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Structures of large fullerenes: C60 to C94
Volume 193, number 4
CHEMICAL PHYSICS LETTERS
29 May 1992
Structures of large fullerenes: CbOto Cg4 B.L. Zhang, C.Z. Wang and KM. Ho Ames Laboratory USDOE. and Department ofphysics and Astronomy, Iowa State University, Ames, IA 50011, USA
Received 2 December I99 1;in final form 6 March 1992
A systematic study of the structures of large carbon fullerenes ranging from CeOto C94 is performed. We first generate the topological networks for candidate structures using an efficient cage network generation scheme. The resultant networks were relaxed with tight-binding molecular dynamics to obtain the ground-state structures for various low-energy isomers. Unlike the buckminsterfullerene C6,,, the large carbon fullerenes prefer structures with low symmetry and several isomers can be very close in energy and may coexist in synthesis.
1. Introduction The breakthrough in the synthesis of carbon fullerenes [ 1 ] has stimulated widespread experimental and theoretical studies on this new form of carbon. The structures of these remarkable molecules have attracted numerous studies [2-51 since the initial conception of the soccer-ball geometry for CbO [ 61. Carbon fullerenes form threefold coordinated cagelike networks consisting of twelve pentagons and a suitable number of hexagons. It is generally accepted that structures with no adjacent pentagons are more stable because this arrangement minimizes the strain energy. With the success in synthesizing macroscopic quantities of the carbon fullerenes, the Ii, structure of C&, the D5,, structure of CT0 and the chiral D2 structure of CT6 predicted from theory [2,3,6-81 have been confirmed by experiments [ 9, lo]. Recently, more large fullerenes have been isolated [ lo- 12 1, and characterizing their structures becomes a challenge for both experimentalists and theorists. Unlike the situation in C&, CT0 and CT6, where isolated-pentagon isomers are unique or small in number, the large fullerenes have many more isomers satisfying the pentagon-isolation rule. Thus, pinpointing their ground-state geometries is a difftCorrespondence to: B.L. Zhang, Ames Laboratory, USDOE, and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA.
Elsevier Science Publishers B.V.
cult task. Although several structural candidates have been proposed for CT8 [ 13 ] and C,, [ 141, none has as yet been confirmed. In this paper, we report on the results of a systematic study of the structures of large carbon fullerenes. Combining tight-binding molecular-dynamics calculations (TBMD) with a very efficient cage network generation scheme, we investigated the groundstate structures of every even-numbered cluster ranging from CbO to Cg4.
2. Method Simulated annealing is a powerful method for the systematic study of fullerene structures. However, previous attempts [ 15,161 failed to generate low-energy cage structures: the systems were trapped in highenergy metastable states because the strong directional carbon bonds create large energy barriers between different cage structures. Instead of looking at the network connecting individual atoms, our network generation scheme concentrates on the ‘facedual’ network obtained by linking the centers of each of the polygonal faces of the cage structure. Since each atom of the fullerene is threefold coordinated, the face-dual network consists of a triangular mesh which can be generated by simulated annealing of a collection of points confined on a predetermined surface (a sphere for example), and interacting with one an225
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other through a repulsive two-body potential. In this scheme, the change from a pentagonal face to a hexagonal face can occur easily with a small energy barrier even for quite a large number of atoms. The topological networks obtained tend to separate the pentagonal faces as far apart as possible. Thus the structures generated are good candidates for low-energy fullerene structures. By varying the shape of the constraining surface and the range of the repulsive force, we can obtain a set of energetically favorable topological networks. In the present study the constrained surfaces were chosen to be ellipsoids in which the variation of three axes were restricted from 0.8 to 1.2 (too sharp a shape will cause stress energy). The step of axes variations is 0.05 which we have tested for several cases without missing possible isomers. The repulsive potential has the form of l/r” with integer n from 2 to 6. We anticipate that a liner variation of axes and repulsive force is needed as the cluster size gets bigger. The resulting face-dual network can then be inverted to obtain the fullerene structures and the ground-state structure is then found by unconstrained simulated annealing of each of the candidate structures using an accurate TBMD scheme. The TB model potential describing the interactions between the carbon atoms is taken from our previous work #’ . The total energy in this model is expressed as occupied
n
+&,,({rJ)
.
(1)
The first term in ( 1) is the electronic energy calculated by a parametrized TB Hamiltonian J&-n ({ri} ), and the second term is a short-ranged repulsive energy representing the ion-ion repulsion and the correction to the double counting of the electron-electron interaction in the first term. The orthogonal sp3 basis TB Hamiltonian HTB(ri) is described by the following parameters: E, and t, are onsite atomic energies, v,, ( rrj), v,,, ( rij), upp,( TiJ) and v,,, ( rij) are overlap parameters which decay rapidly with interatomic distance rik The repulsive energy is in the form of Erep= 1, f(&@(r,.,)) where @(ri,) is ” For details of the tight-binding model, see ref. [ 17 1. 226
29 May 1992
a pairwise interaction and f is a function with argument C, @(rrj). The above parameters and functions were fitted to first-principles results for the electronic band structures and binding energies of graphite, diamond and the linear carbon chain as a function of the carbon-carbon bond length. Though simple, the TB potential includes the essential quantum mechanics of the covalent bonding in carbon systems. This model not only reproduces the binding energies and bond lengths of crystalline carbon with different coordination numbers, but also describes well the properties of the liquid state [ 17 1. The reliability of this potential for cluster calculations is further tested by studying the ground-state geometries of small clusters: In the range 5
3. Results Results for the structural and electronic properties of various fullerene isomers are summarized in table 1. The cohesive energies (per atom) of the groundstate structures with respect to that of bulk graphite are plotted in fig. 1. For C6,, and Cc,, the ground-state structures predicted by our scheme is exactly the same as that proposed by Kroto et al. [ 6,7]. In the range n= 60-70, the highly symmetrical Ih isomer of CbO and D5,, isomer of CT0 are the only two isomers without adjacent pentagons. Due to this distinct structural feature, their energies are dramatically lower than other isomers and their neighbouring fullerenes. For larger fullerenes, the number of isomers with no adjacent pentagons grow dramatically as the fullerene size increases and for a given cluster size, we can often find more than one isolated-pentagon isomer with the same symmetry. For cluster sizes beyond 76, most of the ground-state structures have rather low symmetries and often there are several isomers close in energy to the ground-state isomer. Thus we expect the experimental isolation and char-
CHEMICAL PHYSICS LETTERS
Volume 193, number 4
29 May 1992
Table 1 Structural and electronic data for large fullerenes Cluster size
Energy ‘)
60 62
0.401 0.435
64 66 68 70 72 74 76
0.41 I 0.410 0.399 0.365 0.377 0.357 0.354
78
0.346
AE b’
0.003
0.344
0.335 0.325
88
0.327 0.314
0.038 90
Td(D2d)
h
C D:’ D$ ” D,,(G) C D:: CS
D2d C2 D6h Td
D2 (leapfrog) CI c2 G c2
0.180 0.307 92 94
31 62 16 33 34 5 4 8 19 11 22 8 21 13 8 40 22 20 41 44 21 11 42 5 4 21 86 43 44 45 24 45 23 47 47
D2
0.033 0.263 0.300 1.224 1.902 0.329
1.612 0.378 0.384 0.952 0.471 0.725 1.103 1.388 0.224 0.796 0.096 0.493 0.353 0.545 0.443 1.373 0.076 0.198 0.518 0.649 0.773 0.823 0.844 0.660 1.140 1.489 1.277 0.344 0.351 0.154 0.633 0.629 0.564 0.575 0.679 0.625
c2
0.076
86
Ih C2 G D2 C2 G DSh JLJ
C 2”
0.030 0.090
84
NMR =’
D3,
0.087 0.284 0.324 0.913
82
HOMO-LUMO d,
D2
0.230
80
Symmetry ‘)
0.312 0.304
C C: D2 c2
0.026
c2
1
‘) The energy here is the cohesive energy of the fullerene with respect to that of bulk graphite in the unit eV/atom. b, AE is the cohesive energy of the isomer relative to the ground-state isomer in units of eV/molecule. ‘) If there are two entries for a symmetry, the first one is the topological group and the second is the real symmetry after full annealing. d, HOMO-LUMO energy separation is in the unit of eV. ‘) The value under NMR is the number of distinct NMR lines. ‘) The ground-state isomer for C,* proposed by Fowler et al. [ 131.
acterization of larger fullerenes to become more difficult as the cluster size increases. Among the large fullerenes, CT6, CT8, CS4, C9,, and Cg4 have attracted considerable interest recently and we would like to discuss them in comparison with experiments and existing theoretical calculations. (a) C,,. This fullerene has recently been isolated
and identified by Ettl et al. [ lo] as having a chiral Dz symmetry with double-helical arrangement of edge-sharing pentagons and hexagons confirming the theoretical prediction made by Manolopoulos [8] based on a direct computer search. In our present study, we indeed found the chiral D2 structure to be the ground-state geometry. We also obtained another 227
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70
75
60
85
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Cluster size (N) Fig. 1. The cohesive energy (eV/atom) plotted as a function of cluster size. (Note: the &on here is the cohesive energy relative to bulk graphite.)
isolated-pentagon isomer which has Td symmetry and with energy only 0.23 eV higher than that of the chiral structure. The Td isomer changes to a lower DZd symmetry through Jahn-Teller distortions when the structure is relaxed with TBMD. (b) C,+ Our studies show that the five isolatedpentagon isomers of CT8 are very close in energy. The most stable structure is that of fig. 2a which has CZv symmetry. This structure should give 22 distinct NMR lines. Recently, the structural candidates for CT8 have also been discussed by Fowler et al. [ 13 1. Based on the HOMO-LUMO energy gap estimated from the Hiickel molecular-orbital theory, they predicted that the ground-state structure has D3,, symmetry (fig. 2~). Although our present calculations give the HOMO-LUMO gaps in the same order as predicted by Fowler et al., the total energies from our calculation do not favour the Dji, structure of fig. 2c. (4
Fig. 2. Perspective views of CT8 isomers: ground-state structure.)
228
D3h (a) ground-state
In fact, the energy of the structure in fig. 2c is about 0.91 eV higher than that of the Czv isomer of fig. 2a. (c) C,, . Three structural candidates which have helical D2, Dbh and Td symmetries have been proposed for this fullerene by Fowler [ 14 1. Total energy calculations using MNDO and Hartree-Fock theory have been performed for the D& and Td isomers by Raghavachari and Rohlfing [ 191. After the success in identification of CT6, speculations [ lo] have been made that Cs4 also belongs to the same chiral family with helicity as proposed by Fowler [ 141. However, our calculations show that the helical Dz isomer of Cs4 is energetically unfavorable compared with a number of other possible candidates (see table 1). The energetically most stable structure of Cs4 obtained by our scheme is shown in fig. 3a. Although this isomer and the isomer proposed before [ 141 all have the same point group symmetry, their structures are very different. Whereas the helical isomer (fig. 3f ) consists of a double-helical arrangement of pentagon-hexagon strips, the lower energy Dz structure is characterized by a circular band of hexagons around the waist of the molecule. The simple argument why CS4, unlike CT6, does not prefer a helical structure is that the pentagons would be closer to each other in the helical arrangement. There are two caps with three pentagons around one hexagon in a helical structure. However in our D2 isomer, the largest number of pentagons around one hexagon is two. Using ellipsoidal constraining surfaces, we can obtain 10 out of 24 possible isolated-pentagon isomers for Cs4 constructed by Manolopoulos and Fowler [ 201, including a DZd isomer that is slightly above (0.03 eV per molecule) the ground-state D2 isomer. These two isomers are separated from the rest by 0.3 (c)
(b)
c2v
29 May 1992
LETTERS
D3h
C&, (b) Dji, and (c) D’&,. (Df,, has been proposed
by Fowler et al.
[ 131 as the
29 May 1992
CHEMICAL PHYSICS LETTERS
Volume 193, number 4
D2
D2d
D6h
Td
D2(leapfrog)
Fig. 3. Perspective views of (a) ground-state D2, (b) Dzd. (c) Cz, (d) D6,,, (e) Td and (f ) leapfrog Dz isomer of Cs4. (a)
04
c2
c2
c2v
Fig. 4. Perspective views of (a) ground-state C1, (b) Cz and (c) Czv isomer of CgO.
eV so that the experimental Cs4 extract may be a mixture of D2 and DZd isomers. (d) CPO and C,,. Abundant peaks corresponding to Go and Cgq have also been observed in mass spectra [ 111. We found that the ground-state structures of both fullerenes have C2 symmetry as shown in fig. 4a and fig. 5a. The energy difference between the ground-state C2 isomer and the C2 isomer of Cg4 is only 0.026 eV. This is to be contrasted with Ceo where the second-best isomer is 1.4 eV less stable. So it is likely that the coexistence of different isomers can be observed experimentally.
(ai
c2
c2
Fig. 5. Perspective views of (a) ground-state C2 and (b) C, isomer of C&.
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4. Conclusion
We have studied the structures of large fullerenes ranging from CGOto Cg4. Large fullerenes (II 2 76) have more than one isolated-pentagon isomer, therefore the isolated-pentagon rule alone is not sufficient to determine the ground-state structure. As a general trend, our study shows that large fullerenes prefer low-symmetry structures with separate the pentagons as far apart as possible. For some large fullerenes there are several isomers with energies close to that of the ground-state isomer. This may lead to the possibility that several isomers can coexist in the synthesis of a given cluster size.
Acknowledgement
Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract No. W-740%ENG-82. This work was supported by the National Science Foundation under Grant No. DMR-88 19379, and in part by the Director of Energy Research, Office of Basic Energy Sciences.
References [ 1 ] W. Kratschmer, L.D. Lamb, K. Fostiropoulos and D.R. Huffman, Nature 347 ( 1990) 354.
230
29 May 1992
[2] T.G. Schmalz, W.A. Seitz, D.J. Klein and G.E. Hite, J. Am. Chem.Soc. 110(1988) 1113. [ 31 P.W. Fowler, J.E. Cremona and J.I. Steer, Theoret. Chim. Acta (1988) 1. [ 41 D.E. Manolopoulos, J.C. May and S.E. Down, Chem. Phys. Letters 181 (1991) 105. [ 51D. Bakowies and W. Thiel, J. Am. Chem. Sot. 113 ( 199 1) 3704. [6] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 162. [7] H.W. Kroto,Nature 329 (1987) 529. [8] D.E. Manolopoulos, J. Chem. Sot. Faraday Trans. 87 (1991) 2861. [ 91 R. Taylor, J.P. Hare, A.K. Abdule-Sada and H.W. Kroto, J. Chem. Sot. Chem. Commun. 20 (1990) 1423. [lo] R. Ettl, I. Chao, F. Diederichand R.L. Whetten, Nature 353 (1991) 149. [ 111 F. Diederich, R. Ettl, Y. Rubin, R.L. Whetten, R. Beck, M. Alvarez, S. Anz, D. Sensharma, F. Wudl, K.C. Khemani and A. Koch, Science 252 ( 1991) 548. [ 121 K. Kikuchi, N. Nakahara, M. Honda, S. Suzuki, K. Saito, H. Shiromaru, K. Yamauchi, I. Ikemoto, T. Kuramochi, S. Hino and Y. Achiba, Chem. Letters ( 1991) 1607. [ 131 P.W. Fowler, R.C. Batten and D.E. Manolopoulos, J. Chem. Sot. Faraday Trans. 87 (1991) 3103. [ 141 P.W. Fowler, J. Chem. Sot. Faraday Trans. 87 ( 1991) 1945. [ 151 J.R. Chelikowsky, Phys. Rev. Letters 67 (1991) 2970. [ 161 P. Ballone and P. Milani, Phys. Rev. B 42 ( 1990) 3201. [17]C.H. Xu, C.Z. Wang, CT. Chan and K.M. Ho, to be published. [ 181 K. Raghavachari and J.S. Binkley, J. Chem. Phys. 87 ( 1987) 2191. [ 191 K. Raghavachari and C.M. Rohlting, J. Phys. Chem. 95 (1991) 5768. [ 201 D.E. Manolopoulos and P.W. Fowler, to be published.
CHEMICAL PHYSICS LETTERS
29 May 1992
Structures of large fullerenes: CbOto Cg4 B.L. Zhang, C.Z. Wang and KM. Ho Ames Laboratory USDOE. and Department ofphysics and Astronomy, Iowa State University, Ames, IA 50011, USA
Received 2 December I99 1;in final form 6 March 1992
A systematic study of the structures of large carbon fullerenes ranging from CeOto C94 is performed. We first generate the topological networks for candidate structures using an efficient cage network generation scheme. The resultant networks were relaxed with tight-binding molecular dynamics to obtain the ground-state structures for various low-energy isomers. Unlike the buckminsterfullerene C6,,, the large carbon fullerenes prefer structures with low symmetry and several isomers can be very close in energy and may coexist in synthesis.
1. Introduction The breakthrough in the synthesis of carbon fullerenes [ 1 ] has stimulated widespread experimental and theoretical studies on this new form of carbon. The structures of these remarkable molecules have attracted numerous studies [2-51 since the initial conception of the soccer-ball geometry for CbO [ 61. Carbon fullerenes form threefold coordinated cagelike networks consisting of twelve pentagons and a suitable number of hexagons. It is generally accepted that structures with no adjacent pentagons are more stable because this arrangement minimizes the strain energy. With the success in synthesizing macroscopic quantities of the carbon fullerenes, the Ii, structure of C&, the D5,, structure of CT0 and the chiral D2 structure of CT6 predicted from theory [2,3,6-81 have been confirmed by experiments [ 9, lo]. Recently, more large fullerenes have been isolated [ lo- 12 1, and characterizing their structures becomes a challenge for both experimentalists and theorists. Unlike the situation in C&, CT0 and CT6, where isolated-pentagon isomers are unique or small in number, the large fullerenes have many more isomers satisfying the pentagon-isolation rule. Thus, pinpointing their ground-state geometries is a difftCorrespondence to: B.L. Zhang, Ames Laboratory, USDOE, and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA.
Elsevier Science Publishers B.V.
cult task. Although several structural candidates have been proposed for CT8 [ 13 ] and C,, [ 141, none has as yet been confirmed. In this paper, we report on the results of a systematic study of the structures of large carbon fullerenes. Combining tight-binding molecular-dynamics calculations (TBMD) with a very efficient cage network generation scheme, we investigated the groundstate structures of every even-numbered cluster ranging from CbO to Cg4.
2. Method Simulated annealing is a powerful method for the systematic study of fullerene structures. However, previous attempts [ 15,161 failed to generate low-energy cage structures: the systems were trapped in highenergy metastable states because the strong directional carbon bonds create large energy barriers between different cage structures. Instead of looking at the network connecting individual atoms, our network generation scheme concentrates on the ‘facedual’ network obtained by linking the centers of each of the polygonal faces of the cage structure. Since each atom of the fullerene is threefold coordinated, the face-dual network consists of a triangular mesh which can be generated by simulated annealing of a collection of points confined on a predetermined surface (a sphere for example), and interacting with one an225
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CHEMICAL PHYSICS LETTERS
other through a repulsive two-body potential. In this scheme, the change from a pentagonal face to a hexagonal face can occur easily with a small energy barrier even for quite a large number of atoms. The topological networks obtained tend to separate the pentagonal faces as far apart as possible. Thus the structures generated are good candidates for low-energy fullerene structures. By varying the shape of the constraining surface and the range of the repulsive force, we can obtain a set of energetically favorable topological networks. In the present study the constrained surfaces were chosen to be ellipsoids in which the variation of three axes were restricted from 0.8 to 1.2 (too sharp a shape will cause stress energy). The step of axes variations is 0.05 which we have tested for several cases without missing possible isomers. The repulsive potential has the form of l/r” with integer n from 2 to 6. We anticipate that a liner variation of axes and repulsive force is needed as the cluster size gets bigger. The resulting face-dual network can then be inverted to obtain the fullerene structures and the ground-state structure is then found by unconstrained simulated annealing of each of the candidate structures using an accurate TBMD scheme. The TB model potential describing the interactions between the carbon atoms is taken from our previous work #’ . The total energy in this model is expressed as occupied
n
+&,,({rJ)
.
(1)
The first term in ( 1) is the electronic energy calculated by a parametrized TB Hamiltonian J&-n ({ri} ), and the second term is a short-ranged repulsive energy representing the ion-ion repulsion and the correction to the double counting of the electron-electron interaction in the first term. The orthogonal sp3 basis TB Hamiltonian HTB(ri) is described by the following parameters: E, and t, are onsite atomic energies, v,, ( rrj), v,,, ( rij), upp,( TiJ) and v,,, ( rij) are overlap parameters which decay rapidly with interatomic distance rik The repulsive energy is in the form of Erep= 1, f(&@(r,.,)) where @(ri,) is ” For details of the tight-binding model, see ref. [ 17 1. 226
29 May 1992
a pairwise interaction and f is a function with argument C, @(rrj). The above parameters and functions were fitted to first-principles results for the electronic band structures and binding energies of graphite, diamond and the linear carbon chain as a function of the carbon-carbon bond length. Though simple, the TB potential includes the essential quantum mechanics of the covalent bonding in carbon systems. This model not only reproduces the binding energies and bond lengths of crystalline carbon with different coordination numbers, but also describes well the properties of the liquid state [ 17 1. The reliability of this potential for cluster calculations is further tested by studying the ground-state geometries of small clusters: In the range 5
3. Results Results for the structural and electronic properties of various fullerene isomers are summarized in table 1. The cohesive energies (per atom) of the groundstate structures with respect to that of bulk graphite are plotted in fig. 1. For C6,, and Cc,, the ground-state structures predicted by our scheme is exactly the same as that proposed by Kroto et al. [ 6,7]. In the range n= 60-70, the highly symmetrical Ih isomer of CbO and D5,, isomer of CT0 are the only two isomers without adjacent pentagons. Due to this distinct structural feature, their energies are dramatically lower than other isomers and their neighbouring fullerenes. For larger fullerenes, the number of isomers with no adjacent pentagons grow dramatically as the fullerene size increases and for a given cluster size, we can often find more than one isolated-pentagon isomer with the same symmetry. For cluster sizes beyond 76, most of the ground-state structures have rather low symmetries and often there are several isomers close in energy to the ground-state isomer. Thus we expect the experimental isolation and char-
CHEMICAL PHYSICS LETTERS
Volume 193, number 4
29 May 1992
Table 1 Structural and electronic data for large fullerenes Cluster size
Energy ‘)
60 62
0.401 0.435
64 66 68 70 72 74 76
0.41 I 0.410 0.399 0.365 0.377 0.357 0.354
78
0.346
AE b’
0.003
0.344
0.335 0.325
88
0.327 0.314
0.038 90
Td(D2d)
h
C D:’ D$ ” D,,(G) C D:: CS
D2d C2 D6h Td
D2 (leapfrog) CI c2 G c2
0.180 0.307 92 94
31 62 16 33 34 5 4 8 19 11 22 8 21 13 8 40 22 20 41 44 21 11 42 5 4 21 86 43 44 45 24 45 23 47 47
D2
0.033 0.263 0.300 1.224 1.902 0.329
1.612 0.378 0.384 0.952 0.471 0.725 1.103 1.388 0.224 0.796 0.096 0.493 0.353 0.545 0.443 1.373 0.076 0.198 0.518 0.649 0.773 0.823 0.844 0.660 1.140 1.489 1.277 0.344 0.351 0.154 0.633 0.629 0.564 0.575 0.679 0.625
c2
0.076
86
Ih C2 G D2 C2 G DSh JLJ
C 2”
0.030 0.090
84
NMR =’
D3,
0.087 0.284 0.324 0.913
82
HOMO-LUMO d,
D2
0.230
80
Symmetry ‘)
0.312 0.304
C C: D2 c2
0.026
c2
1
‘) The energy here is the cohesive energy of the fullerene with respect to that of bulk graphite in the unit eV/atom. b, AE is the cohesive energy of the isomer relative to the ground-state isomer in units of eV/molecule. ‘) If there are two entries for a symmetry, the first one is the topological group and the second is the real symmetry after full annealing. d, HOMO-LUMO energy separation is in the unit of eV. ‘) The value under NMR is the number of distinct NMR lines. ‘) The ground-state isomer for C,* proposed by Fowler et al. [ 131.
acterization of larger fullerenes to become more difficult as the cluster size increases. Among the large fullerenes, CT6, CT8, CS4, C9,, and Cg4 have attracted considerable interest recently and we would like to discuss them in comparison with experiments and existing theoretical calculations. (a) C,,. This fullerene has recently been isolated
and identified by Ettl et al. [ lo] as having a chiral Dz symmetry with double-helical arrangement of edge-sharing pentagons and hexagons confirming the theoretical prediction made by Manolopoulos [8] based on a direct computer search. In our present study, we indeed found the chiral D2 structure to be the ground-state geometry. We also obtained another 227
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Cluster size (N) Fig. 1. The cohesive energy (eV/atom) plotted as a function of cluster size. (Note: the &on here is the cohesive energy relative to bulk graphite.)
isolated-pentagon isomer which has Td symmetry and with energy only 0.23 eV higher than that of the chiral structure. The Td isomer changes to a lower DZd symmetry through Jahn-Teller distortions when the structure is relaxed with TBMD. (b) C,+ Our studies show that the five isolatedpentagon isomers of CT8 are very close in energy. The most stable structure is that of fig. 2a which has CZv symmetry. This structure should give 22 distinct NMR lines. Recently, the structural candidates for CT8 have also been discussed by Fowler et al. [ 13 1. Based on the HOMO-LUMO energy gap estimated from the Hiickel molecular-orbital theory, they predicted that the ground-state structure has D3,, symmetry (fig. 2~). Although our present calculations give the HOMO-LUMO gaps in the same order as predicted by Fowler et al., the total energies from our calculation do not favour the Dji, structure of fig. 2c. (4
Fig. 2. Perspective views of CT8 isomers: ground-state structure.)
228
D3h (a) ground-state
In fact, the energy of the structure in fig. 2c is about 0.91 eV higher than that of the Czv isomer of fig. 2a. (c) C,, . Three structural candidates which have helical D2, Dbh and Td symmetries have been proposed for this fullerene by Fowler [ 14 1. Total energy calculations using MNDO and Hartree-Fock theory have been performed for the D& and Td isomers by Raghavachari and Rohlfing [ 191. After the success in identification of CT6, speculations [ lo] have been made that Cs4 also belongs to the same chiral family with helicity as proposed by Fowler [ 141. However, our calculations show that the helical Dz isomer of Cs4 is energetically unfavorable compared with a number of other possible candidates (see table 1). The energetically most stable structure of Cs4 obtained by our scheme is shown in fig. 3a. Although this isomer and the isomer proposed before [ 141 all have the same point group symmetry, their structures are very different. Whereas the helical isomer (fig. 3f ) consists of a double-helical arrangement of pentagon-hexagon strips, the lower energy Dz structure is characterized by a circular band of hexagons around the waist of the molecule. The simple argument why CS4, unlike CT6, does not prefer a helical structure is that the pentagons would be closer to each other in the helical arrangement. There are two caps with three pentagons around one hexagon in a helical structure. However in our D2 isomer, the largest number of pentagons around one hexagon is two. Using ellipsoidal constraining surfaces, we can obtain 10 out of 24 possible isolated-pentagon isomers for Cs4 constructed by Manolopoulos and Fowler [ 201, including a DZd isomer that is slightly above (0.03 eV per molecule) the ground-state D2 isomer. These two isomers are separated from the rest by 0.3 (c)
(b)
c2v
29 May 1992
LETTERS
D3h
C&, (b) Dji, and (c) D’&,. (Df,, has been proposed
by Fowler et al.
[ 131 as the
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CHEMICAL PHYSICS LETTERS
Volume 193, number 4
D2
D2d
D6h
Td
D2(leapfrog)
Fig. 3. Perspective views of (a) ground-state D2, (b) Dzd. (c) Cz, (d) D6,,, (e) Td and (f ) leapfrog Dz isomer of Cs4. (a)
04
c2
c2
c2v
Fig. 4. Perspective views of (a) ground-state C1, (b) Cz and (c) Czv isomer of CgO.
eV so that the experimental Cs4 extract may be a mixture of D2 and DZd isomers. (d) CPO and C,,. Abundant peaks corresponding to Go and Cgq have also been observed in mass spectra [ 111. We found that the ground-state structures of both fullerenes have C2 symmetry as shown in fig. 4a and fig. 5a. The energy difference between the ground-state C2 isomer and the C2 isomer of Cg4 is only 0.026 eV. This is to be contrasted with Ceo where the second-best isomer is 1.4 eV less stable. So it is likely that the coexistence of different isomers can be observed experimentally.
(ai
c2
c2
Fig. 5. Perspective views of (a) ground-state C2 and (b) C, isomer of C&.
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CHEMICAL PHYSICS LETTERS
4. Conclusion
We have studied the structures of large fullerenes ranging from CGOto Cg4. Large fullerenes (II 2 76) have more than one isolated-pentagon isomer, therefore the isolated-pentagon rule alone is not sufficient to determine the ground-state structure. As a general trend, our study shows that large fullerenes prefer low-symmetry structures with separate the pentagons as far apart as possible. For some large fullerenes there are several isomers with energies close to that of the ground-state isomer. This may lead to the possibility that several isomers can coexist in the synthesis of a given cluster size.
Acknowledgement
Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract No. W-740%ENG-82. This work was supported by the National Science Foundation under Grant No. DMR-88 19379, and in part by the Director of Energy Research, Office of Basic Energy Sciences.
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