J. CHEM. SOC. FARADAY TRANS., 1994, 90(2), 253-261 Orientational Ordering in the Solid Fullerene Oxide: C,,O Ailan Cheng and Michael L. Klein Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA, 19104-6323,USA Constant-pressure molecular dynamics simulations have been employed to investigate the orientational order- ing in the solid phases of fullerene oxide, C6,0. A pairwise additive atom-atom intermolecular potential model developed for solid c60 is modified slightly to reflect the functionalized character of the C6,0 molecule. The simulation results indicate that at low temperature the carbon cages are frozen into a Pa3-like structure, as in the pure c60 solid. Most oxygen atoms point randomly to one of the neighbouring octahedral interstitial sites (i.e:(100) direction) but about 20% point to the smaller tetrahedral sites ((111) direction).Above the transition temperature, estimated to be around 210 K compared with the measured value of 278 & 2 K, C600 molecules rotate about the centre-of-mass-oxygen axis. The bridging oxygen atoms tend to wobble in their interstitial sites rather freely but they cannot move from one pocket to another. However, at very high temperature (ca. 800 K), the oxygen atoms are able to hop between different interstitial sites on the molecular dynamics timescale. Fullerene derivatives have recently attracted considerable attention, and already, many new molecules have been reported.'-'' One of the simplest among them is the epoxide C600.5,6In this molecule, the attached oxygen atom breaks the icosahedral symmetry of the c60 molecule and hence it should affect the solid-state properties and both the intra- and inter-molecular dynamics.For example, the far infrared spectrum should be particularly rich. The orientations and motion of the oxygen atoms in the solid are of particular interest. Ab initio calculations have already been carried out to explore the structure of an isolated C600molecule,' '-14 and we recently reported the results of a preliminary study15 of the orientational ordering in the bulk solid. Our earlier inves- tigation used the molecular dynamics (MD) simulation tech- nique and Lennard-Jones atom-atom potential model. The MD results suggested that the solid should undergo a phase transition at a temperature close to that of pure c60.However, even above the transition temperature C6,0 mol-ecules rotated anisotropically ; the oxygen atoms tending to point preferentially to the octahedral interstitial sites that lie along the crystal (100) directions. A subsequent detailed X-ray diffraction study16 found that solid C6,0 indeed exhibits a phase transition at 278 & 2 K (compared with 260 K in pure C60).17-20 At low temperature, the cages of the C6,0 molecules are frozen in a Pa3-like structure analogous to pristine solid c60, but about two-thirds of the oxygen atoms preferentially occupy the tetrahedral sites (( 11 1) direction) and one-third the octahedral sites (( 100) direction).However, in the high-temperature phase the reverse situation obtains, namely, the octahedral sites seemed to be favoured.I6 In this study, we have employed a more refined pairwise additive atom-atom potential model based on one developed for solid c60.2'The latter has been shown to reproduce the experimentally observed Pa3 structure and also correlates with a wide range of properties such as the transition tem- perature, jump in the lattice constant at the transition, phonon density of states, reorientational relaxation time, bulk compressibility and far-infrared spectrum for solidc6,.18.20,22-26 The present simulation results confirm the anisotropic rotation of C6,0 molecules at room temperature.The carbon cages are predicted to rotate most easily about the centre-of-mass-oxygen axis ;the motion of oxygen atoms is mostly confined within the interstitial sites. At high tem- perature, the two-fold axis begins to reorient occasionally. The estimated residence time for an oxygen atom at 796 K in a particular site is of the order of 10-20 ps, whereas the transit time from site to site is usually less than 1 ps. Upon cooling below room temperature the rotation about the two- fold axis freezes-out and the carbon cages orient the same way as in the Pa3 structure of pure c60; a result which agrees with the X-ray data.16 However, we find that most oxygen atoms (ca. 80%) point randomly to one of the neighbouring octahedral sites. Only about ca.15% of the 0 atoms freeze into tetrahedral sites. The disagreement with experiment con- cerning the relative occupancy of octahedral us. tetrahedral sites16 suggests that the intermolecular potential model needs to be refined further. Details of Calculation Structure of C,,O Molecule In the current simulation C6,0 molecules are treated as rigid since the coupling between the intramolecular vibrations and the molecular translational and rotational motions is negligi- ble in the temperature range of interest. The NMR, FTIR and UV-VIS spectra reveal that the C6,0 molecule retains the essential structural and electronic character of c60 .' The same study also found that C6,0 has the epoxide structure (C2J with an oxygen atom bridging across the double bond that fuses two hexagons in the parent C,, molecule.Ab initio calculations" found that the isomer (C,) with a bridging oxygen atom across the bond connecting a pentagon and a hexagon is more stable. However, a more recent Car-Parrinello density functional theory calculation l ' indicates that the C2v epoxide structure has a slightly lower energy than the C, isomer. As in our previous study, the epoxide structure is assumed in the current MD calculations. Since the remaining cage carbon atoms are only slightly disturbed by the presence of the oxygen atom we assume they have the same structure as in a c60 molecule. The distance between two carbon atoms next to the 0 atom obtained from the various theoretical cal- culations is, 1.54," 1.5613 and 1.56 whereas the C-0 bond length is, 1.45," 1.4313 and 1.47 A,14 respectively.The best fit to the X-ray data yields 1.6 0.3 %, for the C-C bond and 1.43 f0.06 A for the C-0 bond.16 In the current study, C-C and C-0 bond lengths are chosen to be 1.532 and 1.47 A, respectively. We will see later that this choice of C-C bond length was almost certainly too small. Potential Model Previously, we have developed a pairwise additive atom- atom potential model which gives a good account of a variety of solid-state properties of pure c60.21 In this model, carbon atoms are considered to interact with each other via a Lennard-Jones (12-6) potential, and extra repulsive sites are placed at the centre of each double bond.In addition, electro- static interactions are modelled by fractional charge, qc = 0.175 e, assigned to carbon atoms, with a neutralizing charge, 4,, = -0.35 e, placed at the centre of double bonds. In the current calculation, the same basic scheme has been adopted with minor modification to the sites in the imme- diate vicinity of the 0 atom. The bond bridged by the oxygen atom is no longer an interaction site. The two carbon atoms adjacent to 0 still interact via a Lennard-Jones (12-6) poten-tial with the same parameters as other carbon atoms. The oxygen atoms are also treated as Lennard-Jones interaction sites with parameters ~(0-0)= 78 K and o(0-0) = 3.17 A, typical of many organic molecules.27 The van der Waals interactions between the unlike pairs are obtained by using the combining rules, which yield ~(0-C) = 46 K and a(0-C) = 3.28 A.Each oxygen atom also carries a charge (40= -0.5 e),27 which is balanced by the charges (4c = 0.25 e) on the two C atoms bonded to the 0 atom. With this charge distribution, the net dipole moment is 3.06 D,? which is larger than the ab initio calculation value, 1.27 D.13 Details of the Simulations The Parrinello-Rahman constant-pressure MD simulation technique has been employed throughout this work. The Newtonian equations of motion for translational motion of the centres-of-mass of the C6,0 molecules are solved by using the third-order Gear predictorxorrector algorithm. The rotational degrees of freedom, represented by quatern- ions, are integrated by a fourth-order algorithm.Owing to the large number of interaction sites (90 per molecule), most simulations are carried out for a 2 x 2 x 2 face-centred-cubic (fcc) lattice, which is replicated using periodic boundary con- ditions to mimic an infinite lattice. Typically, at a given tem- perature the system is allowed to equilibrate for about 25 ps. Then, trajectories (positions, velocities, angular velocities) covering the next 25-50 ps are collected in order to compute the various quantities of interest. Simulation Results Phase Transition and Low-temperature Structure The constant-pressure MD simulations were performed at zero pressure for temperatures ranging from 20 to 800 K. The simulation started at high temperature after which the system was cooled gradually.Various quantities were monitored to characterize the structural and orientational ordering tran- sition. At high temperature the C6,0 molecules are rotating but retain the imposed fcc structure. During the MD cooling run the lattice remained cubic as evidenced by the lengths of the cubic edges of the simulation box [see Fig. 1 (top)]. An abrupt change of configurational energy and average lattice constant (or volume) sets in at around 210 K; behaviour that. signals a structural transformation. The calculated lattice constant is almost identical with that from a previous simula- tion of solid C60.21However, the calculated lattice constant is significantly smaller than that obtained from X-ray scat- -f 1 D z 3.33564 x C m.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 14.3 r I I I 1 4 .t ,I I I I 'I II14.2 t ,I I I 1I I Ic .d i -200 @*.@"4 I-210 ' 0 200 400 600 800 TIK Fig. 1 The three edge lengths of the simulation box (top), average lattice constant (middle) and configurational energy (bottom) as a function of temperature. In the last two panels, the solid circles are for the cooling run and open circles for the heating run. Also, values of the configurational energy for the run started at low temperature in the P2,3 structure are represented by the squares (open symbols for heating and solid squares for cooling). tering experiments. For example, the present simulation yields a = 14.10 A at 300 K, whereas the experimental value is a = 14.185 A.The discrepancy here is likely to be due to the fact that a smaller C-C bond length, and hence c60 radius (R = 3.518 A), is used in the MD simulation than is suggested by the X-ray lattice constant data which are better fitted by R = 3.54 A.16 The recent ab initio calculations also suggest a slight expansion of the molecule when functional- ized by an 0 atom.12 The isobaric thermal volume expansion coefficient, E", can be estimated from the data in Fig. 1. The current simulation gives average values, a, = 6.7 x lo-' K-' and 5.4 x lo-' K-' for temperatures above and far below the transition point. The X-ray experimentI6 obtained the corresponding values, a, = 7.3 x lo-' K-' and 10.9 x lo-' K-'.However, the latter value is for the temperature range just below the transition, and is likely to be much larger than that far below the transition temperature, as is the case in pure C,, where a, = 6.2 x lo-' K-' for T < 245 K and T > 260 K, and 21 x lo-' K-' for 245 K < T/K < 260. Our previous MD study of pure c60 solid2' yielded a, = 7.0 x lo-' K-' above transition and 5.7 x lo-' K-' far below. These values agree with the experimental results rather well. The orientational behaviour of C6,0 molecules can be characterized in a fashion similar to that used in the study of pure c60 solid.28 For this purpose, one defines the order parameters as the ensemble average of the appropriate com- binations of spherical harmonic functions, YIm.If we assume that the 60 carbon atoms to be averaged over maintain icosa- hedral symmetry, the lowest non-zero order parameters are then those for I = 6.These 13 independent parameters indi- cate that the change in the lattice constant and the configu- rational energy shown in Fig. 1 is accompanied by an orientational ordering transition. More accurately, at low J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 temperature, the carbon cages orient the same way as in the low-temperature Pa3 structure of C,,; an observation which suggests that the interactions between cages are dominant. Locations of Oxygen Atoms Fig. 2 shows an instantaneous configuration taken from the simulation at T = 145 K.The snapshot reveals that most of the oxygen atoms occupy randomly the available octahedral sites in the fcc lattice.The orientation of a C,,O molecule can be specified by the Euler angles (p, 6,$) using the crystal axes as a Cartesian frame. Here, we have adopted Goldstein's notation for Euler angles.29 However, the motion of an oxygen atom can be monitored more easily by following the unit vector along the direction connecting the centre-of-mass of C,,O and the atom in question. This unit vector (i)can be characterized by the polar angle, 8, and azimuthal angle, a. These two angles are related to the usual Euler angles p and 4, where the polar angle 8 = B and the azimuthal angle a = 4 -90". The probability distributions of the two Euler angles, p(B) and p(4), are shown in Fig.3. At 796 K (top panel), though the &distribution is roughly uniform over all directions, the B-distribution display broad peaks at 0",90" and 180" which confirm that in the present simulation the oxygen atoms slightly prefer octahedral sites, which lie in the (100) directions. At lower temperature but still well above the freezing temperature, the probability distributions (central panel) suggest that oxygen atoms have well defined orienta-tions with most oxygen atoms favouring the octahedral sites. However, some point to the tetrahedral sites indicated by the satellite peaks at /.? M 50" and 120", even though the tetra-hedral sites lying in the (100) directions are smaller. Below the transition temperature when C,,O molecules stop rotating, the octahedral-site occupancy is still higher than the tetrahedral one. Fig.3 (bottom panel) shows p(B) at T = 126 K,the splitting of peak at 90" is related to the fact that oxygen atoms now point to the edge rather than the centre of the octahedral voids. The deviation from an ideal (001) direction is related to the Pa3 structure preferred by the carbon cages. The average probability for an 0 atom to point to the octahedral and tetrahedral sites was evaluated and the result is shown in Fig. 4. Here, the probability is defined in terms of Fig. 2 Snapshot of C,,O fcc lattice at T = 145 K. The white balls represent the oxygen atoms. Oxygen atoms prefer to occupy octa-hedral interstitial sites but the 0 atom occupying the tetrahedral site (middlerieht of fieure) should be noted.255 0.08 0.042 0.06 0.03 v Q-0.04 0.02 0.02 0.01 0.12 0.08 0.08 0.040.04 0 0 45 90 135 -90 0 90 180 /3/degrees +/degrees Fig. 3 Probability distributions, p(#?) and p(+), for various tem-peratures: 796 K (top),281 K (middle) and 126 K (bottom) the percentage of molecules pointing to a particular direction. At high temperature ca. 60% of 0 atoms prefer the (100) direction and ca. 25% point to tetrahedral voids, which agrees well with the X-ray diffraction data.16 As the system is cooled, the octahedral occupancy increases slightly and reaches 80% at the orientational freezing temperature. However, analysis of the X-ray data suggests that at low tem-perature the occupancy of octahedral and tetrahedral sites becomes reversed at 36 and 64%,respectively.' In an attempt to understand this discrepancy we carried out a second series of MD simulations this time starting on purpose with a P2,3 structure as initial configuration.In the P2,3 structure, all oxygen atoms point to tetrahedral sites. This structure did not seem to be stable. At 100 K about 10% of the 0atoms jumped spontaneously to octahedral sites after only 25 ps of simulation. However, the temperature 100 K is too low to facilitate rapid reorientation and hence find the 'true' ground state in a reasonable time. The system was therefore heated gradually. Between 300 and 400 K,only 25% of the 0 atoms remain in the tetrahedral pockets.Above the transition temperature the occupancy of the two types of 0.8 0.6 >- E 0.4 0 200 (00 600 800 TIK Fig. 4 Average probability for occupancies of octahedral sites (sauares). tetrahedral sites (hexagons) and other orientations (circles) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 (4 Fig. 5 Trajectory plots of C,,O molecules in an fcc lattice at 381 K (a)and 145 K (b).The white traces are associated with rotational diffusion about the two-fold axis. The localized black trajectories represent the confined oxygen atoms. sites agreed with that of the previous MD run. We then son of the two sets of simulations suggests that the run quenched the high-temperature phase first to 200 K and then started with all tetrahedral occupancy has higher configu- further to 100 K.The tetrahedral occupancy increased by rational energy (see Fig. 1). Thus, the low-temperature only 5% relative to the previous MD simulations. Compari- 0-atom orientations obtained from first MD runs is likely to J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 be reasonably close to the equilibrium value for the assumed potential model. Reorientation of C,oO Below the transition temperature, the C,,O molecules are completely frozen on the MD simulation timescale. Essen- tially, molecules only librate about preferred orientations. Around room temperature, the carbon cages rotate about the two-fold i axis rather freely. The spinning of molecules at 381 K is obvious in the trajectory plots of C,,O molecules shown in Fig. 5.However, the rotation is rather anisotropic. Basi- cally, oxygen atoms are still trapped for extended periods in the interstitial sites; they can wobble in their pockets but jumping between sites is relatively rare. The trajectories of oxygen atoms are shown in Fig. 6. Below the freezing temperature they are pinned in either the octahedral or tetrahedral sites. Even above the transition temperature, when the cage undergoes hindered rotation about the two-fold axis, the i axis still cannot reorient easily. As the temperature increases, hopping between different orientations sets in. Though some 0 atoms still stay in one pocket at 796 K, most of them make at least one hop during the 50 ps simulation.Fig. 7 shows the variation of the Euler angles, fl and 4, as a function of time. This picture gives an idea of the residence time in a particular pocket and the transit time between pockets for several molecules. For example, for the molecule labelled A [Fig. 7(a)],the 0 atom stays in an octahedral site for ca. 10 ps, and then flips 180" to another octahedral pocket within 0.5 ps. Molecule B [Fig. 7(b)]experiences two such events: first from a tetrahedral (T) to an octahedral site (0), then, to another octahedral site. The typical residence time is of the order of 10-20 ps and the transit time is usually ca. 0.5 ps with occasional slower events. Several hopping patterns such as 0 -+ T, 0 -P 0,T -+ T and T + 0 are exhibited.Mol- ecules C and D [Fig. 7(c), (41follow the hopping patterns, 0 -+ T, -,T, , and 0 -+ T, respectively. The reorientational relaxation time was examined at various temperatures. In particular, the autocorrelation func- tion, Cl(f),defined in ref. 15 for unit vectors along the three molecular axes 1,9 and f were estimated. Here i is along the centre-of-mass-0 axis, 1 and 3 are perpendicular to i, and 1 is the order of the Legendre polynomial. If we assume that the reorientation is uia rotational diffusion3' one can fit the correlation function to an exponential form, and extract an estimate of the reorientational relaxation time, T~.Only the calculated values for unit vectors 3 and i for I = 1 are summarized in Table 1, since that for the unit vector iis similar to that for 3.There are clearly two timescales. The relaxation time obtained for motion of the unit vector 3 is associated with the timescale of the spinning of carbon cage about the two-fold axis. This timescale is expected to be Table 1 Reorientational relaxation time for the molecular axes of C600 182 103 810 220 38 329 234 65 258 272 52 218 28 1 34 126 374 20 166 381 23 100 572 6 50 659 6 18 796 9 13 Fig. 6 Trajectory plots of oxygen atoms for various temperatures. At 126 K (a) oxygen atoms are locked in the interstitial sites. At 281 K (b), they can move in a particular pocket freely but hop-ping between different sites is rare. This motion is more prevalent at 796 K (c).J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 90 90 0 -90 80 90 0 -90 0 A An A 9010i 1 w -901-4 )!I1 ' -180' ' I 1 0 5 10 15 20 25 0 5 10 15 20 25 tlPS tlPS Fig. 7 Euler angles, B and 4, which describe the motion of an oxygen atom, as a function of time. From top to bottom the curves are for molecules A, B, C and D, respectively. The oxygen atom of molecule A jumps from one octahedral site to another, and that of molecule B hops first from a tetrahedral (T) to octahedral (0),then, to another octahedral site. Molecules C and D follow the hopping patterns 0 -,T, -+ T, ,and 0 -,T, respectively. similar to that of pure c60 solid.28 The unit vector t^ reflects the motion of oxygen atoms.The time taken for an oxygen atom to flip over is much longer. Below 200 K, the reorien- tational relaxation time z is much longer than the current simulation time~cale,~~ and the simulation simply shows a complete frozen phase similar to an orientational glass.32 Assuming that the value of zl at high temperature can be described by an Arrhenius law, z = zo exp(EJk, T), one can fit zi and zf by this form. For I = 1, this fit gives the pre- factors, rg = 2.9 x s and zt = 6.0 x s, and acti- vation energies, Ei = 690 f50 K and E,' = 960 f50 K. The estimated activation energy of 690 K for molecular spinning motion about the t^ axis is in reasonable agreement with experimental values, 480,24 69525 and 400 K,22 reported for pure c60 solid.The activation energy, 960 K, for the reorien- tation of the two-fold i axis is much higher. Even though the C600molecule is only slightly distorted relative to a pristine c60 molecule, the small oxygen bump causes a dramatic increase of rotational anisotropy. As shown above the reorientation time below the tran- sition temperature is much longer than the simulation time- scale. It is simply impractical to observe such motions within a reasonable computing time. However, the dynamics in this temperature range are of great interest. For example, it is known that pure C,o undergoes a 'ratchet' reorientation below the transition temperat~re.~~.~~.~~Recent NMR experiments suggest that C6,0 molecules display a similar type of motion, i.e.infrequent hopping between equivalent orientation^.^' The key question concerning this type of acti- vated motion relates to the barrier height of the associated pathway. To obtain some semi-quantitative information, we have taken one of the configurations generated in the MD simulation at 100 K, and arbitrarily chosen one molecule as a test particle. The static configurational energy is then calcu- lated as the test molecule is rotated while artificially freezing the translational and rotational motion of the remaining mol- ecules. The initial orientation of the test molecule was chosen so that the three molecular axes are aligned with the crystal axes, x, y and z. In this configuration the oxygen atom on the 60008ooo i 200 -11 '3" 3000 t I) 0 90 180 270 360 0 90 180 270 360 yldegrees yldegrees Fig.8 Configurational energy relative to the initial orientation of a test molecu!e as a function of rotation angle about (a) (100), (b) (OlO), (c) (1 10) and (d) (1 11) axes, respectively (see text for details). The configuration is taken from the simulation at 100 K. test molecule points to the (001) direction. The configu- rational energy relative to that of the initial orientation is then calculated while the test molecule is rotated about several axes. Fig. 8 shows the relative configurational energy values as a function of the angle rotated about the axes (loo), (OlO), (il0) and (lll), respectively. The energy profiles show strong angle dependence.When rotating about the (100) and (010) axes, the oxygen atom will bump into the four nearest neighbours along the path (recall that the low-temperature phase has the Pa3 structure). When the protrud- ing oxygen atom approaches the nearest-neighbour molecules lying in the (1 10) directions, the configurational energy increases dramatically. The peaks at 3 15" when rotating about the (100) axis and at 45" about the (010) axis corre- spond to configurations where the oxygen atom on the test molecule hits the 0 atom of the neighbouring molecules. The four wells at 0", 90", 180" and 270" correspond to the four octahedral sites along the (100) directions. The rather flat- bottomed potential wells reflect the large size of the octa- hedral sites, which is consistent with the above finding that the oxygen atoms can wobble rather freely in this type of site (recall Fig.5). When the test molecule is rotated about the (!lo) axis its oxygen atom will pass two types of interstitial site. On flip- ping the oxygen atom by 180", it first moves away from the initial octahedral site to an adjacent tetrahedral site. Then it passes a nearest neighbour to find another tetrahedral site, and eventually it falls into the final octahedral vacancy. The energy barrier between the octahedral and tetrahedral sites is much less than that needed to pass the nearest neighbour. Rotation about the (111) axis reflects the three-fold sym- metry of the lattice. Here, the three prominent peaks also indicate that the energy barriers are lower than that of passing nearest neighbours but higher than that between an octahedral and a tetrahedral site.From this calculation it seems that the energy barrier is lowest when rotating about the (il0) direction, which corre- sponds to jumping from a tetrahedral to an octahedral site or vice versa. As mentioned earlier, we have set up a system, on purpose, in a P2,3 structure and allowed it to relax to its equilibrium state. At 150 K one molecule hopped from a tetrahedral site to an octahedral site during the first 25 ps. The time dependence of the Euler angles, /3 and 4, for this molecule shown in Fig. 9 confirms that the molecule indeed J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0-T I1 t 1 180 I I I I I t 4 got-&t i a OI-LiYd!dd-1 800 5 10 15 20 25 tlPS Fig.9 Euler angles, /? and 4, as a function of time for a molecule taken from the simulation started with the P2,3 structure. This plot suggests that the molecule hops from a tetrahedral site to an octa- hedral site through the valley between the two nearest-neighbour molecules. passes the valley between the two nearest neighbours during its reorientation. In fact, the trajectory plots in Fig. 6 also reveal that even at high temperature most jumps are along the diagonal in the picture, which corresponds precisely to the path along the valley mentioned above. The actual values of the barrier heights vary slightly with configuration and the test molecule chosen. In the real situ- ation, all the surrounding molecules are also moving (librating and vibrating); therefore, the reorientational barrier will be somewhat time dependent.However, the main fea- tures of the energy profiles are likely to be the same. The path through the unfilled space between the two nearest neigh- bours is likely still to be the optimal route. This energy calcu- lation also indicates that the lowest barrier, ca. 3000 K, estimated from Fig. 8, is sufficiently high that thermally acti- vated hopping is unlikely to occur on the simulation time- scale at low temperatures. The energy profile also indicates that for the present model the system has slightly lower energy when the oxygen atom points to the octahedral voids rather than the tetrahedral sites.Again, while this is consis- tent with the higher octahedral occupancy (ca. 80%)relative to the tetrahedral occupancy (ca. 20%) observed in the simu- lation, it is not in accord with low-temperature X-ray data.' At higher temperatures the energy barriers are reduced somewhat owing to the rotation of the molecules and the expansion of the lattice. The elevated thermal motion also facilitates reorientation. From an examination of molecular reorientation in a configuration taken from a simulation at 796 K it is clear that the energy barriers are reduced. However, the (110) pathway still has the lowest barrier. Typically, the energy needed for an oxygen atom to jump from a tetrahedral to an octahedral location is ca.10 kJ mol-' and ca. 20 kJ mol-' for the opposite process. This energy scale is roughly consistent with the average activation energy, 960 K, obtained from an analysis of the reorientation time. Because the oxygen atom must necessarily follow the lowest-energy barrier path during the molecular reorientation the pathway for the C6,0 cage might be different from that of pure c60.Consider an oxygen atom pointing into an octa- hedral site, if it moves so that it is close to the unfilled space between the two nearest neighbour molecules, it can then librate about one of the (I 10) axes and jump to a tetrahedral site. After the hopping of this oxygen atom the cage can rotate about the i axis to rest_ore the original Pa3 orientation.For a c60 molecule in a Pa3 structure one can always find one of the 30 double bonds pointing to one of the neighbour- ing voids (either tetrahedral or octahedJal), so it is always possible for the carbon cage to find a Pa3 orientation for any of the orientations of the oxygen atom pointing to an octa- hedral or tetrahedral site. However, during the reorientation the carbon cage may be temporarily away from the ideal Pa5 configuration. The present MD simulation also suggests that an oxygen atom is likely to go through a tetrahedral site on the way from one octahedral to another octahedral location. This type of pathway is also observed in the simulation at high temperature (see trajectory in Fig. 6). Phonon and Libron Densities of States The densities of states (DOS) for the librational and trans- lational motions have been calculated.The crystal dynamics are of importance because they can provide a stringent test for the intermolecular potential model. Velocity and angular velocity autocorrelation functions (ACF) have been evalu- ated. One simply obtains the density of states by taking the Fourier transform of the ACFs. Fig. 10 gives both the libra- tional and translational DOS at several temperatures together with that for pure c60. In comparison with the c60 data at T = 106 K, both the translational and librational bands of C6,0 are shifted about 5 cm-' to higher frequency and are slightly broader. The gap between the translational and librational bands is also narrower in C,,O relative to the c60 solid.Also, there seems to be a weak coupling between the rotational and translational motion of C,,O molecules. As temperature increases both bands shift to lower frequency. At 374 K, when molecules are rotating, the coupling is enhanced slightly. It is likely that when oxygen atoms try to jump from one interstitial site to another, the nearby mol- ecules may recoil to open up more space and hence facilitate the hopping event. Since the rotation is very anisotropic for C,,O, the angular velocity ACFs about three molecular axes, x, y and z, have 3 2 1 0 2 1 h v c,o 1 0 2 1 0 20 40 60 v/cm -' Fig. 10 Density of states for librational (dotted lines) and trans- lational (solid lines) motions.The curves are for (a) pure C,, at 106 K. and C,,O at 100 (b)374 (c) and 796 K (d). 260 4 3 2 1 0 h 52 u 1 0 4 L 1,, -12 0 10 20 30 40 v/cm -' Fig. 11 Fourier transforms of the angular velocity ACFs, represent- ing the rotation about the three molecular axes, f (solid lines), 3 (dotted lines) and i (dash lines): (a) 100, (b) 374, (c) 796 K been determined separately. The Fourier transform of these three ACFs are shown in Fig. 11 for T = 100, 374 and 796 K, respectively. The libration about the i axis has the lowest frequency, while rotation about the f and 3 axis, which re- flects the tumbling of oxygen atoms, has a much higher frequency. As the temperature increases, the C600 molecule begins to spin about the two-fold axis and a central peak appears.However, even at 796 K, the frequency correspond- ing to the wobbling of oxygen atoms is still peaked at 5 cm-',confirming the previous finding that oxygen atoms are not rotating freely. Note that both the translational and rota- tional DOS show a lot of structure. The small oscillations near the ends of the bands are due to the truncation artifacts 10 5---0-4 L I I I I I I I J I I I 1 J 0 200 400 600 800 TIK Fig. 12 Average librational frequencies corresponding to the rota- tion about the three molecular axes, f (triangles), j (squares) and i (circles). Plateaus at high temperature for f and j components suggest that oxygen atoms are wobbling with confined amplitude in the interstitial sites.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 of the Fourier transform. However, the prominent peaks are independent of the truncation in the calculation. In fact, the choice of trucation only changes the resolution of the spectra but not the overall features. The average frequency and width of the librational band have been estimated as these values can be obtained from inelastic neutron scattering experiments.22 The calculated values are shown in Fig. 12. The abrupt change in the average frequency correlates with the freezing of the rota- tions. The plateau above the transition temperature for f and 9 components is consistent with the picture that C6,0 mol- ecules undergo anisotropic rotation even at high temperature.Conclusion Molecular dynamics studies have been performed for solid C6,0 using a revised pairwise additive atom-atom potential. The present simulation confirms the presence of a phase tran- sition from an anisotropic rotator phase to an orientational glass (ratchet) phase at a temperature close to that found in the pure c60 solid.21 Below the transit_ion temperature, the carbon cages are found to adopt a Pa3-like structure, as in pure c60. Evidently, the effect of the attached oxygen atom was not strong enough to alter the packing of the carbon cages. Oxygen atoms are locked randomly into one of the neighbouring interstitial sites. In the simulation, octahedral occupancy (ca. 80%) is favoured over the tetrahedral one (15- 20%).Just above the transition temperature, C6,0 molecules rotate anisotropically : molecules spin about their i axis almost freely, but on the MD timescale the 0 atoms are mostly restricted to wobbling around in a particular intersti- tial site, i.e. libration about a fixed orientation. As the temperature increases further, hopping of oxygen atoms become possible. However, the residence time is still much longer than the transit time. The reorientation time and activation energy for the spinning about the two-fold axis are consistent with those of c6,. The reorientation of the molecular i axis is much slower and the activation energy is estimated to be ca. 1000 K. Below the transition temperature the lowest-energy barrier for several paths examined in this study is ca.3000 K. The densities of states for the librations and centre-of-mass vibrations (phonons) have been calculated and our results remain to be tested by experiment. The simulation results disagree with the X-ray data16 con- cerning the octahedral and tetrahedral site occupancies at low temperature. The calculated values of ca. 80% and ca. 20%, respectively, are different from the experimental findings of 36% and 64%. This may be due to an unrealistic choice of the oxygen charge and size, and the C-C and C-0 bond lengths, since the site occupancy of the smaller tetrahedral void is likely to be very sensitive to modest variation of these parameters. The estimated transition temperature (210 K) is lower than the experimental value of 278 K.16 This most likely arises because the parameters for the c60 cage were chosen to be the same as in the pure c60.2' The transition temperature and the interstitial-site occupancies appear to provide a stringent test of the potential parameters.Although additional work is needed to refine the intermolecular poten- tial, the present results should provide a useful starting point for this endeavour. We would like to thank Don Cox, Paul Heiney, Amos Smith and Gavin Vaughan for stimulating discussions and sharing their experimental results prior to publication. We are grate- ful to Steve Erwin and Kari Laasonen for supplying details of their numerical work. 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