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Pentagonal rings and nitrogen excess in fullerene-based BN cages and nanotube caps
16 January 1999
Chemical Physics Letters 299 Ž1999. 359–367
Pentagonal rings and nitrogen excess in fullerene-based BN cages and nanotube caps P.W. Fowler a , K.M. Rogers a , G. Seifert b, M. Terrones c , H. Terrones
d
a
b
School of Chemistry, UniÕersity of Exeter, Stocker Road, Exeter, EX4 4QD, UK Institut fur ¨ Theoretische Physik, Technische UniÕersitat ¨ Dresden, Zellescher Weg 17, D-01062 Dresden, Germany c School of Chemistry, Physics and EnÕironmental Science, UniÕersity of Sussex, Brighton, BN1 9QJ, UK d ´ Instituto de Fisica, UNAM, Apartado Postal 20-364, Mexico, D.F. 01000, Mexico Received 8 October 1998
Abstract Classical fullerene polyhedra are considered as candidates for boron-nitrogen cages. Simple arguments show that nitrogen-rich, fullerene-like cages with the general formula B x N xq4 can be constructed to have just six N–N bonds, with full B,N alternation in all hexagonal rings. Systematic density-functional tight-binding calculations indicate special stability for such cages when they contain six isolated pentagon pairs. Closure of BN nanotubes with three isolated pentagon pairs gives an alternative to the usual explanation of their observed flat tips in terms of square rings. Inclusion in polyhedra of various combinations of squares, pentagon–pentagon and pentagon–heptagon pairs would lead to BN cages and shells where nitrogen excess correlates with sphericity. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Several research groups have described the production of boron nitride-based nano-structures, both tube-like w1–5x and particulate w6–8x and there have been numerous theoretical studies of hypothetical ŽBN. x clusters w9–13x. In view of the structural similarity of graphite and bulk boron nitride, it is usual to discuss these new materials in terms of modifications of the carbon fullerene and nanotube models. Most workers, including ourselves w12,13x, have taken the view that the natural generalisation of the carbon fullerene is a trivalent, polyhedral structure with rigorous B,N alternation, and therefore composed entirely of even-membered rings, with six squares replacing the twelve pentagons of the fullerene recipe. Local minima on the potential sur-
face of x B and x N atoms are found for such squarerhexagon clusters, with some magic numbers Že.g. B12 N12 . and intuitively reasonable structural trends such as an isolated-square counterpart of the fullerene isolated-pentagon rule w14,15x. However, another plausible generalisation is to retain the pentagonal rings of the carbon fullerene and to accept a minimal number of forced homonuclear bonds in the B,N decoration of the framework. Simple energetics w16x suggest that NN pairs should in this case be preferred over BB. Calculations on small B x N y rings and small clusters w17x find that NN pairs are often protected by an energy barrier to extrusion and have only moderate effects on overall stability of a mainly alternant framework. It is pointed out in the present paper that the problem of designing B x N y structures based on classical fullerenes
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 2 6 5 - 2
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P.W. Fowler et al.r Chemical Physics Letters 299 (1999) 359–367
with maximisation of alternation, minimisation of NN pairing and total aÕoidance of BB pairs has a unique mathematical solution: it is necessary to use an excess of precisely four nitrogen atoms in a fullerene cage with at least six pentagon adjacencies. The ideal stoichiometry for such cages is therefore B x N xq4 . The purpose of the present paper is to make a systematic exploration of these nitrogen-rich boronnitrogen cage structures based on fullerene polyhedra and to demonstrate that they obey an ‘isolated-pentagon-pair rule’ rather than the isolated-pentagon rule of the carbon fullerenes. As an interesting consequence, it is suggested that the structural pattern of these nitrogen-rich cages may also explain the flat tips observed in BN nanotubes w2,3,5x, and that variations of the face recipe at higher nuclearities may generate near-spherical BN shells with larger nitrogen excess. The paper is organised as follows. The generation and selection of the structures to be considered is outlined ŽSection 2., followed by a brief description of the density-functional tight-binding ŽDFTB. model used to evaluate the energetics and geometries of the structures ŽSection 3.. The results of calculations for the selected BN cages are then presented and their structural and energetic trends discussed ŽSection 4.. The preference which is found for isolated pentagon pairs leads to a proposal for closure of BN nanotubes with flat tips that contain homonuclear bonds ŽSection 5.. Finally, the relationships between face recipe, nitrogen excess and sphericity of BN cages and shells are briefly discussed.
2. Generation and selection of cage structures A fullerene polyhedron C n consists of twelve pentagonal and nr2 y 10 hexagonal rings. As a non-alternant polyhedron it cannot avoid having some homonuclear edges when each carbon atom is replaced by either B or N. The number of such edges varies with the B:N ratio of the decoration and with the underlying structure. An isolated-pentagon fullerene has twelve or more homonuclear edges, as each pentagon must contain at least one. If the pentagons are in contact, it may be possible to reduce this number to six, each at the fusion of two
pentagonal rings. In favourable cases, all hexagonal rings of the fullerene will have full B,N alternation, and the decorated polyhedron will achieve this minimal number of just six homonuclear edges. Suppose now that we insist that all six of these homonuclear edges be NN. It can be proved that the stoichiometry of the B,N-decorated fullerene must be B x N xq4 , where x s nr2 y 2. The argument is as follows. Take the formula to be B x N xq2 y . In an obvious notation, the total number of vertices is Õ s ÕB q ÕN s 2 x q 2 y
Ž 1.
and the total number of edges of the trivalent cage is e s 3Õr2 s e BB q e BN q e NN s 3 x q 3 y.
Ž 2.
Counting edges emanating from B and N sites, respectively, 2 e BB q e BN s 3 x ,
Ž 3.
2 e NN q e BN s 3 x q 6 y
Ž 4.
so that e NN y e BB s 3 y
Ž 5.
and from the fact that there are 12 pentagons, e NN q e BB G 6.
Ž 6.
Thus e NN G 3 q 3 yr2,
Ž 7.
e BB G 3 y 3 yr2.
Ž 8.
In favourable cases, therefore, a value of y s 2 suffices to remove all BB edges and gives e NN s 6. Hence, for closed cages based on classical fullerenes, the formula B x N xq4 is optimal. Structures that achieve this optimum include the ‘even’ subset of the isolated-pentagon-pair ŽIPP. fullerenes. IPP fullerenes are cages where the twelve pentagons fall into six pentalene Žpentagon pair. units, each surrounded by hexagons. Such cages may be either ‘odd’, allowing decorations B x N x with three NN and three BB pairs at the pentagon joins, or ‘even’, allowing conjugate B xq 4 N x and B x N xq4 decorations with respectively six BB and six NN pairs at the joins w18x. Some smaller fullerenes with more highly condensed arrangements of pentagon pairs also achieve this optimum.
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All 30,579 fullerene cages w19x in the range 20 F n F 70 were searched to find the candidates for optimal B x N xq4 cages based on the model described above. The algorithm used follows Ref. w18x and is in two parts. The first finds all isomers that contain a set of six pentagon–pentagon edges which satisfy the rule that each edge in the set must be separated from all others by at least two edges, thus avoiding a homonuclear chain longer than two vertices. The second part involves deciding whether the selected isomers are odd Žallowing a formula B x N x with three N–N and three B–B bonds. or even Žallowing a formula B x N xq4 with six N–N bonds.. In practice, this is done by temporarily breaking the six pentagon–pentagon edges. If the twelve divalent vertices of the resulting alternant cage have coefficients of the same sign in the most antibonding eigenvector of the adjacency matrix, the isomer is even, i.e. all six edges could be NN pairs. Otherwise, the isomer is odd, with three NN and three BB pairs. The rest of the vector is used to finish the ‘decoration’ of the cage. A given B x N xq2 y isomer is generated < G
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Table 1 Enumeration of BN isomers based on classical fullerene structures. Nf is the total number of fullerene isomers found by the spiral algorithm w19x at each nuclearity n. Ne is the number of even isomers of formula B x N xq4 Ž x s n r2y2. with 6 N–N bonds. No is the number of odd isomers of formula B x N x Ž x s n r2. with three N–N and three B–B bonds n 20 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 total
Nf
Ne
No
1 1 1 2 3 6 6 15 17 40 45 89 116 199 271 437 580 924 1 205 1 812 2 385 3 465 4 478 6 332 8 149
1 0 1 2 1 3 1 1 2 3 3 6 4 5 5 8 9 10 8 11 10 17 15 18 17
0 0 1 1 0 3 3 8 7 12 10 20 15 27 26 36 41 59 50 76 68 102 89 126 116
30 579
161
896
molecular mechanics package before full optimisation.
3. Quantum-mechanical method The main tool for the full survey of the selected structures is the density-functional tight-binding ŽDFTB. method w22,23x. This approach has been applied for carbon cages w24–26x and is known to give a realistic description of energetics and structures for BN systems, both solid-state and molecular w12,13,27x. DFTB is a density-functional based non-orthogonal tight-binding approach. It has an LCAO framework, but considers only two-centre integrals. This
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method makes use of universal, two-particle, shortrange repulsive potentials which are derived in a general, almost parameter-free, manner using ab initio results for small reference systems. Hence, the overall scheme can be viewed as being intermediate between ab initio density-functional theory and traditional empirical tight-binding treatment.
4. Results and discussion The results of the calculations on the B x N xq4 cages are summarised in Figs. 1a and 1b, where
binding energies Žper atom. and HOMO–LUMO gaps are shown, respectively. Fig. 2 illustrates the geometry of the most stable isomer at each stoichiometry. It is notable that every isomer in the range compatible with the rule of construction corresponds to a local minimum on the potential surface, and that stability generally increases smoothly from the smaller clusters towards the bulk BN limit with the 1rn relationship expected from the size dependence of the cage curvature w13x. There is no clear overall correlation between the binding energies and HOMO–LUMO gaps ŽFigs. 1a and 1b., but the most stable isomers at each nuclearity are mostly characterised by a relatively large gap.
Fig. 1. Variation of Ža. binding energy per atom Ž EB . and Žb. HOMO–LUMO gap energy Ž D . with cage size n Žs 2 x q 4. for all 161 B x N xq4 isomers. The energies Žin eV. were calculated with the DFTB model. The solid line in Ža. shows the trend for the most stable cage at each nuclearity. The dashed line picks out the family of unstable cylindrical isomers that contain two B 7 N9 six-pentagon caps ŽI in Fig. 3.. The two lines in Žb. correspond to the same families as in Ža.. Isolated-pentagon-pair ŽIPP. isomers are circled in both graphs.
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Fig. 2. Optimised geometries of the best B x N xq4 isomers for nuclearities 20 F n F 70, where n s 2 x q 4. Each isomer is labelled n:m Ž g:h., where n is the nuclearity of the cage, m is the isomer number in the spiral code sequence w19x, g is the maximum point group symmetry of the undecorated fullerene cage and h is the maximum symmetry of the cage when decorated with B and N atoms. Black dots indicate nitrogen atom sites. The structures are drawn from the optimised cartesian coordinates in the DFTB model.
The 28-vertex Td B12 N16 cage has been treated by other authors as a formal derivative of C 28 w28,29x though it has not been seen before as part of a well
defined homologous series of nitrogen-rich species. In fact, the results from the DFTB study actually show the C3 Õ isomer of B12 N16 , which has not been
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considered before, to be more stable than the Td isomer by 0.454 eV. In addition to steric advantages, the four extra nitrogen atoms confer electronic stability on both isomers of B12 N16 by closing the 5A 2 open shell of the fullerene cage in a manner identical to that already suggested for C 24 N4 w30x. A similar shell-closing effect with four extra p electrons is noticeable for nearly all B x N xq4 decorations of the fullerene polyhedra investigated, with the only counterexamples being the 20-vertex B 8 N12 cage and two unstable isomers belonging to the family shown by the dotted line in Fig. 1a. The isomers in this family contain two C3 B 7 N9 caps of six fused pentagons Žcap I in Fig. 3. and are highly strained owing to the narrow tubular shape created by the clustered pentagons and apical boron atoms at both ends. In general, the more stable isomers tend to have nitrogen atoms at apical Ž‘sp 3 ’. sites created by the fusion of pentagons, with boron atoms at flat Ž‘sp 2 ’. sites created by the fusion of three hexagons w30x. Electronegativity arguments also favour this partition of B and N sites, with nitrogen tending to a pyramidal ‘lone-pair’ configuration where possible. The first ‘magic’ isomer which is much more stable relative to others appears at 52 vertices and is a B 24 N28 cage of T symmetry. The outstanding
feature of this cage is that it is the smallest mathematically possible even isomer containing isolated pentagon pairs w18x. The lower bound on an IPP cage is 48 vertices, since each pentalene unit accounts for eight atoms, but the smallest realisable IPP cage has 50 vertices and corresponds to an odd B 25 N25 cage. It is clear from Figs. 1a and 1b that the IPP isomers as a family are particularly stable, both in terms of binding energy per atom and size of HOMO–LUMO gap. Only one IPP isomer exists at each nuclearity 52, 58, 64 and 70, and it turns out that these isomers form a ‘growth series’ with each cage being created from its predecessor by insertion of 3 BN units. This series is based on two C3 caps ŽII and III in Fig. 3.. In fact, cages smaller than the 52-vertex T isomer can also be seen as part of the same series, though they are not large enough to contain isolated pentagon pairs. Despite this, these isomers at 40 and 46 atoms are still relatively stable, as shown by the small peaks in Figs. 1a and 1b. Other series begin with the five IPP isomers at 68 vertices ŽB 32 N36 ., several of which give isomers based on caps of C3 or C3 Õ symmetry ŽIV to VI in Fig. 3.. The 161 isomers selected by the six-NN-pair rule are special in that they all contain six distinct sets of
Fig. 3. Threefold-symmetric caps containing three pentagon pairs. Black circles denote nitrogen and white circles boron atoms. I is the highly-strained B 7 N9 cap present at both ends of the isomers indicated by the dashed line in Fig. 1a. Caps II and III are present in the smallest series of IPP cages, including the first IPP cage, the T isomer of B 24 N28 . Caps IV to VI form the basis of the next few threefold-symmetric series starting with several isomers of B 32 N36 . Note that the perimeters of all the caps alternate with B and N atoms, allowing alternation in all adjoining hexagonal rings.
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are equivalent: each carries 1r6 of the total 4p steradian of curvature required to close a sphere. The squarerpentagon-pair analogy is used here to apply the IPP rule to BN nanotube tips. Extension of the
Fig. 4. Patches of three fused pentagon pairs occurring in small B x N xq4 isomers. Black circles denote nitrogen and white circles boron atoms. Each discrete pair can be distinguished by its N–N bond. Note that the perimeters of all the patches alternate with B and N atoms, allowing fusion with either pentagonal or hexagonal rings.
pentagon pairs, each containing one N–N bond, although the pairs are isolated in only a few cases. Several structural motifs containing two or three fused pentagon pairs commonly occur in B x N xq4 cages, and these are illustrated in Fig. 4. The current investigation has highlighted the role of an IPP rule in nitrogen-rich boron-nitrogen cages. This differs from the situation for the carbon fullerenes themselves, where full isolation of pentagons is preferred, as expressed in the isolated-pentagon rule ŽIPR. w14,15x. Isolation of pentagon pairs in BN cages based on fullerene polyhedra is a compromise between the reduction of steric strain that arises from dispersion of pentagons Žminimising pentagon adjacencies w31x. and the loss of electronic stabilisation caused by the non-alternant odd-membered rings. The specific advantage of an IPP cage is that it is compatible with full alternation in all hexagonal rings. Broadly speaking, different polyhedra are favoured in the two series: a good heteronuclear Že.g. BN. cage will be derived from an energetically poor fullerene parent, and the best fullerenes give poorer heteronuclear derivatives. In this sense, boron nitride cages composed of pentagons and hexagons are ‘anti-fullerenes’.
5. BN nanotubes and larger BN shells The IPP rule is very similar to the isolated-square rule for alternant BN polyhedra w12x in that both require all defects Žsquares or pentagon pairs. to be isolated by hexagons. From the point of view of Euler’s theorem, a fused pentagon pair and a square
Fig. 5. Molecular-mechanics simulations of BN nanotube tips. Ža. A ziz-zag tubule with three square defects included in an otherwise hexagonal framework; Žb. a similar zig-zag tube with three pentagon-pair defects; Žc. an armchair tubule with four square and one octagonal ring; Žd. a similar armchair tube with three pentagon-pairs. In Žb. and Žd. there is an excess of just two nitrogen atoms in the visible portion of the tube. Rotation around the tube axis shows that the flat profile of the tip is present in all cases, but that the two tubes containing pentagon-pair defects have more rounded corners.
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rule to include pentagon–heptagon pairs is also discussed with respect to its effect on the nitrogen excess and sphericity of larger BN cages and shells. The presence of square rings has been invoked by several groups w2,3,5x to explain the observed flat caps of boron nitride nanotubes. We note here that very similar profiles can be obtained by placing three pentagon pairs near the tip of the tube instead of squares. The IPP tubes have flat caps but slightly more rounded corners compared to those made with squares ŽFig. 5.. Grouping of the six pentagons into pairs serves to remove much of the chemical frustration that caused Blase et al. w32x and Loiseau et al. w3,5x to conclude on energy grounds that BN tube caps must be fully alternant and hence contain three squares. Similar factors may be at work in finite polyhedral BN cages and concentric shells. As Fig. 5 illustrates, both zig-zag and armchair type tubes can be closed by introducing three isolated pentalene units, and it can also be envisaged that chiral tubes could be closed in a similar fashion. A direct comparison of the energetic penalty of squares and pentagon pairs would best be achieved by comparing isoelectronic systems. In this respect, the nitrogen-rich cages studied in this investigation cannot be directly compared with the fully alternant squarerhexagon cages composed of equal numbers of B and N atoms. A detailed comparison of squarer hexagon and fullerene-based B x N x cages will therefore require a systematic investigation of the ‘odd’ cages of Table 1, and this will be reported elsewhere. Finally, we note that the calculations described here indicate stability of boron-nitrogen clusters supporting a small excess of nitrogen. However, boron atoms may also be able to bind larger accumulations of nitrogen atoms and pairs. There are several mathematical possibilities for building more NN pairing into the model and the most attractive of these would be to include heptagonal rings in the cages. A B,Ndecorated trivalent polyhedron with 12 q f 7 pentagons and f 7 heptagons must contain a minimum of 6 q f 7 homonuclear bonds, which in favourable cases will all be N–N, whilst retaining full alternation in all hexagons. Such a cage would have a formula B x N xqy with y s 4 q 2 f 7r3. Inclusion of pentagon–heptagon pairs has the favourable side effect of making the polyhedron more spherical in shape, especially at large nuclearities where cages that con-
tain a minimal number of defect rings are highly faceted w33x. Within the purely alternant series of trivalent BN cages, sphericity can be increased at the cost of including pairs of square and octagonal rings, but these carry a much higher steric penalty than pentagon–heptagon pairs. Conversely, a nitrogen excess of just two is possible with a hybrid polyhedron having three squares and three pentalene ŽIPP. units, where the three N–N bonds are confined to the pentagon–pentagon edges and all the even-membered rings are still fully alternant. Thus, it can be seen that the exact composition and shape of a BN cage or shell is critically dependent on the number, type and arrangement of defects. A large number of combinations is possible, and it is likely that the preferred configuration will vary with the overall size of the shell.
Acknowledgements P.W.F., K.M.R. and G.S. acknowledge financial support of this work by EPSRC ŽUK., DFG ŽGermany., DAADrBritish Council ŽARC 868. and the TMR initiative of the EU under contract FMRXCT97-0126 on Usable Fullerene Derivatives. H.T. acknowledges grants CONACYT ŽMexico, 25237-E., DGAPA-UNAM ŽMexico, IN-107296. and TWAS Ž97-178 RGrPHYSrLA. and M.T. thanks the Royal Society and Materials Research Laboratory ŽUCSB, USA. for financial support.
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P.W. Fowler et al.r Chemical Physics Letters 299 (1999) 359–367 w7x L. Boulanger, B. Andriot, M. Cauchetier, F. Willaime, Chem. Phys. Lett. 234 Ž1995. 227. w8x O. Stephan, Y. Bando, A. Loiseau, F. Willaime, N. Schram´ chenko, T. Tamiya, T. Sato, Appl. Phys. A 67 Ž1998. 107. w9x F. Jensen, H. Toftlund, Chem. Phys. Lett. 201 Ž1993. 89. w10x M.-L. Sun, Z. Slanina, S.-L. Lee, Chem. Phys. Lett. 233 Ž1995. 279. w11x H.-Y. Zhu, D.J. Klein, W.A. Seitz, N.H. March, Inorg. Chem. 34 Ž1995. 1377. w12x P.W. Fowler, T. Heine, D. Mitchell, R. Schmidt, G. Seifert, J. Chem. Soc. Faraday Trans. 92 Ž1996. 2197. w13x G. Seifert, P.W. Fowler, D. Mitchell, D. Porezag, T. Frauenheim, Chem. Phys. Lett. 268 Ž1997. 352. w14x H.W. Kroto, Nature 329 Ž1987. 529. w15x T.G. Schmaltz, W.A. Seitz, D.J. Klein, G.E. Hite, J. Am. Chem. Soc. 110 Ž1988. 1113. w16x See, for example, bond energies in: J. Emsley, The elements ŽOxford University Press, New York, 1989.. w17x P.W. Fowler, K.M. Rogers, T. Heine, G. Seifert, to be published. w18x P.W. Fowler, Phil. Trans. Roy. Soc. A 343 Ž1993. 39. w19x As listed in: P.W. Fowler, D.E. Manolopoulos, An atlas of fullerenes, Oxford University Press, New York, 1995. w20x D.E. Manolopoulos, P.W. Fowler, J. Chem. Phys. 96 Ž1992. 7603. w21x D.E. Manolopoulos, J.C. May, S.E. Down, Chem. Phys. Lett. 181 Ž1991. 105.
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w22x D. Porezag, T. Frauenheim, T. Kohler, G. Seifert, R. ¨ Kaschner, Phys. Rev. B 51 Ž1995. 12947. w23x G. Seifert, D. Porezag, T. Frauenheim, Int. J. Quant. Chem. 58 Ž1996. 185. w24x P.W. Fowler, T. Heine, D.E. Manolopoulos, D. Mitchell, G. Orlandi, R. Schmidt, G. Seifert, F. Zerbetto, J. Phys. Chem. 100 Ž1996. 6984. w25x P.W. Fowler, T. Heine, D. Mitchell, G. Orlandi, R. Schmidt, G. Seifert, F. Zerbetto, J. Chem. Soc. Faraday Trans. 92 Ž1996. 2203. w26x P.W. Fowler, D. Mitchell, R. Taylor, G. Seifert, J. Chem. Soc. Perkin Trans. 2 Ž1997. 1901. w27x J. Widany, T. Frauenheim, T. Kohler, M. Sternberg, D. ¨ Porezag, G. Jungnickel, G. Seifert, Phys. Rev. B 53 Ž1996. 4443. w28x E. Kaxiras, K. Jackson, M.R. Pederson, Chem. Phys. Lett. 225 Ž1994. 448. w29x X.-T. Jiang, X.-G. Hu, Q.-S. Li, Chin. J. Chem. 14 Ž1996. 490. w30x P.W. Fowler, J.P.B. Sandall, J. Chem. Res. ŽS. Ž1995. 68. w31x E.E.B. Campbell, P.W. Fowler, D. Mitchell, F. Zerbetto, Chem. Phys. Lett. 250 Ž1996. 544. w32x X. Blase, A. De Vita, J.-C. Charlier, R. Car, Phys. Rev. Lett. 80 Ž1998. 1666. w33x M. Terrones, H. Terrones, Fullerene Sci. Technol. 4 Ž1996. 517.
Chemical Physics Letters 299 Ž1999. 359–367
Pentagonal rings and nitrogen excess in fullerene-based BN cages and nanotube caps P.W. Fowler a , K.M. Rogers a , G. Seifert b, M. Terrones c , H. Terrones
d
a
b
School of Chemistry, UniÕersity of Exeter, Stocker Road, Exeter, EX4 4QD, UK Institut fur ¨ Theoretische Physik, Technische UniÕersitat ¨ Dresden, Zellescher Weg 17, D-01062 Dresden, Germany c School of Chemistry, Physics and EnÕironmental Science, UniÕersity of Sussex, Brighton, BN1 9QJ, UK d ´ Instituto de Fisica, UNAM, Apartado Postal 20-364, Mexico, D.F. 01000, Mexico Received 8 October 1998
Abstract Classical fullerene polyhedra are considered as candidates for boron-nitrogen cages. Simple arguments show that nitrogen-rich, fullerene-like cages with the general formula B x N xq4 can be constructed to have just six N–N bonds, with full B,N alternation in all hexagonal rings. Systematic density-functional tight-binding calculations indicate special stability for such cages when they contain six isolated pentagon pairs. Closure of BN nanotubes with three isolated pentagon pairs gives an alternative to the usual explanation of their observed flat tips in terms of square rings. Inclusion in polyhedra of various combinations of squares, pentagon–pentagon and pentagon–heptagon pairs would lead to BN cages and shells where nitrogen excess correlates with sphericity. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Several research groups have described the production of boron nitride-based nano-structures, both tube-like w1–5x and particulate w6–8x and there have been numerous theoretical studies of hypothetical ŽBN. x clusters w9–13x. In view of the structural similarity of graphite and bulk boron nitride, it is usual to discuss these new materials in terms of modifications of the carbon fullerene and nanotube models. Most workers, including ourselves w12,13x, have taken the view that the natural generalisation of the carbon fullerene is a trivalent, polyhedral structure with rigorous B,N alternation, and therefore composed entirely of even-membered rings, with six squares replacing the twelve pentagons of the fullerene recipe. Local minima on the potential sur-
face of x B and x N atoms are found for such squarerhexagon clusters, with some magic numbers Že.g. B12 N12 . and intuitively reasonable structural trends such as an isolated-square counterpart of the fullerene isolated-pentagon rule w14,15x. However, another plausible generalisation is to retain the pentagonal rings of the carbon fullerene and to accept a minimal number of forced homonuclear bonds in the B,N decoration of the framework. Simple energetics w16x suggest that NN pairs should in this case be preferred over BB. Calculations on small B x N y rings and small clusters w17x find that NN pairs are often protected by an energy barrier to extrusion and have only moderate effects on overall stability of a mainly alternant framework. It is pointed out in the present paper that the problem of designing B x N y structures based on classical fullerenes
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with maximisation of alternation, minimisation of NN pairing and total aÕoidance of BB pairs has a unique mathematical solution: it is necessary to use an excess of precisely four nitrogen atoms in a fullerene cage with at least six pentagon adjacencies. The ideal stoichiometry for such cages is therefore B x N xq4 . The purpose of the present paper is to make a systematic exploration of these nitrogen-rich boronnitrogen cage structures based on fullerene polyhedra and to demonstrate that they obey an ‘isolated-pentagon-pair rule’ rather than the isolated-pentagon rule of the carbon fullerenes. As an interesting consequence, it is suggested that the structural pattern of these nitrogen-rich cages may also explain the flat tips observed in BN nanotubes w2,3,5x, and that variations of the face recipe at higher nuclearities may generate near-spherical BN shells with larger nitrogen excess. The paper is organised as follows. The generation and selection of the structures to be considered is outlined ŽSection 2., followed by a brief description of the density-functional tight-binding ŽDFTB. model used to evaluate the energetics and geometries of the structures ŽSection 3.. The results of calculations for the selected BN cages are then presented and their structural and energetic trends discussed ŽSection 4.. The preference which is found for isolated pentagon pairs leads to a proposal for closure of BN nanotubes with flat tips that contain homonuclear bonds ŽSection 5.. Finally, the relationships between face recipe, nitrogen excess and sphericity of BN cages and shells are briefly discussed.
2. Generation and selection of cage structures A fullerene polyhedron C n consists of twelve pentagonal and nr2 y 10 hexagonal rings. As a non-alternant polyhedron it cannot avoid having some homonuclear edges when each carbon atom is replaced by either B or N. The number of such edges varies with the B:N ratio of the decoration and with the underlying structure. An isolated-pentagon fullerene has twelve or more homonuclear edges, as each pentagon must contain at least one. If the pentagons are in contact, it may be possible to reduce this number to six, each at the fusion of two
pentagonal rings. In favourable cases, all hexagonal rings of the fullerene will have full B,N alternation, and the decorated polyhedron will achieve this minimal number of just six homonuclear edges. Suppose now that we insist that all six of these homonuclear edges be NN. It can be proved that the stoichiometry of the B,N-decorated fullerene must be B x N xq4 , where x s nr2 y 2. The argument is as follows. Take the formula to be B x N xq2 y . In an obvious notation, the total number of vertices is Õ s ÕB q ÕN s 2 x q 2 y
Ž 1.
and the total number of edges of the trivalent cage is e s 3Õr2 s e BB q e BN q e NN s 3 x q 3 y.
Ž 2.
Counting edges emanating from B and N sites, respectively, 2 e BB q e BN s 3 x ,
Ž 3.
2 e NN q e BN s 3 x q 6 y
Ž 4.
so that e NN y e BB s 3 y
Ž 5.
and from the fact that there are 12 pentagons, e NN q e BB G 6.
Ž 6.
Thus e NN G 3 q 3 yr2,
Ž 7.
e BB G 3 y 3 yr2.
Ž 8.
In favourable cases, therefore, a value of y s 2 suffices to remove all BB edges and gives e NN s 6. Hence, for closed cages based on classical fullerenes, the formula B x N xq4 is optimal. Structures that achieve this optimum include the ‘even’ subset of the isolated-pentagon-pair ŽIPP. fullerenes. IPP fullerenes are cages where the twelve pentagons fall into six pentalene Žpentagon pair. units, each surrounded by hexagons. Such cages may be either ‘odd’, allowing decorations B x N x with three NN and three BB pairs at the pentagon joins, or ‘even’, allowing conjugate B xq 4 N x and B x N xq4 decorations with respectively six BB and six NN pairs at the joins w18x. Some smaller fullerenes with more highly condensed arrangements of pentagon pairs also achieve this optimum.
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All 30,579 fullerene cages w19x in the range 20 F n F 70 were searched to find the candidates for optimal B x N xq4 cages based on the model described above. The algorithm used follows Ref. w18x and is in two parts. The first finds all isomers that contain a set of six pentagon–pentagon edges which satisfy the rule that each edge in the set must be separated from all others by at least two edges, thus avoiding a homonuclear chain longer than two vertices. The second part involves deciding whether the selected isomers are odd Žallowing a formula B x N x with three N–N and three B–B bonds. or even Žallowing a formula B x N xq4 with six N–N bonds.. In practice, this is done by temporarily breaking the six pentagon–pentagon edges. If the twelve divalent vertices of the resulting alternant cage have coefficients of the same sign in the most antibonding eigenvector of the adjacency matrix, the isomer is even, i.e. all six edges could be NN pairs. Otherwise, the isomer is odd, with three NN and three BB pairs. The rest of the vector is used to finish the ‘decoration’ of the cage. A given B x N xq2 y isomer is generated < G
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Table 1 Enumeration of BN isomers based on classical fullerene structures. Nf is the total number of fullerene isomers found by the spiral algorithm w19x at each nuclearity n. Ne is the number of even isomers of formula B x N xq4 Ž x s n r2y2. with 6 N–N bonds. No is the number of odd isomers of formula B x N x Ž x s n r2. with three N–N and three B–B bonds n 20 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 total
Nf
Ne
No
1 1 1 2 3 6 6 15 17 40 45 89 116 199 271 437 580 924 1 205 1 812 2 385 3 465 4 478 6 332 8 149
1 0 1 2 1 3 1 1 2 3 3 6 4 5 5 8 9 10 8 11 10 17 15 18 17
0 0 1 1 0 3 3 8 7 12 10 20 15 27 26 36 41 59 50 76 68 102 89 126 116
30 579
161
896
molecular mechanics package before full optimisation.
3. Quantum-mechanical method The main tool for the full survey of the selected structures is the density-functional tight-binding ŽDFTB. method w22,23x. This approach has been applied for carbon cages w24–26x and is known to give a realistic description of energetics and structures for BN systems, both solid-state and molecular w12,13,27x. DFTB is a density-functional based non-orthogonal tight-binding approach. It has an LCAO framework, but considers only two-centre integrals. This
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method makes use of universal, two-particle, shortrange repulsive potentials which are derived in a general, almost parameter-free, manner using ab initio results for small reference systems. Hence, the overall scheme can be viewed as being intermediate between ab initio density-functional theory and traditional empirical tight-binding treatment.
4. Results and discussion The results of the calculations on the B x N xq4 cages are summarised in Figs. 1a and 1b, where
binding energies Žper atom. and HOMO–LUMO gaps are shown, respectively. Fig. 2 illustrates the geometry of the most stable isomer at each stoichiometry. It is notable that every isomer in the range compatible with the rule of construction corresponds to a local minimum on the potential surface, and that stability generally increases smoothly from the smaller clusters towards the bulk BN limit with the 1rn relationship expected from the size dependence of the cage curvature w13x. There is no clear overall correlation between the binding energies and HOMO–LUMO gaps ŽFigs. 1a and 1b., but the most stable isomers at each nuclearity are mostly characterised by a relatively large gap.
Fig. 1. Variation of Ža. binding energy per atom Ž EB . and Žb. HOMO–LUMO gap energy Ž D . with cage size n Žs 2 x q 4. for all 161 B x N xq4 isomers. The energies Žin eV. were calculated with the DFTB model. The solid line in Ža. shows the trend for the most stable cage at each nuclearity. The dashed line picks out the family of unstable cylindrical isomers that contain two B 7 N9 six-pentagon caps ŽI in Fig. 3.. The two lines in Žb. correspond to the same families as in Ža.. Isolated-pentagon-pair ŽIPP. isomers are circled in both graphs.
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Fig. 2. Optimised geometries of the best B x N xq4 isomers for nuclearities 20 F n F 70, where n s 2 x q 4. Each isomer is labelled n:m Ž g:h., where n is the nuclearity of the cage, m is the isomer number in the spiral code sequence w19x, g is the maximum point group symmetry of the undecorated fullerene cage and h is the maximum symmetry of the cage when decorated with B and N atoms. Black dots indicate nitrogen atom sites. The structures are drawn from the optimised cartesian coordinates in the DFTB model.
The 28-vertex Td B12 N16 cage has been treated by other authors as a formal derivative of C 28 w28,29x though it has not been seen before as part of a well
defined homologous series of nitrogen-rich species. In fact, the results from the DFTB study actually show the C3 Õ isomer of B12 N16 , which has not been
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considered before, to be more stable than the Td isomer by 0.454 eV. In addition to steric advantages, the four extra nitrogen atoms confer electronic stability on both isomers of B12 N16 by closing the 5A 2 open shell of the fullerene cage in a manner identical to that already suggested for C 24 N4 w30x. A similar shell-closing effect with four extra p electrons is noticeable for nearly all B x N xq4 decorations of the fullerene polyhedra investigated, with the only counterexamples being the 20-vertex B 8 N12 cage and two unstable isomers belonging to the family shown by the dotted line in Fig. 1a. The isomers in this family contain two C3 B 7 N9 caps of six fused pentagons Žcap I in Fig. 3. and are highly strained owing to the narrow tubular shape created by the clustered pentagons and apical boron atoms at both ends. In general, the more stable isomers tend to have nitrogen atoms at apical Ž‘sp 3 ’. sites created by the fusion of pentagons, with boron atoms at flat Ž‘sp 2 ’. sites created by the fusion of three hexagons w30x. Electronegativity arguments also favour this partition of B and N sites, with nitrogen tending to a pyramidal ‘lone-pair’ configuration where possible. The first ‘magic’ isomer which is much more stable relative to others appears at 52 vertices and is a B 24 N28 cage of T symmetry. The outstanding
feature of this cage is that it is the smallest mathematically possible even isomer containing isolated pentagon pairs w18x. The lower bound on an IPP cage is 48 vertices, since each pentalene unit accounts for eight atoms, but the smallest realisable IPP cage has 50 vertices and corresponds to an odd B 25 N25 cage. It is clear from Figs. 1a and 1b that the IPP isomers as a family are particularly stable, both in terms of binding energy per atom and size of HOMO–LUMO gap. Only one IPP isomer exists at each nuclearity 52, 58, 64 and 70, and it turns out that these isomers form a ‘growth series’ with each cage being created from its predecessor by insertion of 3 BN units. This series is based on two C3 caps ŽII and III in Fig. 3.. In fact, cages smaller than the 52-vertex T isomer can also be seen as part of the same series, though they are not large enough to contain isolated pentagon pairs. Despite this, these isomers at 40 and 46 atoms are still relatively stable, as shown by the small peaks in Figs. 1a and 1b. Other series begin with the five IPP isomers at 68 vertices ŽB 32 N36 ., several of which give isomers based on caps of C3 or C3 Õ symmetry ŽIV to VI in Fig. 3.. The 161 isomers selected by the six-NN-pair rule are special in that they all contain six distinct sets of
Fig. 3. Threefold-symmetric caps containing three pentagon pairs. Black circles denote nitrogen and white circles boron atoms. I is the highly-strained B 7 N9 cap present at both ends of the isomers indicated by the dashed line in Fig. 1a. Caps II and III are present in the smallest series of IPP cages, including the first IPP cage, the T isomer of B 24 N28 . Caps IV to VI form the basis of the next few threefold-symmetric series starting with several isomers of B 32 N36 . Note that the perimeters of all the caps alternate with B and N atoms, allowing alternation in all adjoining hexagonal rings.
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are equivalent: each carries 1r6 of the total 4p steradian of curvature required to close a sphere. The squarerpentagon-pair analogy is used here to apply the IPP rule to BN nanotube tips. Extension of the
Fig. 4. Patches of three fused pentagon pairs occurring in small B x N xq4 isomers. Black circles denote nitrogen and white circles boron atoms. Each discrete pair can be distinguished by its N–N bond. Note that the perimeters of all the patches alternate with B and N atoms, allowing fusion with either pentagonal or hexagonal rings.
pentagon pairs, each containing one N–N bond, although the pairs are isolated in only a few cases. Several structural motifs containing two or three fused pentagon pairs commonly occur in B x N xq4 cages, and these are illustrated in Fig. 4. The current investigation has highlighted the role of an IPP rule in nitrogen-rich boron-nitrogen cages. This differs from the situation for the carbon fullerenes themselves, where full isolation of pentagons is preferred, as expressed in the isolated-pentagon rule ŽIPR. w14,15x. Isolation of pentagon pairs in BN cages based on fullerene polyhedra is a compromise between the reduction of steric strain that arises from dispersion of pentagons Žminimising pentagon adjacencies w31x. and the loss of electronic stabilisation caused by the non-alternant odd-membered rings. The specific advantage of an IPP cage is that it is compatible with full alternation in all hexagonal rings. Broadly speaking, different polyhedra are favoured in the two series: a good heteronuclear Že.g. BN. cage will be derived from an energetically poor fullerene parent, and the best fullerenes give poorer heteronuclear derivatives. In this sense, boron nitride cages composed of pentagons and hexagons are ‘anti-fullerenes’.
5. BN nanotubes and larger BN shells The IPP rule is very similar to the isolated-square rule for alternant BN polyhedra w12x in that both require all defects Žsquares or pentagon pairs. to be isolated by hexagons. From the point of view of Euler’s theorem, a fused pentagon pair and a square
Fig. 5. Molecular-mechanics simulations of BN nanotube tips. Ža. A ziz-zag tubule with three square defects included in an otherwise hexagonal framework; Žb. a similar zig-zag tube with three pentagon-pair defects; Žc. an armchair tubule with four square and one octagonal ring; Žd. a similar armchair tube with three pentagon-pairs. In Žb. and Žd. there is an excess of just two nitrogen atoms in the visible portion of the tube. Rotation around the tube axis shows that the flat profile of the tip is present in all cases, but that the two tubes containing pentagon-pair defects have more rounded corners.
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rule to include pentagon–heptagon pairs is also discussed with respect to its effect on the nitrogen excess and sphericity of larger BN cages and shells. The presence of square rings has been invoked by several groups w2,3,5x to explain the observed flat caps of boron nitride nanotubes. We note here that very similar profiles can be obtained by placing three pentagon pairs near the tip of the tube instead of squares. The IPP tubes have flat caps but slightly more rounded corners compared to those made with squares ŽFig. 5.. Grouping of the six pentagons into pairs serves to remove much of the chemical frustration that caused Blase et al. w32x and Loiseau et al. w3,5x to conclude on energy grounds that BN tube caps must be fully alternant and hence contain three squares. Similar factors may be at work in finite polyhedral BN cages and concentric shells. As Fig. 5 illustrates, both zig-zag and armchair type tubes can be closed by introducing three isolated pentalene units, and it can also be envisaged that chiral tubes could be closed in a similar fashion. A direct comparison of the energetic penalty of squares and pentagon pairs would best be achieved by comparing isoelectronic systems. In this respect, the nitrogen-rich cages studied in this investigation cannot be directly compared with the fully alternant squarerhexagon cages composed of equal numbers of B and N atoms. A detailed comparison of squarer hexagon and fullerene-based B x N x cages will therefore require a systematic investigation of the ‘odd’ cages of Table 1, and this will be reported elsewhere. Finally, we note that the calculations described here indicate stability of boron-nitrogen clusters supporting a small excess of nitrogen. However, boron atoms may also be able to bind larger accumulations of nitrogen atoms and pairs. There are several mathematical possibilities for building more NN pairing into the model and the most attractive of these would be to include heptagonal rings in the cages. A B,Ndecorated trivalent polyhedron with 12 q f 7 pentagons and f 7 heptagons must contain a minimum of 6 q f 7 homonuclear bonds, which in favourable cases will all be N–N, whilst retaining full alternation in all hexagons. Such a cage would have a formula B x N xqy with y s 4 q 2 f 7r3. Inclusion of pentagon–heptagon pairs has the favourable side effect of making the polyhedron more spherical in shape, especially at large nuclearities where cages that con-
tain a minimal number of defect rings are highly faceted w33x. Within the purely alternant series of trivalent BN cages, sphericity can be increased at the cost of including pairs of square and octagonal rings, but these carry a much higher steric penalty than pentagon–heptagon pairs. Conversely, a nitrogen excess of just two is possible with a hybrid polyhedron having three squares and three pentalene ŽIPP. units, where the three N–N bonds are confined to the pentagon–pentagon edges and all the even-membered rings are still fully alternant. Thus, it can be seen that the exact composition and shape of a BN cage or shell is critically dependent on the number, type and arrangement of defects. A large number of combinations is possible, and it is likely that the preferred configuration will vary with the overall size of the shell.
Acknowledgements P.W.F., K.M.R. and G.S. acknowledge financial support of this work by EPSRC ŽUK., DFG ŽGermany., DAADrBritish Council ŽARC 868. and the TMR initiative of the EU under contract FMRXCT97-0126 on Usable Fullerene Derivatives. H.T. acknowledges grants CONACYT ŽMexico, 25237-E., DGAPA-UNAM ŽMexico, IN-107296. and TWAS Ž97-178 RGrPHYSrLA. and M.T. thanks the Royal Society and Materials Research Laboratory ŽUCSB, USA. for financial support.
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