15634
J. Phys. Chem. 1996, 100, 15634-15636
C62: Theoretical Evidence for a Nonclassical Fullerene with a Heptagonal Ring A. Ayuela,† P. W. Fowler,*,‡ D. Mitchell,‡ R. Schmidt,† G. Seifert,† and F. Zerbetto§ Institut fu¨ r Theoretische Physik, Technische UniVersita¨ t Dresden, Mommsenstrasse 13, D-01069 Dresden, Germany; Department of Chemistry, UniVersity of Exeter, Stocker Road, Exeter EX4 4QD, UK; and Dipartimento di Chimica, G. Ciamician, UniVersita´ di Bologna, Via F. Selmi 2, 40126 Bologna, Italy ReceiVed: May 7, 1996; In Final Form: July 31, 1996X
Six different levels of theory agree in predicting a C62 cage with one heptagonal, 13 pentagonal, and 19 hexagonal rings to be of lower total energy than all 2385 “classical” 12-pentagon fullerene isomers. At the LDA level of density functional theory the nonclassical structure, which uniquely has three pentagon-pentagon and five pentagon-heptagon edges, is more stable than its nearest fullerene rival by 36 kJ mol-1.
Introduction All fullerenes so far characterized obey the “classical” definition in that they are trivalent polyhedral carbon cages made up entirely of pentagonal and hexagonal rings. The experimental fullerenes also belong to the subset of classical isomers that have isolated pentagons, which is a possibility only for cages Cn with n ) 60 or n ) 70 + 2k for k g 0. This “isolated pentagon rule”1,2 is part of a more general pattern in which stability of a fullerene isomer is maximized by minimizing e55, the number of pentagon-pentagon fusions.3 However, there is a growing case for extension of the classical definition to include larger rings: bending and coiling of nanotubes4,5 and tube tapering6 appear to require matched heptagon-pentagon pairs and large, open faces have been produced by chemical perforation of C60 and C70 cages.7,8 Totalenergy calculations also point to stability for nonclassical fullerenes.9-11 For example, all 92 C40 cages made up of pentagons, hexagons, and a single heptagon are predicted to lie within the energy range spanned by the 40 classical fullerenes9 though none is lower in energy than the best classical cage on 40 atoms. Stability of these generalized fullerenes is found to increase with e57, the number of pentagon-heptagon fusions: a simple recipe for low energy is therefore to minimise e55 and then maximize e57. These trends suggest that a nonclassical fullerene might overtake the classical structures in stability for some nuclearity where e55 cannot be zero but e57 may be large for the minimal value of e55. The present letter predicts that this actually happens at an atom count of 62. Calculations C62 has 2385 distinct structural isomers obeying the classical definition,12 of which three (2194(0), 2377(0), 2378(0)) have the minimal e55 value of 3. (n(m) denotes the nth isomer in the spiral sequence12,13 for the C62 cages with m heptagons, 12 + m pentagons and 21 - 2m hexagons). Of the 56 950 C62 5-6-7 cages with one heptagon found by the extended spiral algorithm,9 just one (5030(1)) has e55 ) 3. It also has e57 ) 5 (Figure 1). Optimization with the semiempirical QCFF/PI (quantum consistent force field/π) model14 showed this cage to have lower energy than all three fullerenes with minimal e55. Further investigation was indicated. †
Technische Universita¨t Dresden. University of Exeter. § Universita ´ di Bologna. X Abstract published in AdVance ACS Abstracts, September 15, 1996. ‡
S0022-3654(96)01306-8 CCC: $12.00
Exhaustive QCFF/PI optimizations of all 2385 classical structures confirmed the validity of the minimal pentagon adjacency rule (Figure 2) and gave the best fullerene as the asymmetric cage 2194(0), some 23 kJ mol-1 above the nonclassical Cs cage 5030(1) and 1983 kJ mol-1 below the least stable C62 fullerene. Widening the search of nonclassical cages to those with at most four pentagon fusions produce 12 new candidates, but all higher in energy than 5030(1), as expected from the e55/e57 minimax principle mentioned earlier. Although QCFF/PI has had considerable success in describing fullerene stability, structure, and vibrations,9,14-16 it seemed advisable to check such a startling result against other independent methods. The three MOPAC parameterizations17 (MNDO, PM3, and AM1) were applied in optimizations of the 12 best fullerenes, i.e., those with QCFF/PI energies within 100 kJ mol-1 of 2194(0) and the 13 nonclassical candidates. The previous conclusions were confirmed: the three minimal-adjacency fullerenes span a small energy range (in kJ mol-1: 8(QCFF/PI), 12(MNDO), 20(PM3), and 20(AM1)), with the one-heptagon cage 5030(1) lying below the best of them (by 23(QCFF/PI), 72(MNDO), 53(PM3), and 54(AM1) kJ mol-1; see Table 1). A parametrized method of a quite different type, the density functional tight-binding (DFTB) model18-21 was also used in optimization, this time of all 42 C62 fullerenes and 13 oneheptagon cages that have e55 e 4. The DFTB method is parameterized to mimic full density functional calculations and has been found to parallel the qualitative predictions of the QCFF/PI model for carbon cages, though with greater tolerance of nonclassical square and heptagonal rings.9,22 Yet again the nonclassical fullerene 5030(1) is found to have lower energy than any fullerene (by a margin of 44 kJ mol-1), although the detailed ordering of the minimal-adjacency fullerenes differs in that 2194(0) is now 19 kJ mol-1 above the almost isoenergetic 2377(0) and 2378(0) (Table 1). 2378(0) is also the best classical isomer found in previous tight-binding calculations.23 Two further one-heptagon cages, 5083(1) and 5104(1), also lie below 2194(0) according to DFTB (Table 1); both have e57 ) 5 but e55 ) 4. Finally, the results were checked at a higher level of theory by full density functional calculations in the local density approximation performed with the ADF program24,25 using a triple-ζ-plus-polarization basis of Cartesian Slater-type orbitals and the VWN exchange functional.26 Optimizations on the best three classical and best three one-heptagon isomers gave the results listed in Table 1: despite some variation in the energy © 1996 American Chemical Society
Letters
J. Phys. Chem., Vol. 100, No. 39, 1996 15635
Figure 2. Correlation of classical fullerene energies for C62 (computed in the QCFF/PI model, in kJ mol-1) with e55, the number of pentagon fusions. A least-squares fit indicates a penalty per fusion of 112.3 kJ mol-1 with a standard deviation of σ ) 64 kJ mol-1.
TABLE 1: Relative Energies (kJ mol-1) of the Best Three Classical Fullerenes n(0) and the Three Best One-Heptagon Cages n(1) for C62 Calculated by Semiempirical Methods (See Text) and by LDA Density Functional Theorya isomer
QCFF/PI
MNDO
PM3
AM1
DFTB
LDA
2194(0) 2377(0) 2378(0)
23.0 30.9 27.2
72.4 84.5 79.1
53.1 73.2 66.9
54.4 73.6 66.5
63.3 44.6 44.0
58.9 35.9 40.4
5030(1) 5083(1) 5104(1)
0.0 61.1 58.6
0.0 73.6 64.0
0.0 73.6 63.2
0.0 80.3 69.5
0.0 45.6 47.6
0.0 36.1 44.1
a All methods find these to be the six best isomers of the cages examined, and agree that the nonclassical 5030(1) has the lowest energy of all.
Figure 1. Candidates for the stable C62 cage. The three classical fullerenes (a) 2194(0), (b) 2377(0), (c) 2378(0) and the one-heptagon cage (d) 5030(1), that all have the minimal realizable number of pentagon adjacencies. Also included is (e) 5083(1) the second best oneheptagon cage that has one more than the minimal number of pentagon adjacencies.
sequence, the nonclassical isomer 5030(1) remains the most stable, lying some 36 kJ mol-1 (0.37 eV) below its nearest rival. Further refinement of the treatment is unlikely to overturn a difference of this magnitude. Nor would inclusion of zero-point effects affect the conclusion: MNDO frequencies give zeropoint energies differing by ∼1 kJ mol-1 for 2377(0) and 5030(1), for example. The conclusion from this comparison of results from six different methods is therefore that the most stable form of the C62 cage is, surprisingly, not a classical fullerene but rather a heptagon-containing variant stabilized by azulenoid pentagonheptagon contacts. The favored isomer is formally a product of C2 insertion in a hexagonal face of icosahedral C60 and is also reachable by a generalized Stone-Wales rearrangement27 from a minimal-adjacency C62 fullerene (Figure 3). Identification of isomer 5030(1) of C62 in fullerene soot would greatly advance our understanding of the formation of these species.
Figure 3. Formal genesis of the best C62 cage (a) by C2 insertion in icosahedral C60 and (b) by generalized Stone-Wales rearrangement of classical C62 fullerene 2194(0).
Acknowledgment. This work was supported by the European Union Human Capital and Mobility Scheme under the HCM Network of Formation, Stability and Photophysics of Fullerenes. D.M. thanks the University of Exeter for a University Postgraduate Scholarship. References and Notes (1) Kroto, H. W. Nature (London) 1987, 329, 529. (2) Schmalz, T. G.; Seitz, W. A.; Klein, D. J.; Hite, G. E. J. Am. Chem. Soc 1988, 110, 1113. (3) Campbell, E. E. B.; Fowler, P. W.; Mitchell, D.; Zerbetto, F. Chem. Phys. Lett. 1996, 250, 544. (4) Iijima, S.; Ajayan, P. M.; Ichihashi, T. Phys. ReV. Lett. 1992, 69, 3100. (5) Dunlap, B. I. Phys. ReV. B 1992, 46, 1933. (6) Iijima, S.; Ichihashi, T.; Ando, Y. Nature 1992, 356, 776. (7) Hummelen, J. C.; Prato, M.; Wudl, F. J. Am. Chem. Soc. 1995, 117, 7003. (8) Birkett, P. R.; Avent, A. G.; Darwish, A. D.; Kroto, H. W.; Taylor, R.; Walton, D. R. M. J. Chem. Soc., Chem. Commun. 1995, 1869.
15636 J. Phys. Chem., Vol. 100, No. 39, 1996 (9) Fowler, P. W.; Heine, T.; Mitchell, D.; Orlandi, G.; Schmidt, R.; Seifert, G.; Zerbetto, F. J. Chem. Soc., Faraday Trans. 1996, 92, 2203. (10) Murry, R. L.; Strout D. L.; Odom, G. K.; Scuseria, G. E. Nature 1993, 366, 665. (11) Eckhoff, W. C.; Scuseria, G. E. Chem. Phys. Lett. 1993, 216, 399. (12) Fowler, P. W.; Manolopoulos, D. E. An Atlas of Fullerenes; Oxford University Press: Oxford, 1995. (13) Manolopoulos, D. E.; May, J. C.; Down, S. E. Chem. Phys. Lett. 1991, 181, 105. (14) Warshel, A.; Karplus, M. J. Am. Chem. Soc. 1972, 94, 5612. (15) Fowler, P. W.; Manolopoulos, D. E.; Orlandi, G.; Zerbetto, F. J. Chem. Soc., Faraday Trans. 1995, 91, 1421. (16) Austin, S. J.; Fowler, P. W.; Orlandi, G.; Manolopoulos, D. E.; Zerbetto, F. Chem. Phys. Lett. 1994, 226, 219. (17) The calculations used Mopac 6.00: Stewart, J. J. P. Quantum Chemistry Program Exchange, Department of Chemistry, Indiana University, Bloomington, IN 47405. (18) Porezag, D.; Frauenheim, Th.; Ko¨hler, Th.; Seifert, G.; Kaschner, R. Phys. ReV. B 1995, 51, 12947.
Letters (19) Seifert, G.; Porezag, D.; Frauenheim, T. Int. J. Quantum Chem. 1996, 58, 185. (20) Seifert, G.; Jones, R. O. Z. Phys. D 1991, 20, 77. (21) Seifert, G.; Eschrig, H. Phys. Status Solidi B 1985, 127, 573. (22) Fowler, P. W.; Heine, T.; Manolopoulos, D. E.; Mitchell, D.; Orlandi, G.; Schmidt, R.; Seifert, G.; Zerbetto, F. J. Phys. Chem. 1996, 100, 6984. (23) Zhang, B. L.; Wang, C. Z.; Ho, K. M.; Xu, C. H.; Chan, C. T., J. Chem. Phys. 1992, 97, 5007. (24) Baerends, E. J.; Ros, P, Int. J. Quantum Chem. Symp. 1978, 12, 169. (25) Baerends, E. J., J. Comp. Phys. 1992, 99, 84. Technical details are in: Krijn, J.; Baerends, E. J., 1984 Internal Report, Free University of Amsterdam. (26) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (27) Stone, A. J.; Wales, D. J. Chem. Phys. Lett. 1986, 128, 501.
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