6984
J. Phys. Chem. 1996, 100, 6984-6991
Energetics of Fullerenes with Four-Membered Rings P. W. Fowler,*,† T. Heine,‡ D. E. Manolopoulos,§ D. Mitchell,† G. Orlandi,| R. Schmidt,‡ G. Seifert,*,‡ and F. Zerbetto*,| Department of Chemistry, UniVersity of Exeter, Stocker Road, Exeter EX4 4QD, U.K., Institut fu¨ r Theoretische Physik, Technische UniVersita¨ t Dresden, Mommsenstrasse 13, D-01069 Dresden, Germany, Department of Chemistry, UniVersity of Nottingham, UniVersity Park, Nottingham NG7 2RD, U.K., and Dipartimento di Chimica, ‘G. Ciamician’, UniVersita´ di Bologna, Via F. Selmi 2, 40126 Bologna, Italy ReceiVed: October 31, 1995; In Final Form: February 6, 1996X
The energetic cost of introducing square faces to fullerenes with adjacent pentagons is investigated theoretically. Relative energies of all 1735 hypothetical C40 cages that can be assembled from square, pentagonal, and hexagonal faces are calculated within two independent semiempirical models. All isomers are found to lie in local minima on the potential surface. The QCFF/PI (quantum consistent force field/π) and DFTB (density functional tight binding) approaches agree in predicting that no cage with one or more squares is of lower energy than the best classical C40 fullerene but that many such cages are more stable than many C40 fullerenes. Energy penalties of 160-200 kJ mol-1 per square are suggested by the DFTB calculations, and penalties of about twice this size by the QCFF/PI model. The energy variation across the range of fullerenes and pseudofullerenes is steric in origin and correlates well with the normalized second moment of the hexagon neighbor signature: aggregation of hexagons in one part of the cage surface is incompatible with even distribution of curvature and implies crowding of defects elsewhere. QCFF/PI calculations for selected isomers of C62 to C68 also show that though cages with squares may again be more stable than some fullerenes, they are all bettered in energy by the best classical fullerene at each nuclearity.
Introduction All fullerenes characterized so far conform to the classical definition: a fullerene is a carbon cage Cn in the form of a trivalent polyhedron with exactly 12 pentagonal and (n/2 - 10) hexagonal faces. This definition agrees with chemical intuition in that those polyhedra in which all nonhexagonal rings are pentagons represent the minimal departures from the graphite sheet. Rings of other sizes, for example 4 or 7, would be expected to induce extra local strain and/or further loss of π delocalization, and so variations on the fullerene recipe would be expected to carry an energetic penalty. All experimentally characterized fullerenes have a further feature in common. Not only are they pentagon-hexagon polyhedra but their pentagonal faces are isolated one from another,1,2 obeying the “isolated pentagon rule”. Fullerene polyhedra are possible for all nuclearities n ) 20 + 2k (k * 1),3 but isolation of pentagons is possible only for n ) 60 and n ) 70 + 2k (all k including k ) 0).4 Pentagons in contact give rise to an energetic penalty that has been variously estimated at 70-150 kJ mol-1 per adjacency.5-10 It has therefore been suggested that insertion of square (i.e. quadrilateral) faces, with consequent loss of two pentagons per square, could turn out to be energetically favorable in the ranges (n e 58 and 62 e n e 68) where isolated-pentagon classical fullerenes are impossible.11 The present paper makes a systematic study of the cost of introducing squares to a typical lower fullerene. The example chosen is C40. Two established parametrized methods (QCFF/ PI12 and DFTB13,14) are used to derive energies for all 1735 distinct polyhedral C40 frameworks made up of square, pen†
University of Exeter. Technische Universita¨t Dresden. § University of Nottingham. | Universita ´ di Bologna. X Abstract published in AdVance ACS Abstracts, April 1, 1996. ‡
0022-3654/96/20100-6984$12.00/0
tagonal, and hexagonal faces, constructed by a modification of the spiral algorithm.15 An energetic penalty for introduction of square faces is thereby obtained. It will be shown that the two models find the best classical fullerene isomer of C40 to be more stable than any isomer with one or more square faces but also that both models find some cages containing squares to be of lower total energy than most of the classical isomers. Extension of the calculations to the range 62 e n e 68 again reveals some square-containing isomers of low total energy, but none that are more stable than the best classical fullerene at the same nuclearity. Background A trivalent polyhedron with n vertices, e edges, and f faces has, by Euler’s theorem,
e ) 3n/2
(1)
f ) n/2 + 2
(2)
and
If fr denotes the number of faces of size r, then
∑r (6 - r)fr ) 12
(3)
Specifically, then, for a polyhedron with s square, p pentagonal, and h hexagonal faces (f4 ) s, f5 ) p, f6 ) h) we have,
s + p + h ) n/2 + 2
(4)
2s + p ) 12
(5)
and
and so polyhedra with 0, 1, 2, 3, 4, 5, 6 squares and p ) 12-2s ) 12, 10, 8, 6, 4, 2, 0 pentagons are allowed by Euler’s theorem, © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 17, 1996 6985
although some, for example, n ) 20, s ) 1, p ) 10, h ) 0 and n ) 22, s ) 0, p ) 12, h ) 1 are not constructible for other reasons. In effect, addition of a square face to a cage at fixed n is accompanied by loss of two pentagons and gain of a hexagon. Other things being equal, this will tend to allow reduction of the number of pentagon adjacencies. More precisely, in an obvious notation where ss denotes square-square, sp squarepentagon, sh square-hexagon adjacencies, and so on, the six types of edge in a square-pentagon-hexagon polyhedron are related by three conditions, one for each type of face:
2ss + sp + sh ) 4s
(6)
sp + 2pp + ph ) 60 - 10s
(7)
sh + ph + 2hh ) 3n - 60 + 6s
(8)
For 0 e s e 6 there is at least one cage for each s at each fullerene nuclearity, apart from the two exceptional cases at n ) 20 and n ) 22 noted earlier. The isomer counts in the range 20 e n e 60 have been tabulated by Babic´ and Trinajstic´16 using a modification of their algorithm for fullerene generation. To construct the C40 cages in the present work, we used instead a modification of the fullerene spiral program given in the appendix of the fullerene atlas.4 The adaptation of the latter for 4-5-6 trivalent polyhedra is straightforward, although allowance must be made for the possibility of separating triangles in the dual, which can happen when the trivalent polyhedron contains square and/or triangular faces. In general, three mutually adjacent vertices in the dual correspond either to a unique vertex of the trivalent polyhedron or to a separating triangle. If the former, the original coding applies without change; if the latter, one must avoid assigning a new vertex of the polyhedron to them. Thus, one must have a new method for detecting separating triangles so that they can be avoided. This is not difficult to arrange because, by definition, removal of a separating triangle from the dual graph leaves two disconnected subgraphs, each containing at least one vertex. Minor modifications to the loop structure of the program are also necessary to allow for the fact that “defect” faces are either square or pentagonal. Counts generated by the modified spiral program in the range checked (20 e n e 60) agree with those published previously.16 In the particular case of C40 cages, therefore, there are respectively 40(s ) 0), 163(s ) 1), 544(s ) 2), 601(s ) 3), 342(s ) 4), 38(s ) 5), and 7(s ) 6) structural isomers, giving a total of 1735 candidates for the 40-carbon cage under the extended definition of a fullerene. Procedure Initial Cartesian coordinates were generated for each of the 1735 C40 isomers by feeding either the topological coordinates17 derived from eigenvectors of the adjacency matrix or simply the atom connection list through a proprietary molecularmechanics optimizer. A full geometry optimization was then carried out for each isomer using two independently parametrized models. The first is the QCFF/PI (quantum consistent force field/π) model,12 which has been used successfully in a number of studies of fullerene geometries, energies, and vibrational frequencies.9,10,18-20 The second is the DFTB (density functional tight binding) model,13,14 which was designed to reproduce the results of LDA density functional calculations and has been used for hydrocarbon molecules, small carbon clusters, C60, and bulk systems (for reviews see refs 13 and 14). Neither was initially parametrized for fullerenes, and
neither was adjusted in the present study for fullerenes with squares: the substantial measure of agreement to be reported below for this large sample of conventional and unconventional carbon cages, therefore serves to increase our confidence in both models. The QCFF/PI model12 combines a spin-restricted selfconsistent-field calculation of π-electronic structure with a classical harmonic and anharmonic potential for the σ framework. Integrals used in the π calculation are functions of distances, bond angles, and out-of-plane angles. All QCFF/PI stationary points were checked by diagonalization of the Hessian in the diabatic approximation19 and confirmed as minima. DFTB13,14 is a density functional (DF) based nonorthogonal tight-binding (TB) method. It has an LCAO framework, but considers only two-center integrals. This method makes use of universal, two-particle, short-range repulsive potentials which are derived in a general, almost parameter-free, manner using ab initio results for small reference systems. In contrast to the QCFF/PI method, which distinguishes between σ and π electrons, DFTB treats all valence electrons on an equal footing. Most of the DFTB stationary points were confirmed as minima by molecular dynamics simulations started from the suspected minimum with a finite thermal energy. No symmetry constraints were applied in either QCFF/PI or DFTB models, and so the chances of confusing a higher stationary point with a local minimum are in any case small. Results All 1735 topologies generated by the extended spiral algorithm are found to correspond to local minima on the potential surface of C40 in both QCFF/PI and DFTB models. In this sense, not only are all 40 classical fullerenes of this formula plausible candidates for the best structure of the C40 cage but so are all the square-pentagon-hexagon variants (and so, probably, are many other trivalent polyhedra based on 40 vertices). This prodigal multiplicity of minima underlines the versatility of carbon and incidentally serves as a reminder of the precarious nature of predictions based on small sample sets. Figure 1 shows the excellent correlation of total energies in the two models for the 1735 isomers and gives the least-squares linear regression lines for each s ) 0, 1, 2, 3, 4, 5, 6. The gradients of these lines are all greater than unity and increase steadily with s, implying that square rings carry a heavier penalty in the QCFF/PI model and that the cost per square in this model rises faster as more squares are introduced. This difference between the models is only to be expected, as energies of hydrocarbons with square rings were not included in the original parametrization of QCFF/PI; refinement of the parameter set to take better account of such cases would be expected to lower the predicted energies somewhat, bringing QCFF/PI and DFTB predictions closer together. Some difference would probably remain, however, as studies of conventional fullerenes (e.g. the full set of isolated-pentagon isomers of C84) have also shown a larger spread of energies in the QCFF/PI model7 than in a tightbinding approach.8 The range of energies spanned by the full set of C40 isomers is 2523 kJ mol-1 (DFTB) and 4075 kJ mol-1 (QCFF/PI). A breakdown into subsets by value of s (Table 1) shows that while the ranges for different s overlap, the best of all the considered structures remains a classical fullerene whichever model is used. DFTB and QCFF/PI methods agree with each other and with previously published calculations8,9 in predicting the best fullerene isomers of C40 to be those numbered 38(0) and 39(0) in the spiral ordering (the only two with the minimal value pp ) 10), with isomer 38(0) winning by a small margin (53.5 kJ
6986 J. Phys. Chem., Vol. 100, No. 17, 1996
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Figure 1. Correlation of relative energies (kJ mol-1) calculated in the QCFF/PI (∆EQ) and DFTB (∆ED) models for C40 square-pentagonhexagon cages: (a) scatter diagram of all 1735 energies; (b) best-fit lines for s ) 0, 1, 2, 3, 4, 5, 6 square rings. The equations of the lines (with standard deviations σ) are ∆EQ/kJ mol-1 ) a(∆ED/kJ mol-1) + b, where a ) 1.137, b ) 21.7, σ ) 39 (s ) 0); a ) 1.158, b ) 177.0, σ ) 28 (s ) 1); a ) 1.222, b ) 293.3, σ ) 36 (s ) 2); a ) 1.273, b ) 397.0, σ ) 37 (s ) 3); a ) 1.310, b ) 495.8, σ ) 46 (s ) 4); a ) 1.349, b ) 577.5, σ ) 58 (s ) 5); a ) 1.331, b ) 728.0, σ ) 61 (s ) 6), and the global fit gives a ) 1.619, b ) 96.9, σ ) 111 (s ) 0, 1, 2, 3, 4, 5, 6).
TABLE 1: Ranges and Means of Relative Energies (kJ mol-1) of C40 Cages with Different Numbers of Squares (s), Computed in DFTB (∆ED) and QCFF/PI (∆EQ) Modelsa ∆ED
∆EQ
s
Ns
range
mean
range
mean
0 1 2 3 4 5 6
40 163 544 601 342 38 7
0.0-786.3 87.5-837.5 151.4-1719.2 314.8-1525.8 535.2-2333.2 702.9-1719.9 1096.1-2523.1
263.6 422.8 616.6 796.5 1004.6 1131.9 1480.9
0.0-943.5 239.4-1162.0 459.1-2448.6 805.9-2417.2 1202.6-3715.1 1545.7-2891.6 2126.4-4074.5
321.6 666.4 1046.5 1411.1 1812.0 2104.2 2699.2
a N is the number of distinct structural isomers with exactly s square s rings. The energies are referred to that of fullerene isomer 38(0) (the most stable) in the respective model.
mol-1 (DFTB), 32.9 kJ mol-1 (QCFF/PI)). (We use a notation N(s) where s is the number of squares and N is the isomer sequence number in the lexicographic ordering of spirals for that value of s. Thus, the seven isomers with N ) 1 are 1(0), 1(1), 1(2), 1(3), 1(4), 1(5), 1(6) and are distinguished by their s values.) However, the best isomers with nonzero numbers of squares are more stable than the poorer fullerene isomers: for example, in the QCFF/PI model there are 27, 7, 1, 0, 0, 0 classical fullerenes with energies above that of the best isomer at s ) 1, 2, 3, 4, 5, 6, respectively, and in the DFTB model these numbers increase to 36, 32, 12, 2, 1, 0. There is also a significant degree of agreement between the models on isomer ordering within the s sets. For example, with the exception of s ) 4, the models agree on the best two and worst two isomers for each s and, with only minor changes in ordering, on the next few isomers at each extreme. Figure 2 shows the DFTB optimal geometries of the best and worst isomers, and Table 2 lists their energies and topological characteristics. In general, the isomers of lowest energy have a rounded appearance, whereas those of highest energy are elongated, sometimes extremely so. The globally best structure, the classical fullerene 38(0), has a chiral D2 structure in which the 10 hexagons form a continuous closed strip wound in a way reminiscent of the seam of a tennis ball and the 12 pentagons form two staggered semicircles. The best structure with a single square, 162(1), has Cs symmetry with an isolated square and a strip of seven pairwise fused pentagons with two of its 10 pentagons now occurring in an isolated “butterfly” pair and one as an isolated defect.
Figure 2. Optimized geometries of the best and worst isomers of C40 at each value of s. The structures were drawn from the optimized Cartesian coordinates. Two “best” isomers are shown for s ) 4, as the DFTB model predicts 101(4) to lie below 51(4), and the QCFF/PI model reverses this difference.
The best structure with two squares 542(2), has D4h symmetry with two isolated squares at the poles and eight pentagons in four separate butterfly pairs astride the equator.
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J. Phys. Chem., Vol. 100, No. 17, 1996 6987
TABLE 2: Energetic and Topological Characteristics of the Best and Worst Isomers of C40 at Each Value of sa s
N
spiral
G
ss
sp
sh
pp
ph
hh
∆ED
∆EQ
0 0 1 1 2 2 3 3 4 4 4 5 5 6 6
38 1 162 3 542 355 357 56 101 51 83 21 11 7 1
5565665656565656565565 5555556666666666555555 4666656565656556556565 4555666666665665556555 4666656565656565656564 4555566666666666655554 4666655665656664655646 4456666656666666556554 4666646566566566466564 4666646665666564656465 4456566666666666665454 4666646565666664664646 4456666666666466666544 4666646666466646466664 4464666666666666664644
D2 D5d Cs Cs D4h D4h Cs C1 C2 C2V D2h C1 C2 D2 D2d
0 0 0 0 0 0 0 1 0 0 2 0 2 0 4
0 0 0 3 0 8 0 5 0 0 8 0 4 0 0
0 0 4 1 8 0 12 5 16 16 4 20 12 24 16
10 20 7 11 4 8 3 4 1 2 0 0 0 0 0
40 20 36 25 32 16 24 17 18 16 12 10 6 0 0
10 20 13 20 16 28 21 28 25 26 34 30 36 36 40
0.0 786.3 87.5 837.5 151.4 1719.4 314.8 1525.8 535.2 556.0 2333.2 702.9 1719.9 1096.1 2523.1
0.0 943.5 239.4 1162.0 459.1 2448.6 805.9 2417.2 1247.8 1202.6 3715.1 1545.7 2891.6 2126.4 4074.5
a N is the sequence number in the spiral enumeration at fixed s, G is the maximal symmetry group of the molecular graph, and the indices, ss, sp, sh, pp, ph, hh count ring fusions of each type, as in eqs 6-8 of the text. ∆ED and ∆EQ are energies relative in kJ mol-1 to that of the optimal fullerene in DFTB and QCFF/PI models, respectively. Two “best” isomers are shown for s ) 4, as the DFTB model predicts 101(4) to lie below 51(4), and the QCFF/PI model makes the opposite prediction. All isomers listed are illustrated in Figure 2.
A trend that is indicative of an important role for steric interactions is evident from Figure 2 and Table 2: the best structure at each value of s has no square-square contacts (ss ) 0), no square-pentagon contacts (sp ) 0) for any s, and fully isolated defects (ss ) pp ) sp ) 0) for s ) 5 and 6. The energy data can be used to assign a penalty for square rings in C40 fullerene-like cages in several ways. Taking simple differences between the best isomers at s ) 0 and s ) 1 yields values of 87.5 kJ mol-1 (DFTB), 239.4 kJ mol-1 (QCFF/PI), and halving the differences for the best isomers at s ) 0 and s ) 2 yields an average penalty per square of 75.7 kJ mol-1 (DFTB) or 229.6 kJ mol-1 (QCFF/PI). The spread of energies at each s shows the relative energy to be critically dependent on the details of the cage topology and the total number of squares and pentagons. It may be better therefore to take the differences of the mean energies at each s value, giving penalties per square of 159.2 kJ mol-1 (DFTB), 344.8 kJ mol-1 (QCFF/ PI) for s ) 0 f 1, and 176.5 kJ mol-1 (DFTB), 362.5 kJ mol-1 (QCFF/PI) for s ) 0 f 2, rising to 202.9 kJ mol-1 (DFTB), 396.3 kJ mol-1 (QCFF/PI) for s ) 0 f 6. Some dependence on the size of the cage is also expected as each square introduced is accompanied by an extra hexagon, as noted earlier, and this perturbation of the hexagon count might be expected to become less important as the number of vertices increases. Although it may be difficult to assign a unique numerical value to the square penalty, the values derived above are somewhat larger than the pentagon adjacency penalty derived in equivalent ways. A simple difference between the best classical isomer with pp ) 11 (31(0)) and the globally best isomer (38(0) with pp ) 10) gives 65.1 kJ mol-1 (DFTB), 72.2 kJ mol-1 (QCFF/PI). These are at the low end of the range of published estimates for this penalty. The means for classical fullerene isomers at pp ) 10 and pp ) 11 give 91.5 kJ mol-1 (DFTB), 134.5 kJ mol-1 (QCFF/PI). However they may be derived, the cost of a pentagon adjacency is smaller than that of a square derived in the same way. Electronic Factors Further insight into the factors governing cage stability can be gained by considering π-electronic and topological characteristics of the 1735 cages. Traditional measures of stability for π systems are Eπ, the delocalization energy, and ∆, the HOMO-LUMO gap, which can both be derived in the crudest simple Hu¨ckel approximation from the adjacency matrix of the
TABLE 3: Simple Hu1 ckel Properties of Square-Pentagon-Hexagon Isomers of C40 with s Square, 12 - 2s Pentagonal, and 10 + s Hexagonal Facesa ∆
Eπ s
range
mean
range
mean
0 1 2 3 4 5 6
60.400-61.127 60.757-61.191 60.546-61.380 60.735-61.418 60.564-61.265 60.271-61.074 60.528-60.928
60.846 60.994 61.065 61.052 60.948 60.836 60.684
0.000-0.373 0.018-0.331 0.000-0.568 0.000-0.580 0.001-0.550 0.000-0.521 0.017-0.420
0.115 0.149 0.174 0.200 0.182 0.222 0.243
a Eπ is the total π energy and ∆ the HOMO-LUMO gap. Both are both in units of |β|, where β is the common parameter assumed to be appropriate for all bonds. The delocalization energy of a trivalent Cn cage is found by subtracting nβ from Eπ.
isomer and generally have a similar information content. Table 3 gives a breakdown of the variation of these properties for the 1735 isomers of C40. If Eπ values were taken as literal measures of stability, Hu¨ckel theory would imply that many cages with squares should actually be favored over the classical fullerenes. Hu¨ckel HOMO-LUMO gaps also appear to favor introduction of squares (Table 3). Many cages with squares turn out to have properly closed π shells, in contrast to the classical C40 fullerenes, where three isomers have open shells and the rest are only pseudoclosed. Only five cages with s * 0 have an open shell. Similar conclusions would be reached for other sizes of cage.16 However, neither Eπ nor ∆ correlates with computed overall stabilities for these isomers, a fact which is not surprising in view of the poor performance of purely π-electronic measures of the fullerenes themselves. As Babic´ and Trinajstic´ point out,16 part of the spurious stabilizing effect of squares in Hu¨ckel theory comes from the six-circuits present in fused pairs (and triples) of squares. In previous work on sets of classical fullerenes and related molecules21 it has been noted that Hu¨ckel theory is often a reasonable guide to the size of the HOMO-LUMO gap, even though the gap itself may not correlate well with overall stability. In the present case it is found that the computed gap itself does not correlate well with stability, nor is there any particular association between Hu¨ckel and computed gaps, perhaps signalling that the local environments are now so various that the purely topological single-R, single-β Hu¨ckel model has broken down completely.
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Figure 3. Comparison of ∆EQ and torsional components (Eφ) of the energy of square-pentagon-hexagon C40 cages in the QCFF/PI model. All energies are given in kJ mol-1 and referred to the most stable classical isomer. (a) Scatter diagram for all 1735 isomers; (b) least-squares fitted lines for different values of s (s increasing from the bottom of the graph). The equations of the lines (with standard deviations σ) are ∆EQ/kJ mol-1 ) a(Eφ/kJ mol-1) + b, where a ) 1.154, b ) 41.3, σ ) 37 (s ) 0); a ) 1.156, b ) 278.4, σ ) 36 (s ) 1); a ) 1.195, b ) 497.3, σ ) 38 (s ) 2); a ) 1.167, b ) 741.8, σ ) 37 (s ) 3); a ) 1.226, b ) 935.5, σ ) 47 (s ) 4); a ) 1.190, b ) 1189.1, σ ) 60 (s ) 5); a ) 1.258, b ) 1377.9, σ ) 85 (s ) 6). The difference in vertical separation between the nearly parallel lines in (b) is another measure of the energetic penalty for introduction of square faces and is essentially equivalent to the differences between the best isomers at each s value, as used in the text.
Steric Factors In previous work on classical fullerenes, a much better indicator of relative stability was found in the torsional contribution to the QCFF/PI energy.9,10,22 Figure 3 shows that this correlation survives for s * 0, but with parallel curves for the different values of s. Steric effects are therefore the main contributions to the isomer preference in this extension of the fullerene set, and correlations with topological descriptors of these effects can be expected to be significant. The most successful energetic correlation found in our previous study9 of the classical fullerene isomers of C40 was with pp, the number of pentagon adjacencies: the more pentagon fusions in a structure, the higher its energy. Taken in conjunction with the topological data for the best isomers of each class (Table 2), this suggests that in the present case the three most important parameters of the six edge types will be those describing defect fusions, i.e., ss, sp, and pp. Together with s, these form a set of four independent parameters. Least-squares fitting to the functional form
∆E/kJ mol-1 ) a‚ss + b‚sp + c‚pp + d‚s + e
(9)
gives a ) 289.4, b ) 144.8, c ) 70.79, d ) 272.5, and e ) -662.0 with a standard deviation of σ ) 66 (DFTB) and a ) 386.2, b ) 186.5, c ) 87.34, d ) 471.6, and e ) -819.2 with a standard deviation of σ ) 76 (QCFF/PI). It is clear from the coefficients of the fit that fusions of defects (i.e. nonhexagonal rings) lead to high energies. In previous work on the 1812 classical fullerene isomers of C60, a good correlation of energy with pp was obtained, but an even closer fit was found with a function of the hexagon neighbor signature.22 This signature is a set of numbers {hk} (k ) 0, 1, 2, 3, 4, 5, 6) where hk is the number of hexagonal faces in the cage that have exactly k hexagonal neighbors. Raghavachari23 suggested that the most stable isomers of higher fullerenes would be those in which the hexagon environments were most uniform, i.e. those in which the dispersion of the hexagon signature was smallest. For example, this criterion selects three isomers of C84, of which two are the experimentally observed D2 and D2d isomers that are also calculated to have the lowest energy and to be nearly isoenergetic in many models of electronic structure. The quantity used in our earlier paper22
for the isomers of C60, nearly all of which have adjacent pentagons, was the second moment of the hexagon signature,
∑k k2hk
(10)
Here, since the number of hexagons changes with changes in s, the normalized second moment,
H ) ∑ k2hk k
/∑
hk
(11)
k
is used. This measure has the advantage over any involving squares or pentagons of remaining well-defined over the whole range of s, regardless of the presence or absence of squares or pentagons. It is effectively a mean square coordination number of the hexagonal faces of the cage. Figure 4 shows correlations of computed relative stabilities of the 1735 isomers of C40 with the variable H. The high quality of the correlation shows H to be a suitable variable to quantify steric strain in all the cage types of the sample set. The variation of H with s is broken down in Table 4: as with the energy, there is a rising trend, but the ranges overlap for successive values of s; the isomers with extreme energies are close to extremal in H. The fits of energy against H yield an energetic penalty for increase in H, which may seem surprising at first since large values of H occur when there are many contiguous fused hexagons, which would be expected to increase the π aromaticity of the cage. However,22 the locally flat hexagonal regions contributing to the increase in H also imply crowding of defects elsewhere on the surface, and therefore high values of H are associated with sterically strained cages. Discussion The main result of the present study is an unambiguous prediction by two independent methods confirming that the fullerene as classically defined gives a C40 cage of lower energy than any incorporating squares, pentagons, and hexagons. As far as we are aware, only one previous study has addressed this question for C40. Gao and Herndon11 compared the energies of 39(0) and 542(2) in several empirical and semiempirical approaches including Hu¨ckel theory, molecular mechanics,
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J. Phys. Chem., Vol. 100, No. 17, 1996 6989
Figure 4. Correlation of relative energies of square-pentagon-hexagon cages of C40 with the normalized second moment, H, of the hexagon neighbor signature. The zero for the energy scale in each model is defined by the most stable classical fullerene isomer and subscripts D and Q refer to DFTB and QCFF/PI, respectively. The fitted lines are ∆E/kJ mol-1 ) aH + b, with coefficients a and b and standard deviation σ (DFTB) of a ) 67.344, b ) -223.3, σ ) 34 (s ) 0); a ) 80.243, b ) -342.8, σ ) 52 (s ) 1); a ) 82.640, b ) -360.1, σ ) 99 (s ) 2); a ) 94.164, b ) -545.7, σ ) 99 (s ) 3); a ) 117.126, b ) -972.2, σ ) 142 (s ) 4); a ) 128.269, b ) -1326.3, σ ) 96 (s ) 5); a ) 242.618, b ) -4024.3, σ ) 132 (s ) 6), giving an overall best fit of a ) 82.588, b ) -374.8, σ ) 113 (s ) 0, 1, 2, 3, 4, 5, 6). The equivalent results for the QCFF/PI model are a ) 78.495, b ) -245.9, σ ) 38 (s ) 0); a ) 94.660, b ) -236.8, σ ) 59 (s ) 1); a ) 102.501, b ) -165.0, σ ) 122 (s ) 2); a ) 123.318, b ) -346.7, σ ) 121 (s ) 3); a ) 155.054, b ) -804.8, σ ) 187 (s ) 4); a ) 181.906, b ) -1382.1, σ ) 108 (s ) 5); a ) 326.022, b ) -4712.0, σ ) 166 (s ) 6); with an overall best fit of a ) 142.129, b ) -624.0, σ ) 149 (s ) 0, 1, 2, 3, 4, 5, 6).
TABLE 4: Variation of the Normalized Second Moment of the Hexagon Neighbor Signature (H) with s, the Number of Squares, in Square-Pentagon-Hexagon Isomers of C40a s
range
mean
H(best)
H(worst)
0 1 2 3 4 5 6
4.0-16.0 5.8-15.3 7.3-22.7 10.6-21.2 13.0-25.1 16.0-23.6 20.5-26.5
7.2 9.5 11.8 14.3 16.9 19.2 22.7
4.0 5.8 7.3 10.8 13.0 16.4 20.5
16.0 14.2 22.7 19.5 25.1 23.6 26.5
a H(best) and H(worst) are the values for the best energetically and worst (DFTB) isomers at the particular value of s.
π-SCF, and MNDO procedures, the latter two in both spin restricted and unrestricted versions. The two isomers11 are what we have now shown to be the second-best classical fullerene and the best two-square alternative. Hu¨ckel theory, as discussed earlier, gives no guide to overall energy. All others except the MNDO calculation showed an energetic preference for the classical fullerene. The MNDO-RHF results are not reported by Gao and Herndon,11 but we find that at the MNDO-RHF level,24 542(2) is more stable than 39(0) by 5 kJ mol-1, but less stable than the best classical fullerene 38(0) by 37 kJ mol-1; at the MNDO-UHF level, the two-square cage was found to be more stable than 39(0) by a much larger amount11 (over 80 kJ
mol-1), but we find that most of this difference is attributable to the relative instability of 39(0) with respect to the best classical fullerene. At the UHF level the two-square cage is predicted to be more stable than 38(0), but only by 26 kJ mol-1. All UHF calculations are heavily spin-contaminated (〈S2〉 g 9.3), a fact that makes their reliability questionable. It can be argued that both MNDO-UHF and Hu¨ckel calculations suffer from the same tendency to overemphasise the π-electronic energy at the expense of a proper consideration of steric strain. There is thus no convincing evidence against the current conclusion that inclusion of squares is an energetically unfavorable solution to the strain associated with pentagon adjacencies in C40 cages. The other range where pentagon adjacencies are forced in classical fullerenes is 62 e n e 68. An estimate of the benefits or costs of squares faces in this range can be made as follows. The isomer counts for the classical fullerenes in this range are large:4 2385 (n ) 62), 3465 (n ) 64), 4478 (n ) 66), 6332 (n ) 68). Introduction of a single square face leads to even larger numbers: 10 323 (n ) 62), 13 504 (n ) 64), 18 215 (n ) 66), 23 019 (n ) 68), according to the extended spiral algorithm. The best fullerenes at each n are taken to be those with the minimal value of pp, and the best nonclassical candidates to be those with s ) 1, sp ) 0, and fewer pentagon adjacencies than the corresponding fullerene. These criteria select relatively
6990 J. Phys. Chem., Vol. 100, No. 17, 1996
Fowler et al.
TABLE 5: Classical Fullerenes and Other Candidates for the Most Stable Carbon Cages with 62-68 Atomsa n
N(s)
spiral
G
∆EQ
62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 64 64 64 64 66 66 66 66 66 66 66 66 66 66 66 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68
2194(0) 2378(0) 2377(0) 9620(1) 8255(1) 9899(1) 9618(1) 10323(1) 9117(1) 8256(1) 9611(1) 8157(1) 8156(1) 9345(1) 8542(1) 9568(1) 9118(1) 7622(1) 8108(1) 9057(1) 8253(1) 7754(1) 8265(1) 9116(1) 7758(1) 7786(1) 7765(1) 7275(1) 7666(1) 9058(1) 3451(0) 3452(0) 3457(0) 9902(1) 4169(0) 4348(0) 4466(0) 14298(1) 13383(1) 13362(1) 13385(1) 13390(1) 14245(1) 15853(1) 15847(1) 6290(0) 6328(0) 6198(0) 6270(0) 6094(0) 6146(0) 6148(0) 6195(0) 6073(0) 6269(0) 6149(0) 15749(1) 16804(1) 15441(1) 16743(1) 15669(1) 15343(1) 17686(1) 15011(1) 19704(1)
1,2,4,12,15,17,20,23,26,28,30,32 1,2,9,11,13,15,17,21,25,28,30,32 1,2,9,11,13,15,17,21,25,27,30,33 1,7,11,14,17,19,22,25,27,29,31 1,6,9,16,19,21,23,25,27,29,31 1,7,14,17,19,20,22,23,26,28,32 1,7,11,14,17,19,22,24,27,29,32 1,2,9,12,15,17,20,22,24,30,33 1,7,9,11,14,21,24,26,28,30,32 1,6,9,16,19,21,23,25,27,30,32 1,7,11,14,17,18,22,25,28,29,31 1,6,9,15,19,21,23,26,27,30,32 1,6,9,15,19,21,23,26,27,29,31 1,7,9,14,19,21,22,25,27,29,32 1,6,10,15,17,20,22,25,27,29,31 1,7,11,14,16,19,22,26,27,29,31 1,7,9,11,14,21,24,26,28,31,33 1,6,8,17,19,21,23,25,27,29,31 1,6,9,15,18,21,23,26,28,30,32 1,7,9,11,13,22,24,26,28,30,32 1,6,9,16,19,21,23,24,27,29,32 1,6,9,11,15,21,24,26,28,30,32 1,6,9,16,19,22,23,25,27,29,30 1,7,9,11,14,21,24,26,27,30,33 1,6,9,11,15,21,25,26,28,30,31 1,6,9,11,16,21,24,26,28,30,31 1,6,9,11,15,22,25,26,28,29,31 1,6,8,10,20,23,25,27,30,32,33 1,6,8,18,20,22,24,26,28,31,33 1,7,9,11,13,23,25,27,29,30,32 1,2,9,12,14,17,20,23,25,28,30,33 1,2,9,12,14,17,20,23,26,28,30,32 1,2,9,12,14,18,21,23,25,27,29,33 1,6,9,11,15,22,25,27,29,31,33 1,2,4,13,17,19,22,24,26,28,30,33 1,2,9,10,13,16,21,24,27,29,31,33 1,2,9,11,15,17,22,24,26,29,30,32 1,6,10,15,18,20,23,26,30,32,34 1,6,9,15,19,21,23,26,29,32,34 1,6,9,15,19,20,23,26,30,32,34 1,6,9,15,19,21,23,27,29,32,33 1,6,9,15,19,21,24,27,29,31,33 1,6,10,15,17,20,23,27,30,32,34 1,7,9,14,19,21,25,27,29,31,32 1,7,9,14,19,21,24,27,29,31,33 1,2,9,12,14,20,23,25,27,29,31,35 1,2,9,13,17,20,22,25,26,29,31,33 1,2,9,11,15,18,22,24,26,30,32,35 1,2,9,12,14,17,21,23,26,29,34,36 1,2,9,11,13,16,20,24,27,31,33,35 1,2,9,11,14,17,22,25,28,30,32,34 1,2,9,11,14,17,22,26,28,30,32,33 1,2,9,11,15,17,22,24,28,30,32,34 1,2,9,11,13,15,17,28,30,32,34,36 1,2,9,12,14,17,21,23,26,28,35,36 1,2,9,11,14,17,22,26,28,31,33,36 1,6,8,19,21,24,26,28,30,32,34 1,6,9,16,20,22,25,27,30,32,35 1,6,8,17,19,21,24,29,31,33,34 1,6,9,16,19,21,24,29,31,33,34 1,6,8,18,21,24,26,29,30,32,34 1,6,8,15,19,21,27,29,31,33,34 1,6,10,15,17,22,28,30,32,34,35 1,6,8,11,23,25,27,29,31,33,35 1,7,9,11,14,24,27,29,31,33,35
C1 C2 C1 C2V C1 Cs C1 C2V Cs C1 C1 C1 C1 C1 C2 C1 Cs Cs C1 C2V C1 C1 C1 C1 C1 Cs Cs C1 Cs C2V D2 Cs C2 Cs Cs C2V C2 C2V C1 C1 C1 C1 C1 C1 C1 C2 C2 C1 C1 Cs C2 C1 C2 C2V D2 C2 Cs C1 C1 C1 C1 Cs C1 Cs C1
0.0 4.2 7.9 142.3 239.3 298.7 299.6 303.3 303.8 310.9 313.8 334.3 342.3 343.5 343.5 346.9 351.5 364.4 364.8 369.0 374.5 384.1 413.8 416.7 417.1 430.5 433.0 443.5 458.6 495.8 0.0 23.4 44.8 492.0 0.0 24.3 33.9 371.5 373.6 397.5 412.5 414.6 416.7 425.9 438.1 0.0 3.3 33.5 37.2 40.6 45.6 67.4 70.3 79.5 84.9 101.7 294.6 390.8 392.0 397.5 413.4 436.8 446.0 501.7 502.1
a The fullerene candidates are those with the minimal achievable number of pentagon adjacencies; the other cages have only one square face, fewer pentagon-pentagon adjacencies, and no square-pentagon adjacencies, as described in the text. n is the number of atoms, N(s) denotes the position of the isomer in the spiral ordering of cages with s squares, the third column gives the positions of the nonhexagonal rings in the face spiral (with square positions in bold), G is the point group, and ∆EQ is the energy calculated within the QCFF/PI model relative to the most stable candidate at the given value of n.
Figure 5. Square face introduced into a trivalent polyhedron by bevelling an edge. When the original edge connects two pentagons on a fullerene, the result is a pseudofullerene with one square and two fewer pentagonal faces.
Figure 6. Best of the C62 cages with a single square face, 6290(1) in the spiral ordering, according to the QCFF/PI model calculations. Views down the three principal axes are shown.
small numbers of isomers: 3(s ) 0), 27(s ) 1) for n ) 62, 3(s ) 0), 1(s ) 1) for n ) 64, 3(s ) 0), 8(s ) 1) for n ) 66, 11(s ) 0), 9(s ) 1) for n ) 68. QCFF/PI calculations (Table 5) show that the one-square cages are never more stable than the best classical fullerene. The energy differences between the classical fullerenes are small, ranging from 8 kJ mol-1 (n ) 62) to 45 kJ mol-1 (n ) 64), and the effective penalties for the square faces vary from 143 kJ mol-1 (n ) 62) to 492 kJ mol-1 (n ) 64). The best C62 cage with s ) 1 in Table 5, although not of lower energy than the best C62 fullerene with pp ) 3, is nevertheless interesting because of its relationship to the icosahedral C60 cage. The square face is introduced by “bevelling” a hexagon-hexagon edge of the C60 polyhedron (Figure 5). This general operation can be applied to any bond in a fullerene whose extremities lie in distinct pentagons to yield a cage with two more vertices, two more hexagons, one more square, and two fewer pentagons. The relaxed geometry of C62(9620(1)) is essentially that of C60 itself with a bump raised on one side of the ball (Figure 6). Although square rings have not proved to be energetically favorable, it will be interesting to explore other departures from the classical fullerene recipe in view of the evidence for heptagonal defects in nanotubes,25 proposals for heptagonal and octagonal rings in novel allotropes of carbon26 and octagons in the C48 cage,27 and the recent chemically produced ring expansions in derivatives of C6028 and C70.29 Acknowledgment. This work was carried out with some financial assistance from the European Union as part of a Network Project on Formation, Stability and Photophysics of Fullerenes under the Human Capital and Mobility Scheme. D.M. thanks the University of Exeter for the award of a Postgraduate Research Studentship. 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) Gru¨nbaum, B. ConVex Polytopes; Wiley: New York, 1967. (4) Fowler, P. W.; Manolopoulos, D. E. An Atlas of Fullerenes; Oxford University Press: Oxford, 1995. (5) Yi, J.-Y.; Bernholc, J. J. Chem. Phys. 1992, 96, 8634. (6) Murry, R. L.; Strout, D. L.; Odom, G. K.; Scuseria, G. E. Nature (London) 1993, 366, 665. (7) Fowler, P. W.; Zerbetto, F. Chem. Phys. Lett. 1995, 243, 36. (8) Zhang, B. L.; Wang, C. Z.; Ho, K. M.; Xu, C. H.; Chan, C. T. J. Chem. Phys. 1992, 97, 5007; 1993, 98, 3095. (9) Fowler, P. W.; Manolopoulos, D. E.; Orlandi, G.; Zerbetto, F. J. Chem. Soc., Faraday Trans. 1995, 91, 1421.
Energetics of Fullerenes with Four-Membered Rings (10) Austin, S. J.; Fowler, P. W.; Orlandi, G.; Manolopoulos, D. E.; Zerbetto, F. Chem. Phys. Lett. 1994, 226, 219. (11) Gao, Y.-D.; Herndon, W. C. J. Am. Chem. Soc. 1993, 115, 8459. (12) Warshel, A.; Karplus, M. J. Am. Chem. Soc. 1972, 94, 5612. (13) Porezag, D.; Frauenheim, Th.; Ko¨hler, Th.; Seifert, G.; Kaschner, R. Phys. ReV. 1995, B51, 12947. (14) Seifert, G.; Porezag, D.; Frauenheim, Th. Int. J. Quantum Chem., in press. (15) Manolopoulos, D. E.; May, J. C.; Down, S. E. Chem. Phys. Lett. 1991, 181, 105. (16) Babic´, D.; Trinajstic´, N. Chem. Phys. Lett. 1995, 237, 239. (17) Manolopoulos, D. E.; Fowler, P. W. J. Chem. Phys. 1992, 96, 7603. (18) Orlandi, G.; Zerbetto, F.; Fowler, P. W. J. Phys. Chem. 1993, 97, 13575. (19) Negri, F.; Orlandi, G.; Zerbetto, F. Chem. Phys. Lett. 1992, 196, 303. (20) Orlandi, G.; Zerbetto, F.; Fowler, P. W.; Manolopoulos, D. E. Chem. Phys. Lett. 1993, 208, 441.
J. Phys. Chem., Vol. 100, No. 17, 1996 6991 (21) Fowler, P. W.; Mitchell, D. J. Chem. Inf. Comput. Sci. 1995, 35, 874. (22) Austin, S. J.; Fowler, P. W.; Manolopoulos, D. E.; Orlandi, G.; Zerbetto, F. J. Phys. Chem. 1995, 99, 8076. (23) Raghavachari, K. Chem. Phys. Lett. 1992, 190, 397. (24) The calculations used Mopac 6.00: Stewart, J. J. P. Quantum Chemistry Program Exchange; Department of Chemistry, Indiana University: Bloomington, IN, 47405. (25) Iijima, S.; Ichihashi, T.; Ando, Y. Nature (London) 1992, 356, 776. (26) Mackay, A. L.; Terrones, H. Philos. Trans. R. Soc. London Sec. A 1993, 343, 113. (27) Dunlap, B. I.; Taylor, R. J. Phys. Chem. 1994, 98, 11018. (28) Hummelen, J. C.; Prato, M.; Wudl, F. J. Am. Chem. Soc. 1995, 117, 7003. (29) Birkett, P. R.; Avent, A. G.; Darwish, A. D.; Kroto, H. W.; Taylor, R.; Walton, D. R. J. Chem. Soc., Chem. Commun. 1995, 1869.
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