Charging and equilibration of fullerene isomers

8 September 1995

ELSEVIER

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 243 (1995) 36-41

Charging and equilibration of fullerene isomers P.W. Fowler a, F. Zerbetto b a Department of Chemistry, University of Exeter, Exeter EX4 4QD, UK b Dipartimento di Chimica "G. Ciamician", UniversiM di Bologna, Via E Selmi 2, 40126 Bologna, Italy

Received 6 July 1995; in final form 19 July 1995

Abstract

Variation of the total charge is shown to affect the computed relative stabilities of fullerene isomers. Within the QCFF/PI model, the 80 kJ/mol pentagon-adjacency penalty in neutral C60 is reduced by a factor of ten in the dianion, and icosahedral C80 moves from least to most stable of the seven isolated-pentagon isomers of this fullerene as the charge varies from -t-2 to -4. The two experimentally isolable isomers of C84, however, remain almost isoenergetic in this model over the same charge range. Energy-charge isomerisation maps are presented for all isolated-pentagon isomers of C84. Implications for fullerene formation and annealing are discussed.

Fullerenes are still poorly understood. This is not to say that we cannot isolate, modify, or even use them, rather that our understanding of the growth mechanisms of these molecules is still imperfect. Despite the great efforts of recent years [ 1-5 ], it is not known why one and only one of the 1812 [6] possible structural isomers of C60 is produced by the arc synthesis. Chemical common sense rejects the other 1811 isomers on the basis of the presence of pentagon-pentagon adjacencies which entail energy penalties [7-10] with respect to Ih C60 where no pentagons share an edge. Pentagon isolation is not the only criterion, however. C84 has 24 possible isomers with isolated pentagons [ 1 1 ] but so far only the two isomers of lowest energy [ 12] have been observed [13], in the statistical ratio predicted by their D2 and Dzd symmetries. For these fu!lerenes, the evidence points towards the presence of equilibria established during or at the end of a growth stage. For higher clusters, kinetic considerations seem to be dominant, and tubules are usually produced, with leng.ths that increase with the propen-

sity of the metallic catalyst for high oxidation numbers [ 14]. The fullerene self assembly process appears to be highly sensitive to the conditions in the reaction chamber [ 15]. How does the process work? How is it possible, for instance, to anneal all the pentagon-pentagon edges that may be formed accidentally at the high temperatures of the synthesis? The answer must involve equilibration but, at present, it is not clear how such equilibria are established and whether there is any way of directing the growth into or away from a kinetic regime. In this Letter, a simple possibility is explored, namely that the order of stability of fullerenes is affected by their charges. The study is carried out in three steps through quantum chemical calculations. The first is the comparison of the optimized energies, for several charges, of two isomers of C60, namely Ih C60 and its energetically nearest and topologically most closely related neighbour isomer 1809 [9]. The second step is a similar comparison for the set of the

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PW. Fowler, E Zerbetto/Chemical Physics Letters 243 (1995) 36-41

37

Table 2 Relative energies, in kJ/mol, of the optimized structures of the 7 isolated-pentagon isomers of C80 (isomers I to 7 ordered by their face-spiral codings [ 16] )

2 0 -2 -4 -6

Dsd

D2

C2v

D3

C2v

Dsh

Ih

1675.8 1.3 -217.5 933.3 3249.2

1707.1 0.0 -210.7 893.9 3230.0

1676.0 11.4 -336.6 772.2 3099.0

1678.5 75.6 -242.0 850.3 3103.6

1679.1 29.3 -357.5 647.8 2961.9

1696.0 42.9 -380.9 625.9 2824.1

1847.4 143.8 -269.9 591.5 2731.1

seven [ 16] isomers of C80 that satisfy the isolated pentagon rule (IPR). Finally, we study the two experimentally isolated isomers of C84 which are usually referred to as isomer 22 (D2) and isomer 23 (D2d) [ 1 1] and place them within the whole set of 24 isolated-pentagon isomers. Full geometry optimization is carried out for five different oxidation numbers (+2, 0, - 2 , - 4 , - 6 ) of all 33 fullerenes. Confirmation that charges induce a variation of the stability order of fullerenes does not automatically imply that the annealing of either the pentagon-pentagon defects or of the IPR satisfying fullerenes actually takes place when the molecules are charged, but does add a new candidate to the short list of proposals aimed at explaining the growth of these systems [ 1-5]. All the calculations were performed with the quantum consistent force field for 7r electrons (QCFF/PI) [17]. This model has been widely used to study fullerenes in general [ 10,18-25], and anionic C60 in particular [26-28 ]. It employs an empirical potential for the o--electron framework and uses a quantum chemical scheme for the 7r electrons. The o" framework is simulated by harmonic, Morse and torsional oscillators which describe the bond stretchings, the in-plane bending deformations and the out-of-plane pyramidalisations and torsions. For the 7r electrons, Table I Relative energies, in kJ/mol, of the optimized structures of Ih C60 and isomer 1809 (the C2v isomer with 2 pairs of fused pentagons)

+2 0 -2 -4 -6

lh C6o

1809

1927.4 0.0 8.4 1387.7 4106.0

2027.8 162.3 24.4 1388.2 4078.9

both the one-electron (t, or hopping) integrals and the two-electron (U, or Coulomb) integrals are functions of the interatomic distances. All systems studied here have even numbers of electrons and their geometric parameters are optimised consistently at the RHF level for a closed-shell singlet state: any JahnTeller distortion that may occur for a given charge is thereby automatically taken into account. In Table 1, the relative energies of the two most closely related C60 isomers are reported for five oxidation states. Interestingly, the energy difference is largest when both molecules are neutral. Whilst for isolated Ih C60 the preferred oxidation state is zero, the calculation shows that isomer 1809 prefers an oxidation state of - 2 , as expected from its pseudo-closed [29] configuration in Hiickel molecular-orbital theory. For the three negatively charged states of the two isomers of C60 considered here, the calculations give very similar results. It is plausible to assume that if equilibration between the two isomers takes place it happens for the charged systems. Neutral Ih C60, because of its stability, would then act as a funnel or a scavenger, dominating the final product. In Table 2, the relative energies of the 7 IPRsatisfying isomers of C80 are given. For each isomer in the vacuum, the most stable species is predicted to be the doubly charged anion. This is in keeping with the generally high electron affinity and 'electrondeficient' chemistry of fullerenes [ 30]. The five oxidation numbers considered here span a smaller energy range than for C60. This is reasonable, since C80 is a larger cluster and can better accommodate the excess charge. At odds with the previous case, where the gap was largest for the neutral molecules, for C80 the range of calculated energies is largest for the hexaanion. Another perspective can be obtained by plotting the energies relative to the most stable isomer at each

P W. Fowler, E Zerbetto / Chemical Physics Letters 243 (1995) 36-41

38 E(kJ/mol) 6O0

1 500 2

4O0

4 3

300

5 200

I00

6

7

I -6

[ ~1

r -2

I 0

I +2 Charge

Fig. 1. Variation of the order of stability of C8o isomers with net charge. Each curve represents the energy of a given isomer, in kJ/mol, relative to that of the most stable at a given charge, with the labels I to 7 corresponding to columns in Table 3.

Table 3 Relative energies and energy differences, in kJ/mol, of the optimized structures of C84 isomer 22 (D2) and C84 isomer 23 (D2d)

+2 0 -2 -4 -6

Isomer 22

Isomer 23

1768.8 0.0 -- 199.2 775.4 3004.8

1778.2 1.9 --202.9 743.4 3031.0

zl 9.4 1.9 --3.7 -32.0 26.2

Fig. 2. Variation of the stability of C84 isomers on the main StoneWales map as a function of net charge. The bottom layer of the diagram shows the isolation-preserving transformations between IPR isomers and the columns in the other layers represent energies calculated in the QCFF/PI model for each charge (Table 4) and taken relative to the most stable isomer at that charge (indicated in the NE comer of the layer). The colours yellow, red, green, blue mark isomers that are respectively 0, 1, 2 and 3 SW steps from the mathematical centre of the map, as defined in the text.

s s

i

P.W. Fowler, E Zerbetto/Chemical Physics Letters 243 (1995) 36-41

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Table 4 Relative energies, in kJ/mol, of the optimized structures of the 24 isolated-pentagon C84 fullerenes at each total charge (isomers ordered by their face-spiral codings [ 16] ), expressed with respect to the best neutral isomer Isomer

G

+2

0

-2

-4

-6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

D2 C2 Cs D~ D2 C2v C2v C2 C2 Cs C2 Cl C2 Cs Cs Cs C2v C2v D2v Td D2 D2 D2d D~

2050.3 1935.7 1777.1 1836.1 1846.6 1800.7 1806.4 1750.4 1762.1 1763.8 1782.6 1769.6 1796.0 1814.0 1758.7 1794.3 1812.3 1837.8 1791.8 1956.1 1770.9 1768.8 1778.2 1813.1

255.3 185.3 155.1 Ilia 119.0 74.0 101.0 121.0 120.8 153,2 34.5 51,5 102,6 68.4 52,6 28.4 78.5 60.3 28.9 123.2 69.5 0.0 1.8 24.7

174.6 -40.8 -181.0 -105.2 -114.4 -154.4 -157.5 -207.7 -241.2 -235.4 -190.2 -224.5 -222.0 -127.4 -205.7 -181.0 -211.5 -141.6 -212.3 57.6 -201.0 -199.2 -202.9 -125.7

1246.1 1033.2 840.9 911.1 818.2 892.2 781.8 827.0 815.7 761.3 828.2 765.9 788.9 909.4 796.4 852.1 896.4 957.9 850.0 1169.8 739.5 775.4 743.4 889.3

3484.5 3301.0 3092.7 3165.0 3114.4 3152.9 3028.2 3044.1 3106.8 3003.5 3100.5 3059.6 3053.7 3181.7 3009.3 3119.8 3183.4 3236.5 3140.7 3428.9 2955.4 3004.8 3031.0 3105.6

oxidation number (Fig. 1). It is apparent that the stability order is easily modified by changing the charge. In particular, Ih C80 becomes the most stable isomer for the tetraanionic and hexaanionic charge states. This again is consistent with the simple HiJckel picture: neutral Ih C80 has a g2 open-shell configuration that closes at charge - 6 to give a g8 configuration with a sizeable HOMO-LUMO gap of 0.9874fl. With the present availability of experimental results, it is not easy to find a simple test to disprove the possibility, indicated by the present data, that the assembly of fullerenes or equilibration of isomers proceeds through charged species. Isolation of Ih C80 would be strongly supportive of this proposal. A possible counterexample might be provided by C84, since here the experimentally isolated neutrals are known to be almost isoenergetic [ 12], the most stable of the 24 IPR isomers, and to be produced in statistical ratio [ 13]. In Table 3, the energies of the five charge states (-t-2, 0, - 2 , - 4 , - 6 ) of these 2 isomers are given together with their energy difference. It is seen that they re-

main nearly degenerate for all the oxidation numbers considered here. The degeneracy starts to lift for the tetraanion and the hexaanion, although an energy gap of the order of 30 kJ/moi is still small compared with those that occur for C80. Equilibration of neutral or charged species is therefore equally compatible with the experimental evidence. A fuller picture of isomer equilibration in C84 requires the Stone-Wales (SW) map of interconversions. Twenty-one of the 24 IPR isomers of C84 can in principle interconvert by sufficiently long sequences of Stone-Wales [ 31 ] transformations that preserve pentagon isolation at each step; these form the main SW map. Isomers 1, 2 and 5 form the second and minor map [ 16,32]. Table 4 lists relative energies of all 24 IPR isomers as a function of charge, and Fig. 2 illustrates the energetics of the main map at each charge. In constructing the figure, we use a graph-theoretical characterisation that has been applied elsewhere to general SW transformations ofC4o [ 33] and C60 [ 10]. The map has vertices, each representing an isomer

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P W. Fowler, E Zerbetto/Chemical Physics Letters 243 (1995) 36-41

or pair of enantiomeric isomers, and edges, each representing an isolation-preserving SW step. For every pair of vertices there is a shortest path linking the pair, and so for every vertex there is an eccentricity defined by the length of the longest such minimal path. The largest eccentricity is the diameter of the map and the smallest is the radius; vertices with eccentricity equal to the radius are said to lie at the centre of the map. The main SW map for IPR C84 has 21 edges and 31 edges, with a radius of 4 and a diameter of 7 SW steps. The mathematical centre of the map is a set of 5 isomers linked as the carbon skeleton of 2-methylbutane. Fig. 2 is colour-coded to show the distance of each vertex from the centre: one experimental isomer (23) is part of this centre but the other (22) lies one step away from it, so that at most 5 steps are needed to reach each of the experimental isomers from any point on the map. Inspection of the energy dimension of the figure shows that the map is flatter overall for charged fullerenes, presumably making equilibration easier: on the neutral map 8 out of 21 isomers lie more than 100 kJ mol -~ above the best, but at charge - 2 there are only 4 above this threshold, and for charge +2 there is only one above a threshold 88 kJ mo1-1. Interestingly, the tetrahedral leapfrog isomer, which would be predicted to be most stable of all on pure or-electronic considerations, is of exceptionally high energy at all non-zero charges considered here. The other leapfrog ( 1 ) lies on the minor SW map, which is a simple downhill path 1 ~ 2 ~ 5 with its lowest point some 80-160 kJ mo1-1 above the best isomer on the main map at each charge state considered. The experimental isomers 22 and 23 remain most stable or nearly so of all 24 for any reasonable charge. It is tempting to suggest from this that even the final C84 products may be formed in the charged state; there is not the obvious funnel effect down onto the neutral map noted for C60. Variation of stability with charge has been invoked in the shell-filling model of endohedral metallofullerenes [ 34] based on qualitative molecular-orbital theory; the present work confirms that it is also predicted with a more explicit model of electronic structure. In the light of these calculations, it appears that modelling of annealing processes in fullerenes should consider both neutral and charged species. The results also suggest that fullerene isomers which

are proving elusive so far might be prepared by suitable variation of the experimental conditions. For example, Kappes has recently found higher yields of the smaller fullerenes when La203 is mixed with the graphite starting material before arc processing [35]. We thank Robin Batten (Exeter) for expert help with the graphics, and the EU Human Capital and Mobility Programme for financial support via the Network on Formation, Stability and Photophysics of Fullerenes.

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[29] EW. Fowler, J.E. Cremona and J.l. Steer, Theoret. Chim. Acta 73 (1988) 1. [30] P.W. Fowler and A. Ceulemans, J. Phys. Chem. 99 (1995) 508. [31] A.J. Stone and D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 32] P.W. Fowler, D.E. Manolopoulos and R. Ryan, J. Chem. Soc. Chem. Commun. ( ! 992) 408. [33] P.W. Fowler, D.E. Manolopoulos, G. Orlandi and E Zerbetto, J. Chem. Soc. Faraday Trans. 91 (1995) 1421. [34] P.W. Fowler and D.E. Manolopoulos, Nature 355 (1992) 428. [ 35 ] M. Kappes, private communication.