C36, a hexavalent building block for fullerene compounds and solids

5 February 1999

Chemical Physics Letters 300 Ž1999. 369–378

C 36 , a hexavalent building block for fullerene compounds and solids P.W. Fowler b

a,)

, T. Heine

a,b

, K.M. Rogers a , J.P.B. Sandall a , G. Seifert b, F. Zerbetto c

a 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 Dipartimento di Chimica, ‘G. Ciamician’, UniÕersita´ di Bologna, Õia F. Selmi 2, 40126 Bologna, Italy

Received 3 September 1998; in final form 2 December 1998

Abstract Structures and energies are calculated at the DFTB level for C 36-based fullerenes, hydrides, oligomers and solids. The two fullerenes with minimal pentagon adjacencies are isoenergetic. The isomer implicated in recent experiments has C6 Õ broken symmetry, a small HOMO–LUMO gap and can gain or lose up to six electrons. C 36 forms stronger inter-cage bonds than larger fullerenes. A favoured s-bonding pattern rationalises a dimer with ten times the stabilisation of ŽC 60 . 2 , a linear polymer, a ‘superbenzene’ oligomer, a ‘supergraphite’ layer and a hexagonal close-packed solid with a monomer ˚ . compatible with the experiments. q 1999 Elsevier Science B.V. All stabilisation of 522 kJ moly1 and a d-spacing Ž6.82 A rights reserved.

1. Introduction The intensive exploration over the past decade of the chemistry, physics and materials science of fullerenes was made possible by the ready availability of macroscopic quantities of C 60 , C 70 and some higher cages. In contrast to their higher homologues, the lower fullerenes, where unavoidable pentagon adjacencies are expected to lead to high reactivity and poor stability, have remained largely of theoretical interest. A new report of the synthesis of a fullerene below the C 60 threshold suggests that this

) Corresponding author. Fax: q44 1392 263 434; e-mail: [email protected]

situation may be about to change. Piskoti et al. w1x interpret their electron diffraction and solid-state NMR measurements on material produced by a modified Kratschmer–Huffman technique w2x as evidence ¨ for a fullerene solid composed of highly-symmetrical Ž D6 h . C 36 molecules. The solid is relatively tightly bound but can be sublimed, dissolved in organic solvents Žthough less readily than C 60 ., doped with alkali metals and chemically attacked to form addition compounds C 36 X m . The current paper reports systematic calculations on C 36 carbon cages that place this ‘D6 h’ isomer in the context of 15 classical and 73 non-classical fullerenes Žsee Section 2.. An explanation of the main electronic and geometric features from more sophisticated calculations is provided by a reinterpre-

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 3 8 5 - 2

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P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

tation of the Huckel model Žsee Section 3. which ¨ leads to a qualitative picture of this isomer as chemically reactive, ready to gain or lose up to six electrons. A low-energy candidate for the C 36 H 6 molecule is identified Žsee Section 4. and its pattern of 1,4-addition across equatorial hexagons is used as a template for construction of a ŽC 36 . 2 dimer, ŽC 36 . n linear polymer, ŽC 36 H 2 . 6 ‘superbenzene’ oligomer, and finally ‘supergraphite’ and hexagonal closepacked ŽC 36 .` solids which are compared with the rhombohedral structure already proposed by Cote ˆ ´ et al. w3x Žsee Section 5.. Structural data calculated for the hypothetical C 36 solids are compatible with the electron diffraction measurements w1x, though our calculated NMR spectra for C 36 itself and the C 36 H 6 molecule raise questions about the interpretation of the experimental solid-state NMR spectrum, as discussed in Section 6.

2. Energy calculations for C 36 isomers Existing theoretical treatments of C 36 give somewhat different accounts of the electronic and geometric structure to be expected of this fullerene. Of the 15 mathematically possible pentagon and hexagon cages w4x, two have the minimal count of 12 pentagon adjacencies: the D 2 d and D6 h isomers Ž36:14 and 36:15 in the spiral notation.. Simple Huckel ¨ calculations give a significant HOMO–LUMO gap for 36:14 but a zero gap for the D6 h 36:15, apparently indicating an open-shell triplet state for this isomer w4,5x. Semi-empirical calculations w6,7x predict the D 2 d isomer to be the lowest in energy of all 15 classical fullerenes, in accordance with the general tendency to minimise pentagon adjacencies w6x, but place the D6 h isomer considerably higher in energy Žby 134 kJ moly1 at the SAM1 level w7x., above some non-classical isomers. Density-functional calculations, on the other hand, predict the D 2 d and D6 h isomers to be quasi-isoenergetic w8x. The calculations reported in this section treat the set of C 36 fullerenes using the density-functional tight binding ŽDFTB. method w9,10x and, for comparison, the QCFFrPI Žquantum-consistent force fieldrp . model w11x, MNDO and its variants AM1 and PM3 w12x to find optimised geometries. All calculations treat closed-shell singlet configurations,

which are expected to be the ground states under full geometric relaxation. Single-point calculations are also carried out at the DFT level using the VWN functional w13x and a DZVP basis set w14x with the AllChem program w15x as a check on the DFTB energetics. The set of 15 classical fullerenes is extended to include all those trivalent cages that contain either one square or one heptagonal ring in addition to the pentagons and hexagons of the fullerene recipe, bringing in respectively 57 and 16 extra isomers. Earlier work on the classical fullerenes suggested a ‘universal’ linear correlation of total energy with number of pentagon adjacencies w6x, and optimisation with all five of the above methods confirms this trend ŽTable 1.. All methods identify D 2 d 36:14 as the isomer of lowest energy but differ in their estimates of the energy cost per pentagon adjacency. A more significant difference between Hartree–Focklike semi-empirical methods ŽQCFFrPI, MNDO, AM1, PM3, SAM1. and the DFT-based DFTB approach is in their predictions for the relative energies of 36:14 and 36:15. The former methods all attribute a much higher energy to the second isomer Ž D E s EŽ36:15. y EŽ36:14. is 158 ŽQCFFrPI., 119 ŽMNDO., 108 ŽAM1., 95 ŽPM3. and 134 ŽSAM1. w7x in kJ moly1 . whereas DFTB finds the two isomers to be essentially isoenergetic ŽTable 1., in agreement with full DFT calculations at the same geometries, and with published symmetry constrained optimisations using LDA and GGA functionals w8x. DFTB and AllChem DFT calculations at the AM1 optimal geometries also show only small energy differences between 36:14 and 36:15, confirming that the Hartree–FockrDensity-Functional dichotomy reflects a genuinely electronic effect. HOMO–LUMO gaps are, as usual, larger in the SCF-like than the DFT-like calculations Že.g. in the QCFFrPI model the calculated gaps for the classical fullerenes range from 3.6 to 5.6 eV and fit Ž DQC FFreV. s 2.2Ž D DFTB reV. q 3.5 with a standard deviation of 0.5 eV.. All calculations where symmetry breaking was allowed agree that 36:15 as a closed-shell singlet should have a modest HOMO–LUMO gap and a broken C6 Õ symmetry. The overall geometric distortion in 36:15 is small ˚ in D6 h-equivalent bonds. but has conseŽF 0.01 A quences for its 13 C NMR spectrum Žsee Section 6..

P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

371

Table 1 Energetic and topological characteristics of C 36 fullerenes and related cages. The 15 possible classical fullerenes and the three best non-classical one-square and one-heptagon cages are listed. A spiral code w4x is given for each cage, labelled by N, the position in the lexicographic ordering of spirals for the particular face recipe; G is the maximal point group of the structure and e55 the number of pentagon adjacencies. D ED , D EQ and D EA are the energies Žin kJ moly1 . computed in the DFTB, QCFFrPI and AM1 models, respectively, and given relative to the energy of 36:14 in the same method. D E L are relative energies for single-point DFT calculations Žwith the AllChem program w15x. at the DFTB optimal geometries. D is the HOMO–LUMO gap of the optimised structure Žin eV. calculated in the DFTB model N

Spiral

G

e55

D ED

D EQ

D EA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

55555656666665655555 55555665666656655555 55555666666555655655 55555666666556556555 55556656655665665555 55556656656566565555 55556656665565655655 55556656665566556555 55556666665555555566 55565656655665656555 55565656656566556555 55565666655655555656 55566566655565556565 55656656556555656565 55656665556555566565

C2 D2 C1 Cs D2 D2 d C1 Cs C2 Õ C2 C2 C2 D3 h D2 d D6 h

16 18 15 16 16 14 14 14 13 14 13 13 15 12 12

329.9 466.4 247.7 331.9 415.4 153.0 166.7 118.1 43.9 204.7 72.4 41.7 173.3 0.0 11.6

379.9 532.2 253.5 339.3 469.0 124.3 161.5 145.2 43.9 241.8 122.6 94.5 218.8 0.0 157.7

426.9 606.6 295.3 418.6 567.4 115.4 178.6 161.5 36.4 278.9 141.3 102.3 286.7 0.0 108.3

50 51 54

46666555556566665555 46666555565566656555 46666555656565655556

Cs C1 C1

10 10 10

150.3 147.6 118.1

282.4 318.0 333.5

75.4 146.9 159.2

6 7 16

55556656756565555556 55565657565665556555 55565656655756565555

C1 Cs C1

16 16 16

198.9 165.7 224.5

257.3 235.1 298.3

Slanina et al. w7x raise the possibility of nonclassical C 36 fullerene cages and in fact find a one-square candidate that is bettered in energy at the SAM1 level by only two conventional fullerenes. Our DFTB calculations find 44 one-square and 12 one-heptagon cages within the 466 kJ moly1 energy range of the classical fullerenes, with the best onesquare cage 118 kJ moly1 and the best one-heptagon cage 166 kJ moly1 above 36:14 ŽFig. 1.. All the non-classical fullerenes considered here have small calculated HOMO–LUMO gaps. The various methods agree on the identities of the best three one-square structures, but not on their relative energies; the penalty for introduction of a square is highest in the QCFFrPI model Ž282 kJ moly1 . and lowest in the MNDO-type models Že.g. 75 kJ moly1 in AM1., but no method places a non-classical cage lower in energy than the best classical fullerene ŽTable 1..

D EL

40.6

0.0 17.6 110.2

D 0.691 0.864 0.674 0.570 0.765 0.771 0.611 0.677 0.620 0.749 0.602 0.586 0.525 0.601 0.432 0.613 0.513 0.716 0.839 0.601 0.701

Classical cages are also much more stable than open structures such as rings and bowls, according to LDA and GGA calculations w8x. For the remainder of the paper, we concentrate on the cylindrical isomer 36:15 and its derivatives.

3. Qualitative molecular-orbital theory Many of the features of the calculations described in the previous section can be rationalised by a closer analysis of the simplest level of treatment, namely simple Huckel theory. ¨ The frameworks of the two quasi-isoenergetic fullerene cages 36:14 and 36:15 have respectively D 2 d and D6 h maximal w16x point-group symmetries. Their Huckel spectra are illustrated in Fig. 2. The ¨

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P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

and four sets of bonds, respectively containing 12, 12, 24 and 6 members. Calculation at the crudest Huckel level of approximation, where Coulomb inte¨ grals a are equal for all atoms, resonance parameters b are all equal for all s-bonded pairs and all more distant interactions are neglected, produces a substantial HOMO–LUMO gap for the D 2 d isomer ŽFig. 2a., in accordance with the low total energy found for 36:14 in more explicit calculations. However, the D6 h isomer is found to have a fourfold-degenerate set of frontier orbitals occupied by 6 electrons ŽFig. 2b. and hence a zero gap, in apparent contradiction w5x of the stability of this isomer predicted in high-level calculations. This discrepancy can be resolved as follows. Within the D6 h maximal symmetry any degeneracy greater than two must be ‘accidental’, and indeed classification of the eigenvectors of the adjacency matrix according to Ž2. gives a breakdown of the frontier orbitals into symmetry-distinct sets B1 u q B2 g q E1 g ŽFig. 2b and Fig. 3.. If the assumptions of the crudest Huckel model are relaxed, by letting ¨

Fig. 1. Optimised structures for C 36 and C 36 H 6 cages. The three isomers of lowest energy within their class are shown for the classical fullerenes, the one-square and one-heptagon non-classical fullerenes, labelled by isomer number, symmetry and energy as in Table 1. The six C 36 H 6 cages shown are the five of lowest energy found in the survey of 1,885 isomers, plus a D 3 d isomer ŽC 36 H 6 :5. that is much less stable but that forms the template of the proposed rhombohedral C 36 fullerene solid w3x. Energies of the bare carbon cages are referred to isomer 36:14, energies of the C 36 H 6 molecules to the favoured D 3 h isomer ŽC 36 H 6 :2.. All structures are computed in the DFTB model and energies reported in kJ moly1 .

vertex permutation representation for the D 2 d isomer 36:14 is Gs Ž Õ, D 2 d . s 5 A1 q 4 A 2 q 4 B1 q 5B2 q 9E. Ž 1 . In the D6 h point group, 36:15 has three sets of 12 equivalent vertices spanning the permutation representation Gs Ž Õ, D6 h . s 3 A1 g q B1 g q 2 B2 g q 3 E1 g q 3 E2 g q 3 A 2 u q 2 B1 u q B2 u q 3 E1 u q 3 E2 u Ž 2.

Fig. 2. Huckel molecular orbital energy level diagrams for the two ¨ C 36 fullerenes with minimal pentagon adjacencies. Ža. Isomer 36:14 and Žb. isomer 36:15. Levels are labelled according to the symmetries in the maximal point groups and the vertical scale is the dimensionless eigenvalue of the adjacency matrix Ž e y a .r b , where e is an orbital energy, a is the Coulomb integral and b the resonance integral. The energy level marked with an asterisk is the eigenvalue Ž'2 y1., which in the equal-a , equal-b Huckel ¨ model is the HOMO level in both isomers.

P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

Fig. 3. The D6 h symmetry-adapted frontier p orbitals of the cylindrical C 36 cage 36:15, with Huckel eigenvector coefficients ¨ represented by scaled circles, black for positive, white for negative values. All four eigenvectors have eigenvalue Ž'2 y1. in the equal-a , equal-b Huckel model of the cage. The large positive ¨ coefficients in the HOMO Žb. are at the sites of hydrogen addition in the favoured C 36 H 6 isomer.

inequivalent bonds take distinct b values, the E1 g pair splits from the B1 u q B2 g pair of levels, which themselves remain degenerate. Perturbation analysis gives the orbital energy difference e Ž E1 g . y e Ž B1 u , B2 g . as 3 2'2 q 8

3 b 11 y Ž 2'2 q 2 . b 12 q 2'2 b 23 y b 33

q'2 a 1 y Ž '2 q 1 . a 2 q a 3

Ž 3.

with a r and br s labelled by the orbits of atoms: set 1 contains the atoms in the polar hexagons, set 2 those exo to the polar hexagons and set 3 those close to the equator of the cylinder. Eq. Ž3. gives the direction of the splitting: diagonalisation of the original equal-b Huckel Hamiltonian gives p bond or¨ ders p 11 s 0.5406 ) p 23 s 0.5293 ) p 33 s 0.4822 ) p 12 s 0.4270, and self-consistent variation of < br s < according to < pr s < would therefore cause the E2 g level to fall below the frontier B1 u q B2 g pair. The B1 u and B2 g orbitals are in-phase and outof-phase combinations of identical hemispherical caps separated by a 12-atom equatorial nodal belt

373

ŽFig. 3. and are distinguished in energy only when three-bond interactions are allowed into the model, destabilising B2 g relative to B1 u . In more sophisticated treatments this latter splitting would be amplified by s – p interactions. When restricted to full D6 h symmetry, the Huckel configuration of 36:15 is ¨ therefore not an open but a closed p shell e14 g b 12u b 20 g with a very small HOMO–LUMO gap. Such a configuration would be subject to a strong second-order Jahn-Teller effect, provoking a loss of symmetry and opening up a wider gap. The minimal distortion is the reduction D6 h ™ C6 Õ , under which HOMO and LUMO take on the same symmetries Ž B1 u , B2 g ™ B1 ., mix and move apart in energy. The final conclusion of the Huckel model is ¨ therefore that 36:15 should have a closed p shell, probably still with a modest HOMO–LUMO gap, and should have a geometrical symmetry of C6 Õ in contrast to the topological D6 h symmetry. These expectations are all borne out by comparison with the calculations that include all the valence electrons. In SAM1 w7x, MNDO, AM1, PM3, QCFFrPI and DFTB models, 36:15 is found to have a small C6 Õ distortion of the D6 h framework. Symmetry-restricted AM1 calculations, for example, give a second-order Jahn-Teller stabilisation energy of ; 50 kJ moly1 and indicate that the distortion proceeds only as far as the C6 Õ epikernel w17x of the original point group. HOMO and LUMO in all the models have the B1 symmetry produced by the orbital mixing, with a doubly-degenerate E1 pair lying below the HOMO. The qualitative MO diagram ŽFig. 2b. carries further implications for the chemical character of isomer 36:15. As expected on general group-theoretical grounds for fullerenes w18x, the spectrum has a total of six levels floating in a large gap; four are the quasi-degenerate frontier orbitals and two the LUMOq 1 E2 g Ž E2 in C6 Õ . pair. Loss of six electrons from the p system to localised exo s bonding with addends, or gain of two or six electrons from reducing species are therefore predicted to lead to a stabilised closed-shell system. These predictions are consistent with the reported tendency of the C 36 solid to pick up six hydrogen atoms w1x, and its ability to take up alkali metal atoms. Trends in bond 6y lengths on going from the neutral C 36 to the C 36 ion can be rationalised in terms of partial bond orders

374

P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

6y within the formally non-bonding E2 g pair. The C 36 ion might participate in Zintl-like w19x phases A 6 C 36 with alkali metals; in the DFTB model, the fulleride anion would have full D6 h symmetry with some rearrangement of the pattern of long and short bonds compared to the neutral cage. The possible identity of the C 36 H 6 compound will be considered in the next section.

4. Possible structures for C 36 H 6 Although it lies in a local minimum on the potential energy surface for 36 carbon atoms, the fullerene cage 36:15 is expected to be highly reactive. There are the above-mentioned electronic-structure arguments that favour addition of six addends, and some experimental indications. Piskoti et al. w1x recorded mass spectra of films prepared by sublimation of their C 36 soot and observed a signal at mrz s 438 which they attributed to C 36 H 6 produced by reaction with moisture. A systematic study of possible structures for C 36 H 6 isomers based on the cylindrical 36:15 cage is reported here. Direct enumeration using methods similar to those with which C 60 Br24 isomers were counted w20x shows that here there are 82, 123 distinct addition patterns for six hydrogen addends, which can be classified by maximal point group as 3 D 3 h , 3 D 3 d , 3 C6 Õ , 4 C3 Õ , 21 C2 h , 46 C2 Õ , 4 C3 , 57 Ci , 439 C2 , 1, 305 C s and 80, 238 C1. It is not feasible to perform electronic structure calculations on the whole set of 82, 123 isomers, but the subset with non-trivial symmetry is a manageable sample. Explicit DFTB optimisation on all 1, 885 of these isomers gave a spread of 647 kJ moly1 , with 35 appearing within 100 kJ moly1 of the best isomer. The candidate of lowest total energy has the six hydrogens attached 1,4 across three nonadjacent equatorial hexagons. It has a HOMO– LUMO gap of 2.29 eV and full D 3 h symmetry. All of the best five candidates identified in the search include such 1,4 pairings ŽFig. 1.. The arrangement of addends in the best candidate corresponds to attachment of hydrogen to half the atoms of the orbit of 12 that has the highest coefficient in the Huckel HOMO ŽFig. 3b., and it is also ¨ the pattern that would be produced by a mechanism of attack alternately at positions of highest free

valence and highest spin density. All bare carbon atoms of the functionalised cage participate in a 30-electron p system comprising five linked benzenoid hexagons and analogous to the ‘spheriphane’ decoration of fullerenes discussed elsewhere w21,22x. DFTB-computed bond lengths are consistent with the Huckel bond orders of this spheriphane system. ¨ Grossman et al. w8x discuss a C 36 X 12 ŽX s Cl. molecule that includes the H 6 addition pattern as a subset but hexavalent attack is already sufficient to remove the quasi-open shell character of the underlying fullerene cage, creating a molecule with a large HOMO–LUMO gap, whilst steric strain of the bare fullerene cage is also relieved by the introduction of one sp 3-hybridised carbon into each of the 12 pentagonal rings. Analogy suggests that the same disposition of s bonds could provide inter-fullerene binding in dimers, oligomers and solids. Calculations testing this idea are described in the next section.

5. C 36 – a building block for oligomers, polymers and solids In the calculations described above, we identified the most active sites of the 36:15 cage Ži.e. the 1,4-paired positions in the equatorial hexagons. and found this isomer to be hexavalent. The present section considers the consequences of this reactivity for the formation of dimers, oligomers and solids from C 36 . All the following calculations are carried out within the DFTB model and the results summarised in Table 2. The known fullerene dimers and dimeric oxides w23–25x have cyclic bridged structures. If dimers of Table 2 Characteristics of oligomeric and solid structures based on C 36 . l is the length of inter-cage bonds and d is the centre-to-centre ˚ .. EB is the binding energy per distance between cages Žboth in A C 36 monomer Žin kJ moly1 . and D the band gap Žin eV. Žor HOMO–LUMO gap in the case of the dimer.

dimer linear chain supergraphite hcp rhombohedral

l

d

EB

D

1.60 1.61 1.62 1.53 1.57

6.74 6.72 6.78 6.82 6.56

152.5 276.6 375.3 522.0 355.1

0.75 1.28 2.09 1.62 0.31

P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

36:15 are constructed on the same principle, face-toface 1,4-connection of hexagons leads to five topologically distinct structures wequatorial–equatorial Ž2., equatorial–polar Ž2., polar–polar Ž1.x. Of these, just one optimises to a bound dimer. The D 2 h dimer shown in Fig. 4a is bound by 305 kJ moly1 , about ten times higher than the calculated binding energy of ŽC 60 . 2 w23x and has a higher HOMO–LUMO gap Ž0.75 eV. than does C 36 itself. The inter-cage bond ˚ and the cage geometry is consistent length is 1.60 A, with the presence of sp 3 carbons at the bridging sites

375

and is very similar to that of the homologous C2 Õ C 36 H 2 isomer. As a next step, an infinite linear array of the dimers was constructed, forming a linear polymer. Geometry optimisation was carried out in the DFTB model using periodic boundary conditions. The super cell contained 72 atoms Žone dimer., but in the relaxed geometry all monomers became equivalent ŽFig. 4b.. The calculated binding energy is 277 kJ moly1 per C 36 , which is less per bridge than in the dimer, and corresponds to a slightly longer inter-cage

Fig. 4. Hypothetical oligomeric and solid-state structures based on C 36 . Ža. Side view of the C 36 dimer; Žb. top view of the ŽC 36 .` linear polymer, with bridging rings as in Ža.; Žc. ‘superbenzene’ ŽC 36 H 2 . 6 ; Žd. a sheet of the ‘supergraphite’ C 36 solid, viewed at an angle to illustrate the monomer links; Že. two views of the close-packed hexagonal form of solid C 36 , showing the chair-like hexagonal large channel and the end-to-end stacking of C 36 monomers. All structures are based on DFTB optimisations and include bridges based on the active sites of the equatorial hexagons of the C 36 monomer.

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P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378

˚ but a larger gap Ž1.28 eV.. A bond length of 1.61 A ‘zig zag’ chain with bridging angles of 1208 was found to be unstable, dissociating back to dimers on optimisation. Trigonal coordination using all six active sites of 36:15 is possible, however, if C 36 H 2 units are linked. ‘Superbenzene’ ŽFig. 4c. is a stable oligomer with six cyclic bridges, a gap of 2.08 eV, a binding energy respective to six C 36 H 2 units of 1,434 kJ moly1 and a D6 h symmetric structure. The bond ˚ length between the C 36 H 2 units is 1.62 A. A two-dimensional hexagonal structure can be constructed from units of superbenzene ŽFig. 4d.. This ‘supergraphite’ ŽC 36 .` solid is found to be stable, with a structure and HOMO–LUMO gap Ž2.09 eV. similar to that of superbenzene and a binding energy of 375 kJ moly1 per C 36 unit Ži.e. 250 kJ moly1 per bridge.. The bridging bond length ˚ Thus, in comparison between C 36 units is 1.62 A. with the dimer, the energy per bridge is lower and the bridging distance greater, but there is a significant gain of energy per C 36 unit ŽTable 2.. This calculated stability for a two-dimensional layer lattice solid is compatible with the intuitive simple Huckel picture of C 36 as a hexavalent building block. ¨ Perhaps surprisingly, supergraphite is more stable than the fully three-dimensional rhombohedral structure proposed by Cote ˆ ´ et al. w3x ŽTable 2.. Van der Waals interactions between layers would presumably further enhance the stability of supergraphite. The ˚ . would calculated d-spacing of supergraphite Ž6.78 A be compatible with the TEM images recorded by ˚ .. Piskoti et al. w1x Ž d-spacing 6.8 A The rhombohedral close-packed lattice composed of C 36 units w3x is metallic and less stable than supergraphite ŽTable 2.. The DFTB properties reported for this solid in Table 2 and the computed density-of-states distribution Žnot shown. are in good agreement with those from LDA and GGA methods given in ref w3x. Each monomer has six bonds to neighbours, arranged in a D 3 d pattern similar to that in C 36 H 6 :5 ŽFig. 1., a building block which in itself is less stable by 242 kJ moly1 than the isomer C 36 H 6 :2 that forms the template for superbenzene and supergraphite. The centre-to-centre spacing is somewhat shorter than the experimental d-spacing. Finally, a hexagonal close-packed Žhcp. structure ŽFig. 4e. was constructed using the same disposition

of s bonds as in the most stable hydrogenated isomer, C 36 H 6 :2. The active equatorial sites are connected to each other by inter-cage bonds of length ˚ and the fullerene monomers form stacks 1.53 A, ˚ .. This connected by longer, weaker bonds Ž1.65 A hexagonal close-packed structure is more stable than either rhombohedral or supergraphite forms and again geometrically compatible with the experimental TEM ˚ .. images w1x Žcalculated d-spacing 6.82 A The relative stabilities of the solid phases, as those of the C 36 H 6 hydrogenated fullerenes, are rationalised by the simple Huckel picture: the unsat¨ urated cylindrical monomer forms six strong s bonds at the orbit of active sites around the equator to either three Žsupergraphite. or six Žhcp. neighbouring cages.

6. NMR shielding calculations The main piece of evidence advanced in Ref. w1x for a D6 h structure of C 36 is a solid-state 13 C NMR powder spectrum with peaks at 146.1 and 135.7 ppm in 2:1 intensity ratio. Calculations by Grossman et al. w8x had predicted a spectrum of three lines of equal intensity, at 160.0, 159.4 and 137.5 ppm, which within the experimental resolution would appear as a 2:1 pattern. If the discrepancy in position of the major peak is attributable to solid-state effects, the experimental result is therefore compatible with a D6 h structure for C 36 . Given this uncertainty and the intrinsic resolution of only ca. 10 ppm at half-peak height, the experimental data does not necessarily exclude all other isomers of C 36 from consideration, but if the attribution to 36:15 is accepted, the DFTB calculations on the free molecule and the solids suggest two interesting questions. What is the effect of the D6 h ™ C6 Õ symmetry breaking on the 13 C NMR spectrum of the monomer? What chemical shifts and splitting pattern are to be expected for the solid-state spectrum of a strongly interacting lattice of C 36 units? These questions can be explored by calculation of shieldings for C 36 itself and the C 36 H 6 template for the solids. The NMR calculations were performed with the deMon-NMR program w26x within the IGLO approach using the deMon-KS program w27x to obtain the electron density for an ii-iglo basis w28x and

P.W. Fowler et al.r Chemical Physics Letters 300 (1999) 369–378 Table 3 Calculated 13 C NMR chemical shifts for C 36 and C 36 H 6 Žin ppm.. G is the point group imposed in the calculation, and the numbers in brackets signify relative intensities of the peaks. Details of the calculations are given in the text Cage

G

Chemical shifts

C 36 :15 C 36 :15

D6 h C6 Õ

C 36 H 6

D3 h

138.4 Ž12., 161.8 Ž12., 164.6 0Ž12. 138.4 Ž6., 141.1 Ž6., 157.0 Ž6., 157.7 Ž6., 165.0 Ž6., 171.1 Ž6. 72.7 Ž6., 107.3 Ž6., 160.4 Ž6., 163.4 Ž12., 201.6 Ž6.

VWN functional w13x. Shieldings are converted to chemical shifts by referring them to methane in the same model, and hence to TMS Ž d ŽCH 4 . s y14.3 w8x.. Table 3 lists the results. When the DFTB geometry is taken from an optimisation carried out within D6 h symmetry constraints, the calculated shieldings are in excellent agreement with those reported by Grossman et al. w8x and conform to the pseudo-2:1 pattern. In the fully relaxed C6 Õ geometry, the three lines of the D6 h spectrum become six, with the larger cluster of lines now covering a range of 32.7 ppm. A solution spectrum, if such became available, would therefore be expected to differ considerably from the 2:1 pattern. In the various oligomeric and solid forms of C 36 considered here, the monomeric units are connected by Žlong. s bonds and the 13 C shifts would be expected to include some low values characteristic of sp 3 bonding. Direct calculations on the solid-state spectra of supergraphite and hcp-C 36 are not possible with our current programs but as a model for the condensed phases, the shifts were calculated for the D 3h C 36 H 6 molecule. As can be seen from Table 3, they cover a much wider range than those of the C 36 monomer, and in particular the six sp 3 carbons appear at just over 70 ppm. Similar values for chemical shifts of bridging carbons would be expected in the oligomeric and solid forms of C 36 ; the measured 13 C NMR spectrum of the C 60 dimer, for example, contains a signal at 76.22 ppm which is assigned to the ˚ inter-cage sp 3 carbons involved in the 1.575Ž7. A w x bonds 29 . Although the calculated inter-cage bonds in the hypothetical C 36 solids are in most cases slightly longer than in ŽC 60 . 2 , they would still be

377

expected to produce chemical shifts well outside the range of 140 " 10 ppm seen in the spectrum assigned in Ref. w1x to C 36 . In this respect, close agreement between the measured spectrum and that calculated for an unperturbed, symmetry-restricted C 36 monomer would be an argument against the proposed assignment.

7. Conclusion The current study was stimulated by the experiments on C 36 -containing materials w1x, which have been explained in terms of a particular fullerene isomer, 36:15. Semi-quantitative calculations at the DFTB level have been used to predict structures and energetics of this C 36 fullerene molecule, its hydride C 36 H 6 , dimer, polymer and hypothetical solid forms. The solid-state structures are compatible with the limited experimental evidence from TEM measurements, though they would suggest a wider range of 13 C NMR shifts than is observed for the powder identified as C 36 in ref w1x. The pictorial Huckel ¨ molecular orbital method gives an interpretation of the bonding in these different materials that is based on a C 36 building block with six valencies.

Acknowledgements We acknowledge financial support of this work by EPSRC ŽUK., DFG ŽGermany., DAADrBritish Council ŽARC 868., and the TMR initiative of the EU under contract FMRX-CT97-0126 on Usable Fullerene Derivatives. We also thank D.R. Salahub, A.M. Koster, V.G. Malkin and O.L. Malkina for ¨ their support with the DFT and deMon-NMR program packages. The computations were performed at the University of Exeter and Technische Universitat ¨ Dresden.

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