Table 3 4,5,6-fullerene isomers selected as the most stable by the RE(CC) criterion n

Spiral code

E~/e

R E(C C )/ e *

TRE/e

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

444666666444 4466665545546 44666656465464 444666666666444 4465665665665644 44566665665665644 446666666666446644 4465665666665665644 44666666666646644664 445666656666656666544 4466665666566566656446 44566665666666665665644 446666665665656656666644 4465666666666566665656446 46646666666666666664466644 446666666666666666664466644 4466665666666566665656666644 44666656666666656656666566446 446666666666666666664666446664 4466665666566666666666665646564 44666656666666666656666656564466

1.4794 1.4880 1.4857 1.4901 1.5077 1.5076 1.5125 1.5215 1.5138 1.5179 1.5261 1.5280 1.5281 1.5332 1.5311 1.5319 1.5385 1.5383 1.5362 1.5405 1.5423

0.3588 0.3360 0.3844 0.4571 0.4282 0.4758 0.5237 0.5338 0.5721 0.5799 0.6249 0.6184 0.6510 0.6626 0.6844 0.6939 0.7060 0.7151 0.7256 0.7418 0.7461

- 0.05555 - 0.04127 -0.04517 - 0.04330 - 0.02065 - 0.02069 - 0.01924 - 0.00663 - 0.01734 - 0.01008 - 0.00219 0.00013 - 0.00005 0.00534 0.00094 0.00180 0.01053 0.01037 0.00644 0.01262 0.01446

D. Babid, N. Trinajsti~ / Chemical Physics Letters 237 (1995) 239-245

assume a similar correlation between TRE and RE(CC) in these cases. 5,6-fullerenes with isolated pentagons are characterized by the absence of fourand eight-membered cycles in their structure. Thus it seems that the discrepancy could arise from the different weights associated with 4n-cycles in the two resonance energy models. An additional characteristic is shown by the results discussed below. Table 2 gives the spiral codes [16], TRE and RE(CC)/e values of the most stable isomers selected by using the TRE model. Plausible geometries of the same isomers are drawn in Fig. 4. One should note that 4,5,6-fullerenes include 5,6-fullerenes as a subset, and that up to C60 the isomer selected as the most stable has at least one four-membered ring in the structure. The number of quadrangles in two

thirds of the listed isomers is at most equal to 2, and there seems to be a slight trend to isomers with a lower number of quadrangles as the size increases. Note that buckminsterfullerene is the selected isomer for C60. The results obtained with RE(CC) as the stability criterion are shown in Table 3 and the corresponding drawings are given in Fig. 5. Quadrangles are much more abundant in these isomers. In many cases the number of quadrangles is the maximal possible - 6, and there are no pentagons in the structure. In almost all selected structures the number of quadrangles is even, and they are arranged in pairs. This is probably connected with the fact that two adjacent four-membered rings also form one more six-membered cycle. In this way the negative effect of four-membered

@ c2o

%2

%4

c~

C28

G

C30

C32

C3,t

C36

@o C38

C,,6

0@ C54

C4o

C4s

%6

243

C42

%0

Css

C44

%2

6 C~°

Fig. 4. Plausible geometries of the most stable isomers of 4,5,6-fullerenes as selected by TRE.

244

D. Babi6, N. Trinajsti6 / Chemical Physics Letters 237 (1995) 239-245

C20

C22

C24

C30

C32

C3s

C~

C26

C34

C42

C~s

Cs4

C28

C36

C44

C52

C56

C58

C60

Fig. 5. Plausible geometries of the most stable isomers of 4,5,6-fullerenes as selected by RE(CC).

rings tends to be partially compensated. Evidently, this is much less effective in the TRE model.

5. Conclusions

The aim of the present study was to perform a preliminary screening for eventually stable isomers among fullerene-like cages with four-membered rings in their structures. Due to the high number of isomers, some simple models for estimating the stability of these conjugated molecules are desirable. CC and TRE are convenient models, although loaded with the serious drawback of ignoring the molecular geometry. The structures selected as the most stable by using two well-known topological models for conjugated systems are presented in Tables 2 and 3, as well as in Figs. 4 and 5. The predictions made by the two methods are in large disagreement. Since neither of the two models

accounts for the geometry, which is quite an important factor for stability, no definite conclusion about the validity of the models considered can be obtained by comparing with experimental facts, which are meaningful at least in the case of C60 isomers. The more realistic predictions seem to occur for the TRE model, but do not prove that it is better founded than the CC model. Nevertheless, from a practical point of view, the TRE model could be better suited for estimating the stability of fullerenes. Probably, combination with some simple method for assessing the distribution of strain in these molecules could largely improve the present predictions. The fact that fullerenes with four-membered rings have not yet been found should not be considered separately from the rather special currently realized means for the preparation of fullerenes. Kinetic factors certainly play a significant role in the formation of fullerenes. Once a rational synthesis becomes a reality, a number of structures not existing today

D. Babi~, N. Trinajsti6/ Chemical Physics Letters 237 (1995) 239-245 should b e c o m e accessible. S o m e recently reported experimental results indicate the possible presence of f o u r - m e m b e r e d rings in the structure of C n 9 [17].

Acknowledgement Stefano Bassoli (University of Parma, Parma) enabled the generation of isomers with more than 50 atoms on D E C Station 3000 A l p h a A X P 400, and we thankfully a c k n o w l e d g e his help. W e are also grateful to Professor Douglas J. K l e i n (University of Texas, Galveston) for reading the manuscript and g i v i n g m a n y useful suggestions. The w o r k is supported b y the Ministry of Science of the Republic of Croatia through grant 1-07-159.

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