Since the discovery of a method to produce fullerenes in large amounts [ 1 ] many studies have been performed on these exciting all-carbon molecules. First experimental results on the structure of high purity solid CT0 were presented by Vaughan et al. [ 21 over a year ago. Recently, experimental data on the crystal structure and dynamics of Cl0 single crystals have become available [ 3,4]. The CT0 molecule can be described as two half Cbo molecules joined together by a belt of hexagons [ 5 1, as experimentally confiied by Johnson et al. [ 61 using ZD-NMR. At high temperatures, above 350 K, the elongated-soccerball shaped CT,-m, olecules can rotate quasi freely in the crystal lattice. At these temperatures there is a coexistence of two crystal structures, face centered cubic (fee) and hexagonally close packed (hcp), of which the former one seems to be favoured at still higher temperatures [2]. Crystals grown by sublimation
at 870 K have predominantly the fee growth habit, although a minor fraction of the crystals grown has the hcp morphology. X-ray diffraction [ 3 ] shows that the hcp-grown crystals also exhibit microscopically an ideal hcp structure at high temperatures with rotations around both the long and short molecular axes. Upon cooling these crystals to room temperature the rotations around the short axes freeze out and the long axes of the molecules align along the hexagonal c axis, resulting in a deformed hcp structure.
Somewhat below room temperature the orientational disorder around the long axis freezes out
as well. For this phase an ordered superstructure with four molecules per unit cell is observed. It should be noted that no well defined low temperature crystal structures are obtained when fee-grown crystals are cooled down. In this Letter, harmonic lattice dynamics calculations on the low temperature monoclinic structure of solid C,,, are presented. For the calculations we used rigid CT0 molecules with only van der Waals interaction between them. The structure resulting from a lattice energy minimization is compared to the experimentally determined structure and insight is gained in the orientational ordering of the molecules. The frequencies of the lattice modes are calculated for a variety of interaction potentials.
Lattice dynamics calculations
To perform lattice dynamics calculations on the low temperature monoclinic phase of solid CT0 we applied a standard harmonic lattice dynamics program for molecular crystals, details of which can be found elsewhere [ 71. Rigid CT0 molecules are used, with the atomic positions within each molecule deduced from the five, experimentally determined, nonequivalent atomic positions [ 81 by applying the symmetry operations of the I&s point group. The calculations are performed for zero temperature where an orientational ordering exists, i.e. the molecules do not rotate but have only small amplitude librations around a well-defined equilibrium structure. The interaction between the molecules is modeled as the sum over atom-atom interactions. To describe the atom-atom interaction several potentials known from literature, both of the Lcnnard-Jones and of the expd type, are used #‘. It should be explicitly noted that no Coulomb interaction terms have been included, as will be discussed later. The lattice sum is performed over all molecules within a 25 A range. The lattice energy is minimized with respect to the lattice parameters and with respect to the positional coordinates and the Euler angles of the four molecules in the unit cell. The experimental values are used as the initial values of the 30 minimization parameters. No symmetry restrictions have been imposed on any of these parameters. For the energetically most favourable lattice structure thus obtained the frequencies and eigenvectors of the librational and translational inter-molecular modes are computed.
The results calculated for wavevector R = 0 can be directly compared with experimental Raman data. The transformation properties of the eigenvectors under the symmetry operations of the P2,/m space group enable us to make an unambiguous symmetry assignment of the modes. The experimentally observed structure that is used as starting input for the structure optimization is a monoclinic structure with the unique b,,, axis along the former hexagonal c axis and an angle p close to, but not necessarily identical to, 120”. The a, axis is identical to the hexagonal a axis. It is observed experimentally that the monoclinic unit cell is doubled along the c, direction, which corresponds to one of the former a axes. This doubling of the unit cell is ascribed to an orientational ordering in the former hexagonal plane. It is assumed that in the low temperature structure all CT0 molecules are aligned with their long axes parallel to the b, axis. Given this assumption, which is shown to be valid in the calculations, the orientation of the molecules can be described by the angle @ by which the molecules are rotated around their long axis. At @=O’ the comer points of the top (and bottom) pentagons of the CT0 molecules are directed along the minus h axis. From the experimental data a model describing the orientational ordering was proposed [ 3) in which the molecules in a row along the am axis are oriented identically and molecules in two adjacent rows differ by 180” [ mod.72” ] in orientation. It was argued that this arrangement of molecules would yield the deepest lattice energy minimum.
Results and discussion
The crystal structure obtained via minimization of the lattice energy is in good agreement with the experimentally determined structure. In table 1 the results obtained for various potentials are listed, together with the experimental results. The values for the lattice parameters u, b and c agree to within two percent with the experimentally determined values. The angles LY and y are always found to be 90”. The angle /3 is found to be close to 120”, as can be seen in table 1. These results are in agreement with similar calculations by Guo et al. [ 1 1 ] . The positions of the molecules within the unit cell deviate only slightly ( G 1%) from the ideal lattice positions. The orientations of the long axes of the molecules are found to be parallel to the b, axis, so the calculations justify the assumption mentioned earlier. The energetically most favourable lattice structure is a monoclinic structure, with a doubling of the unit cell along the c, axis due to the orientational ordering, which involves a rotation of the molecules around their long axis. It is important to note that the calculated lattice structure obeys the experimentally observed monoclinic symmetry; the angles #i of molecules i (is 1, . . . . 4) are related via & (~~)=180°[mod.72”]+@2 ($1). The 2,/m screw axis is parallel to the b, axis and intersects the a,+,, plane at position (x, z) with x=0.1663 and y=O.3390, still reflecting the apparent hexagonal symmetry of the high temperature phase. The spacegroup is therefore P2,/m. The molecules are located at positions with site symmetry m. The mirror planes perpendicular to the screw axis
cut through the middle of the & molecules at (0, angle &,, as indicated in fig. 1. This angle @a, as ob- (0, 0) and (0.322, 0.178, 0.500). tained for various potentials, is also given in table 1, As seen from table 1 the calculated lattice energies for the various potentials differ considerably. The aim of the calculations was primarily to understand the CT0 crystal structure and the inter-molecular mode spectrum. These quantities are mainly sensitive to the shape of the interaction potential around the equilibrium position and not that much to the absolute value of the binding energy. No experimental data on the absolute value of the lattice energy for low temperature monoclinic C& are available. Comparing the lattice energy with the experimentally determined high temperature heat of sublimation of 45 kcai/mol [ 12 ] shows that the LJ( 1) potential gives the most realistic lattice energy value. From the calculations some additional conclusions on the orientational ordering can be drawn. The ordering found by the calculation, shown schematically in fig. 1, differs somewhat from the simple model proposed by Verheijen et al. [3] mentioned above. Considering only the P2Jm symmetry two independent angles, & and q&, are needed to unambiguously describe the structure. In the model of Verheijen et al. it is assumed that these angles are $i=O”[mod.72”] and d~=180”[mod.72”]. From the calculations it follows, however, that the difference @i - g2 is not equal to 180” [mod.72” 1. Instead, it is found that the (in principle) independent angles $, and & are related via &=180° [mod.72”] -$,.
“double” bonds are present, which can be modeled by extra electron charge density at the center of the double bonds compensated by electron charge deficiency at the single bond centers. Although similar, but considerably more complex, models have been proposed for CT0 [ 141 the present calculations indicate that including the van der Waals interactions alone is sufficient to correctly describe the orienta- . . . tional ordering m solid Cto. After the minimization the frequencies and eigenvectors of the lattice modes are calculated. The librational (f!) and translational (t ) inter-molecular modes in P2,/m CT,, are classified as I% 2% + 4B;+2B:t4A: and P=4%t+B;t2B:tA:. Of these 2 1 modes only the 12 gerade modes are Raman
active [ 4 1, whereas the IR spectrum will originate from the 9 ungerade modes. The calculated frequencies and the symmetry assignment for these modes are given in table 2 for the various potentials used
It is observed that the librational modes span the 15- 25 cm-’ range and that the translational modes are separated from these, spanning the 25-60 cm-’ range. In the first column of table 2 the experimentally determined frequencies are given. Although an assignment of these modes in the low frequency part (the “crowded” part) of the spectrum is not unambiguous, the overall agreement between the theoretical predictions and the experimental observations is good. Again the LJ ( 1) potential gives the best
fitting results. This is explicitly demonstrated in fig2 where the experimental Raman spectrum and the stick spectrum deduced from these data, as reported by van Loosdrecht et al. 141, are compared to the theoretical Raman spectrum obtained with the LJ ( 1)potential (see table 1). The predicted IR spectrum
calculated from the same LJ ( 1) potential is also shown in fig. 2, and awaits experimental conlirmation.
Conclusions
Lattice dynamics calculations on the low temperature structure of solid CT0 have been performed, using simple atom-atom van der Waals potentials to describe the interactions between the rigid CT0 molecules. The theoretically obtained crystal structure agrees with the experimentally determined low temperature structure; the symmetry is found to be P2,/ m. The calculations yield more insight in the orientational ordering, which involves a fixed rotation
of the four molecules in the unit cell around their long axes, and which can be described by a single angle
with 96% 6.5 ‘. The calculated frequencies of the 12 Raman active lattice modes agree quantitatively with the recently observed Raman spectrum. The frequencies of the nine complementary IR active lattice modes are predicted 
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